The Inverse-Optics Problem

Continuing from Chapter 6: Let There Be Light!

Visual perception is a constructive, or generative function that interprets the visual stimulus by constructing a three-dimensional model of the configuration of objects and surfaces in the external world that is most likely to have been the cause of the given stimulus. We can observe this constructive function in visual illusions like the Kaniza figure, Tse’s spiral worm, Idesawa’s spiky sphere, and Tse’s Loch Ness Monster. Each of these illusions are perceived as three-dimensional objects and surfaces, with a distinct experience of depth and surface orientation at every point on every visible surface. In other words, the experience itself is a three-dimensional structure, a model of external reality expressed in an explicit spatial code, a volumetric image projection mechanism.

Visual perception is analogical in nature: We perceive objects by constructing spatial analogies of them and experiencing those analogies, which we come to believe to be the objects themselves, perceived out where they lie beyond the sensory surface. Visual perception will always seem profoundly paradoxical until we see through the Grand Illusion and recognize the world of experience for what it is: A miniature virtual-reality replica of the external world in an internal representation.

The primary function of visual perception is the construction of this volumetric spatial interpretation of objects and surfaces in the world that are most likely to have been responsible for the visual stimulus. This is known as the inverse-optics problem, i.e. to undo the optical projection from the three-dimensional world to the two-dimensional retinal projection. This problem is mathematically under-constrained, because there is an infinite range of three-dimensional spatial interpretations that correspond to any given visual stimulus. For example a rectangle on the retina could correspond to any of the infinite range of irregular quadrilaterals spanning different depths, whose corners correspond to those of the retinal image. How does the visual system select from this infinite set the one that we perceive?

I propose that the only feasible way to solve such a computationally intractable problem is by way of a parallel analog wave-like algorithm that essentially constructs, or reifies, every possible interpretation simultaneously and in parallel, before selecting from that infinite set the one (or more) most stable interpretation. And the selection criterion seems to involve Gestalt principles of simplicity and symmetry. This is where mathematics enters perception.

Hochberg & Brooks(1960) found that the probability that a line-drawing is interpreted as 2-D as opposed to 3-D depends on the simplicity of the interpretation in 2-D compared to 3-D. For example figure A below is perceived as a cube with equal sides and all right angles, whereas as a 2-D pattern  there are 7 tiles with different angles and side lengths. Figure D on the other hand makes a simple pattern in 2-D, and thus is perceived easily as a 2-D figure with six identical equilateral triangles, whereas in 3-D it represents an unlikely singular viewpoint along a diagonal axis, which is evidently more difficult to perceive.

The fact that most of these figures can be perceived as 2-D or 3-D, with a particular preference for one over the other, and the fact that the ambiguous cases can be observed to flip bistably between two states, suggests that the visual system has constructed both alternatives, and is weighing them against each other in real-time to see which is the simpler interpretation.

Indeed this is the same principle in evidence in the illusions introduced earlier. A triangle occluding three perfect circles in the Kanizsa figure trumps an interpretation as three pac-man features perfectly aligned.  Peter Tse’s volumetric worm tends to be perceived with a simple cylindrical body, ends capped with perfect hemispheres, bent into a regular spiral around a regular white cylinder. Likewise with Idesawa’s spiky sphere, and Tse’s sea monster. This is a visual system that is based on symmetry, the perceptual equivalent of Occam’s Razor: In the absence of evidence to the contrary, the most likely interpretation is the most symmetrical one. Not because everything in the world is symmetrical. Far from it! But because symmetry is the primitive operational principle of the visual system: we perceive shapes by their symmetries and by their their violations of symmetry.

The metric found by Hochberg & Brooks to correspond with the perceptual preference for a 2-D or 3-D interpretation involved the number of edges of the same length, and the regularity of the angles between edges. This corresponds to the Gestalt principle of prägnanz, or simplicity, the simplest interpretation is the most stable in perception. This also corresponds to a metric involving resonance in a 2-D and 3-D context.

Start with A, inverse-project into depth, but also within xy plane

Mark centers of symmetry

Extrapolate to implications

Competition between different depths

various examples

Next chapter: Visual ornament

 

 

 

 

Consider Peter Tse’s volumetric worm depicted below in figure A. The computational function of perception can be defined as the expansion of that 2-D stimulus into a full 3-D experience as shown below in figure B.

 

 

 

 

The Kanizsa figure is actually a three-dimensional perceptual experience as the foreground triangle is perceived to occlude three complete circles that complete behind it. There are two aspects to this illusion: There is the amodal contours of the hidden sectors of the three circles that are perceived “invisibly” behind the triangle, and then there are the modal contours of the triangle that take on an actual brightness difference across the illusory edge.

 

 

 

 

 

 

 

The significance of symmetry is apparent in the phenomenon of the kaleidescope, whose symmetrical patterns are instantly perceived before we have time to think about it. When you think of the computational algorithm required to detect this kind of symmetry it is a combinatorial nightmare, as every piece of the image must be matched against every other piece at every different orientation.  The phenomenon of the kaleidescope strongly implicates a parallel analog wave-like computational principle whereby global patterns of symmetry emerge spontaneously from the simultaneous action of innumerable local forces. This is the Gestalt principle of emergence.

The emergent symmetry-detection system must operate in a full 3-D context because the simplicity of the 3-D perceptual interpretations is not at all apparent in 2-D projection.

 

 

 

 

 

The way that the visual system addresses this vast expansion of possibilities is to construct, or reify, every possible interpretation of the stimulus simultaneously and in parallel.

We can see this simultaneous parallel competition between equally likely interpretations in carefully contrived visual illusions that give exactly equal weight to alternative interpretations, resulting in a bistable, or multi-stable percept that alternates randomly between alternative interpretations.

The way that the visual system addresses this overwhelming expansion of possibilities into the third dimension is to essentially construct, or reify, every possible spatial interpretation simultaneously and in parallel, a computational function only possible in a parallel analog wave-based system. Competition between alternative spatial interpretations results eventually to the emergence of the single (or more) interpretation(s) of the stimulus that is (are) consistent with the given stimulus.

The primary function of visual perception

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Let There Be Light!

Continuing from Chapter 5: Ray-Tracing Algorithms

Ray-tracing algorithms highlight the vital importance of lighting to the interpretation of a scene, whether it be the stark bright light of the sun, the more diffuse glow of the blue sky, indirect light reflected by other surfaces, or light glowing or shining from luminiferous objects. We see the light not merely as a feature in the scene, but as a directed, causal process, that intimately links the light source with the patch of the world that it illuminates: we perceive the light as the causal source of the illumination of the scene.

However at the same time we also perceive the reverse inference:  The illumination of the scene implicates a light source illuminating it. And the nature of the illumination reveals information about the light source, even if the source itself is currently invisible, or out of our field of view.

An overall color cast across the whole scene implies that it is the illumination source that accounts for the color across the scene. The color that is distributed across the whole scene is attributed to a single causal agent, the light source.

The direction of the shadows in the scene point backward from the occluder back toward the light source, even if the light source is outside the field of view, giving a distinct amodal percept of the invisible light source that can be localized with some precision.

This spatial inference is exposed by smoke or mist in the air that reveals the invisible beams of light as a glowing shaft of illumination slanting through empty space, making explicit that which is perceived only implicitly in clear air. The haze turns the invisible amodal percept of the directed beams of light into a visible modal experience.

Multiple shadows cast from every object implicate multiple light sources, perhaps with different colors and properties. Sharp stark shadows indicate a point-like light source. Softer shadows indicate an extended source.

The causal connection between a light source and the world it illuminates becomes even more explicit in the case of local light sources that illuminate only local patches of the world.

We are not at all surprised to see the patch of illumination vanish as the lamp disappears. But we are puzzled if the lamp disappears but its illumination remains! There is no clearer example of the perception of direct causality than that which connects the light source to the patch of the world that it illuminates.

Consider this ray-traced scene of a dollhouse, from the POV-Ray gallery. Notice how each illuminated patch on a wall is accounted for, or “explained” by the light source pointing at it. Without the lamp, the bright patch on the wall might be misinterpreted as an actual discoloration, or bleached stain on the wall. The presence of the lamp helps explain each otherwise anomalous patch of brightness, and at the same time, the patches of brightness reveal and explain the presence of the lamps that are causing them. The cause reveals its effect, at the same time that the effect implicates its own cause.

Clearly what is happening in perception is a factoring of the intrinsic color or brightness of an object, and the pattern of illumination shining on it. When all the surfaces pointing in one direction are brighter than the surfaces pointing in other directions, that suggests that those surfaces are oriented toward the light, thus implicating the direction of illumination. Knowing the direction of illumination, in turn, helps disambiguate the perception of otherwise ambiguous forms.

This exemplifies a recurring theme of one of the most puzzling aspects of perception, of a kind of circular inference, where A implies B, but only if B also implies A, and thus the result cannot be computed in a single pass, but must emerge by a kind of resonance between all parts of the scene, to equilibrate to a globally consistent state. This reciprocal causality is demonstrated most explicitly in the ray-tracing model of mirroring, for example with two mirrors pointing at each other to create an “infinite tunnel” of reflections. In the first iteration each mirror records an image of the other by reflection. In the second iteration, each mirror sees its own reflection in the other mirror, and thus each mirror contains the other mirror in their reflections. In the third iteration each mirror sees the reflection of the other mirror in their own reflection producing three reflections in each mirror, and so on through more iterations potentially to infinity. Performed in analog with two real mirrors and actual light, the computation occurs virtually instantaneously, literally at the speed of light. In the ray-tracing algorithm this kind of cyclic reflection can require some of the most computation intensive processing. In fact the user must specify a limit to the number of cycles if the computation is ever to come to an end. And yet in perception, our automatic and instinctive, almost “unconscious” computation of the illumination source when viewing a scene, also seems to occur virtually instantaneously, thus strongly implicating a parallel analog wave-like computation not unlike the actual light reflecting back and forth between two real mirrors.

The perception of light propagating through space is a necessary prerequisite to the reliable perception of objects in that space, if the sensory evidence for those objeects is the result of their illumination by a light source, and their optical projection on the retina.

Continued Chapter 7: The Inverse-Optics Problem

Ray-Tracing Algorithms

Continuing from Chapter 4: The Language of the Mind

The computational function performed by mental imagery can be better understood by comparison with computer ray-tracing applications that perform a very similar computational function. Figure 1 shows an example of a ray-traced image of a scene from Friedrich A. Lohmueller’s most excellent web site  generated by the free open-source ray-tracing program POV-Ray, consisting of a sphere floating in space above a ground plane, under the dome of a synthetic sky. A light source is modeled in the sky, whose rays illuminate the sunny side of the sphere, which casts a shadow on the ground plane below. This is exactly the kind of image that might be evoked in mental imagery by the words “Imagine a sphere floating in the air”. Any artist worth their salt could compose such a picture on demand using paint and canvas to record a mental image that corresponds to those words. The image itself is two-dimensional, a mere projection of the mental image that it depicts. But it requires consideration of the full three-dimensional spatial configuration to correctly compute the scene, especially the patterns of light and shadow on the sphere, as well as the shadow it casts on the ground below.

The computational algorithm used in ray-tracing involves tracing every ray in straight lines radiating outward from the light source, and following its path through empty space until it strikes the surface of an object in the scene. If the surface is opaque, with a certain color, then the modeled ray is absorbed and re-emitted in that color in all directions from every point on the illuminated surface, as if the whole illuminated surface were glowing with that color.  If the object is shiny, then a portion of the ray is traced through a specular reflection as in a reflection through a mirror, creating a shiny highlight, and if the object is transparent, the ray-tracing algorithm traces the ray through the transparent object, modeling the proper refraction and partial absorption, or filtering, of the transparent, possibly colored medium. Every single ray radiating from every light source is traced in this manner through any number of reflections, refractions, and re-emissions through the scene. Any rays that happen to stray in the direction of the viewpoint or ‘camera’ position, if they fall within the frame of the rendered image, leave a trace on the image where the accumulated rays paint out the ray-traced image of the imagined scene. This is an extraordinarily computation-intensive process that takes many billions of computer cycles to compute. (Actual ray-tracing applications reduce the computational load considerably by starting at the final image plane and tracing the light rays backward toward the sources, thus only having to compute those rays that end on the output image)

The remarkable thing is that the human mind can perform this kind of computation virtually instantaneously, in a flash of mental imagery, strongly implicating a parallel analog computational strategy that solves the problem by spatial analogy. Although it takes an artist a certain amount of training and practice to learn to paint realistic scenes with the proper shading, shadows, and reflections, even young children can immediately interpret the result, even with a scene as complex as that in Figure 2, (“Balcony” from the POV-Ray gallery) with multiple cast shadows, reflections, refractions, and complex geometrical shapes. The compelling appeal of ray-tracing algorithms is not so much that computers are capable of performing the computations, which is amazing enough in itself, the real amazing thing is that people can correctly interpret the results, and they do so, even young children, instantaneously and effortlessly, without even any awareness that they are doing a computation at all.

The ray-tracing algorithm demonstrates explicitly the computational function served by the faculty of mental imagery, although what can take many hours of crunching in a digital computer seems to occur near-instantaneously in the human mind.

Ray-tracing applications use a geometrical code to define the structures to be rendered. For example the sphere at the center of the scene in Figure 1 is defined by a POV-Ray function sphere{ } which is given parameters that define its location scale, orientation, and the color of its surface.  The ground plane is defined with the function plane{ }, which is also assigned a location and orientation and color, or patterns of color. These functions serve the same purpose as the concept node in visual perception, representing the general concept in abstract invariant terms, while providing for parameters that can specify any particular cube or plane while conforming to the invariant formula. Friedrich A. Lohmueller’s excellent POV-Ray tutorial demonstrates the great variety of different shapes that can be defined by simple geometrical functions, illustrated here, from spheres and cylinders and cones, through boxes and prisms.

Shapes and surfaces can also be defined by any mathematical function that specifies a location for every point in the surface, for example using  paraboloids and hyperboloids, as shown here, polynomial, exponential, sinusoidal, parametric, any mathematical function that defines a surface or volume. 

Compound shapes can be built up by Constructive Solid Geometry (CSG) operations, logical set-theoretic operations between volumetric solids, such as unions, intersections, difference, etc.  For example the union of the red and yellow spheres defines a shifted double-sphere. The yellow sphere subtracted from the red sphere cuts a

spherical bite,  or void from the red sphere, while the intersection of the two spheres is the volume of their geometrical intersection, a lens-shaped volume that perfectly models a spherical lens.

Shapes can also be defined by extrusion, by translating a three-dimensional shape along a three-dimensional path through space, thereby sweeping a volume through that space as shown in these examples. For example the torus (shown above) is defined by sweeping a flat circle around a circular path normal to its surface. The shape, size, and orientation, and even color of the moving shape can change as a function of position along the path, resulting in shapes that swell and shrink in characteristic ways.

Compound shapes, or arrays of shapes can be constructed by repeating copies of a basic shape, each one translated, rotated, scaled, or  otherwise reshaped to create ever more complex compound shapes.

You can define smoother blob objects using some kind of distance metric from some reference frame, which you can imagine as a kind of haze whose density varies with distance from the frame. A density threshold defined in that haze will in turn define a surface that encompasses a volume. For example the pink triangular blob shown in the upper-left is defined on a triangular frame between three vertex points, with a larger distance metric around the points, and a smaller metric around each triangular side. The two metrics are blended with a nonlinear equation, resulting in a smooth blobby shape that necks continuously from spherical to cylindrical form.

The surfaces of objects can be painted with patterns of  ‘pigment’, colors and textures, which are defined in the very same language, i.e. Euclidean geometry, as the language of three-dimensional shape.

If you modulate regular patterns with some kind of randomized turbulence, you can generate a great variety of different patterns that bear striking resemblance to a great variety of natural phenomena.

Randomized functions and fractals generate more realistic random patterns.

The shapes themselves can also be modulated or replicated in regular arrays, to define compound objects composed of a multitude of sub-units. Random or fractal functions can make these structures quasi-periodic, more similar to certain natural phenomena.

POV-Ray is a truly magnificent tool for defining three-dimensional scenes from your imagination. Its dimensions are those of the artist, color, space, form, and light, the dimensions of conscious experience. But its genius is in the fact that, like virtually all other ray-tracing apps, it uses the natural language of visual perception to express its geometrical structure, because that is the code we all understand. It is the way we think about shapes, that is the way we break down compound shapes into their geometrical primitives. We use this visual code not only in free-wheeling visual imagination, but also in perception. Visual perception is the act of hallucinating a scene that is consistent with all the sensory evidence. Perception is as much an act of creation, as it is an act of detection. If an artist were given any of the two-dimensional ray-traced scenes above, and commissioned to construct a three-dimensional painted sculpture of the scene, the computational function that he is comissioned to perform is exactly the computational function of perception: The input is a two-dimensional image not unlike a retinal image: The output is the rich three-dimensional experience that you have, immediately and unconsciously, as soon as you just glance at any of the ray-traced images above.

There is a general principle encapsulated in this mathematical approach to defining geometrical shapes, and it is a foundational principle that underlies all of mathematical thought. Every mathematical function, whether a simple linear or planar function, a circular or cylindrical or spherical arc or shell, or something more complex like a polynomial, exponential or logarithmic, or sinusoidal function in one, two, or three dimensions. It is that whatever the formula that defines the shape, that shape is defined to infinite precision. The function defines a pattern that is independent of scale. It is a pure and perfect descriptor of shape to essentially infinite resolution. The appearance of this concept in ray-tracing software reflects the deep intuitive basis of this way of representing shape in the human mind. Ray-tracing software is designed by people to be used by people, and that is why it uses basic geometrical concepts starting with points, lines, and planes, to define the shapes that create the imaginary scenes. This is the way that we think about shape, whether in perception or in mental imagery and imagination.The reason why ray-tracing algorithms employ the primitives of euclidean geometry is the same reason why Euclid chose those same primitives in the first place, because that is the way we conceptualize shape. Euclidean geometry was not an invention, but more of a discovery of the basic elements of geometrical thought, and successive generations of geometry students accept euclidean geometry not as dogma, but because they find it consistent with their natural intuitions about shape, a product of the long evolution of our perceptual and conceptual systems.

Continued Chapter 6: Let There Be Light!

The Language Of The Mind

Continuing from Chapter 3: Amodal Perception

Images are the primary language of the mind. We think and imagine in terms of scenes and views of objects and events that we conjure into existence and examine from different angles, testing in our mind whether this couch would fit in that corner, or whether this car would fit in that parking space. We simply will the image into existence, and we maneuver it invisibly into position in the real world, to see how much clearance there is and how tight the turns will have to be to get it there. We do this so automatically and instinctively that we are hardly aware of it, or at least aware of it as anything real, due to its amodal invisibility. But we can “see” the mental image of the couch, or car, clearly enough to estimate the clearances, sometimes assisting our mental image with sweeping motions of our hands to indicate the spatial location of the mental image as it is maneuvered into position relative to the real world. Our mind is generating 3-D images continuously in real time, both modal and amodal, in both perception and cognition. The primary function of the mind, what makes it spatially conscious, is its ability to project three-dimensional images into a spatial world of conscious experience, and that volumetric moving colored image, and the invisible amodal framework that supports it, is our only window onto the objective external world beyond the mind.

 

But the mental image is more than just an image. It is an image that is constructed to a particular formula that give meaning to its shape. A mental image of a cube is tied to the abstract concept of a cube, a concept that can attach to any cube as long as it is cubical. But the cube can vary in spatial location, it can rotate in orientation, it can zoom up and down in scale, it can even stretch or morph elastically (within limits) while still maintaining its perceived identity as a cube. The concept of a cube is distinct from the image of a cube, because the concept is invariant to the rotations and translations and scalings of the cube. And yet the concept is intimately connected to the image in the sense that the concept inevitably “lights up” or becomes activated in our mind whenever a cube, or cubes, are present in our experience wherever it, or they, are located. That node provides the ‘handle’ that connects imagery to language, allowing us to report verbally whether we are experiencing the image of a cube. This is the bottom-up process of visual recognition as expressed in neural network models, where the concept is represented by a “node” whose activation represents the recognition of its corresponding image in the sensory stimulus. Bottom-up recognition at least for simple shapes, exhibits an invariance to rotation, translation, and scale.

 

 

 

The exercise of mental imagery, or imagination, reveals that the concept is capable of more than just bottom-up recognition. It is capable of actually generating cubes in mental imagery, and it can do so through that same invariance relation. I can imagine my cube at any location, orientation, and scale, or I can imagine it rotating, translating, and scaling continuously while I am imagining it, especially if I help stabilize the image by following its corners with my fingertips as I move it through space. Mental imagery is a most extraordinary faculty that forms the foundational basis of mathematical thought. Although we have no idea how we perform this extraordinary mental calculation, we can describe the computational function of mental imagery at least in functional terms.

The most basic feature of mental imagery is the mental image space, the space of our imagination, which is extended in three dimensions, within which the images of our imagination appear. Attached to this mental image space is a conceptual representation which can be described as an abstract or symbolic code, an array of concept nodes, that represent shapes such as spheres and cubes and pyramids, invariant to their location, orientation, or scale. The computational function of mental imagery can be described as the top-down projection of an abstract spatial concept into a three-dimensional mental image through an invariance relation, assigning to the imagined object a specific location, orientation, and scale, by a deliberate act of will.

Chapter 5: Ray-Tracing Algorithms

Amodal Perception

Continuing from Chapter 2: The Schema As A Mental Image

Amodal perception was first discovered as a property of perception, where it occurs so unconsciously and effortlessly that we tend not to notice it at all. For example if we view two pencils crossed, one pencil occluding the other, the occluded pencil has a great gap of missing data, and yet we perceive that pencil as complete and continuous through the occlusion. Furthermore, we only see the exposed side of each pencil as a colored hemi-cylindrical surface, and yet we perceive each pencil as a whole cylindrical form, complete through its invisible volume all the way to its hidden rear surfaces. The modal percept appears as a colored surface, whereas the amodal percept appears as a solid volume occupying a specific region of space, where it is perceived, but remains completely invisible.

When I see a box on the floor before me, I generally see only three exposed surfaces, whose straight edges are usually tilted by perspective so as not to appear at right angles from my viewpoint, and yet I perceive the box as a right-angled rectangular volume, complete with an interior, which can be perceived as hollow or solid, and with hidden rear surfaces. I can easily reach back behind the box and demonstrate by morphomimesis the exact location and orientation of those hidden surfaces as if I were seeing them transparently through the box. I do not decide to form this mental image, or to make it rectangular. The amodal image forms immediately and automatically based only on the appearance of the visible surfaces, assisted by my past experience with boxes. It is my mind’s way of constructing the simplest three-dimensional explanation for the two-dimensional stimulus on my retina. The invisible image has exactly the shape that I perceive the object to have.

One reason why this amodal percept is so easily overlooked, is that it is not really a visual experience as such, even though it is often informed by a visual stimulus. When I encounter a box in pitch darkness, or with eyes closed, and feel it with my palms, I get the same amodal experience of a rectangular volume in a specific location in my space, and I perceive the whole box, through to its rear surfaces, even though I palm only selected faces or corners of the box at a time. And when I turn on the lights, or open my eyes, the tactile texture of the box felt by my palms is experienced on the very same rectangular volume as are its color and brightness that I perceive visually. The hardness or flatness of the ground that we feel underfoot even with eyes closed, are experienced as properties of the very same amodally perceived ground that carries the color and brightness and texture that we perceive visually with eyes open. In other words, the spatial structure that is our amodal experience of the world is the common ground, or lingua franca, that unites all sensory experience in a modality-independent structural representation of the world, and that amodal structure represents our perceptual and cognitive understanding of  the world.

 Furthermore, our picture of the world is incomplete without an amodal experience of our own body at the center of our space, which we also experience, even with eyes closed, as a three-dimensional spatial structure with four limbs and a head, attached to a trunk. And we can feel our own pose or posture even without looking using the sense of proprioception, which is experienced as specific patterns of articulation of the explicit spatial structure of our body.  Close your eyes and observe the experience of proprioception. The information content of that experience is equal to the information content of a wooden marionette, whose limbs and torso are bent into whatever is the current posture of your body. That is the shape of your experience of posture.

The amodal experience of our own body by proprioception, and the amodal experience of the hidden rear faces of perceived objects, are expressed in the same pure geometrical form, like an outline drawing hanging in three dimensional space, printed in invisible ink that we can magically see or feel. And it is that invisible structure that supports the modal colored surfaces that we experience visually on the exposed surfaces of colored objects, and that same amodal structure supports the sensory experience of tactile contact.

When I hold a baseball in my hand, the round shape that I feel in the cup of my hand, is perceived as part of the same amodal sphere that supports the white leathery surface that I perceive visually. The amodal percept is the lingua franca, or common ground that unites all sensory experience in a modality-independent structural representation of the world, that is our cognitive understanding of our self and our world.

The existence of amodal perception compels us to take stock anew of the nature of visual experience. From the earliest days of childhood we have been working under the naive realist assumption that the world we see in our experience is the world itself, viewed directly out there where it lies around us. The fact of amodal perception requires us to revise that view, to acknowledge that the amodal percept is not the object itself, as suggested by naive perception, but rather, the amodal percept is a data structure constructed by our mind based on sensory evidence. And that data structure is expressed in explicit spatial form: It is a solid three-dimensional structure that occupies specific volumes in the space of our experience, and the very existence and shape of that invisible structured experience is identically equal to our understanding of the structure of the world.

Amodal perception is the connecting link between perception and cognition. It has the automatic and instantaneous nature of perceptual recognition, as in the perception of the volume of a box, and yet it is also susceptible to cognitive manipulation – it is possible to modify the shape of your amodal percept by an act of will, for example by imagining (or believing) a sealed box to be full or empty, or by imagining its hidden rear faces to be missing, or punctured, or dented, if we choose to imagine them so. Amodal perception is the earliest, most basic form of mental imagery, one that we certainly share with almost all visual creatures. But amodal perception is also the door that opened perception to cognition and free-wheeling mental imagery, and true human intelligence, connecting the world of direct experience to the world of imagination.

Continued: The Language of the Mind

The Schema As A Mental Image

Continuing from: Chapter 1: The Perceptual Origins of Mathematics

The schema, it seems, is a kind of mental image. But what is mental imagery? There has been considerable debate in the literature over whether mental images even exist as actual images, many insisting that they do not see mental images as pictures in their mind. Surely this is a question best left to introspection. When you instruct yourself for example to imagine a table in your mind’s eye, what, if anything do you see? Well, in the first place I see absolutely nothing in the usual sense: the space before my eyes remains an empty void, with nothing in it. And yet at the same time I do see a totally invisible mental image of a table in that same empty space, or at least in a space that is somehow superimposed on my normal visual space. I can both see the mental image, in an invisible ghostly way, and at the same time I don’t see anything at all. Furthermore, what little I see of the imagined table does not always have a specific location, nor a specific size or scale, nor a specific viewing angle, nor a specific color or furniture style. The “image” of the table (if it can be called such) often appears either fleeting and unstable in location, scale, and orientation, or it appears totally abstract, non-spatial, as if expressed only in some symbolic non-spatial code, like a node in a neural network model that is labeled “table”. It is this fleeting evanescence and instability of the mental image that allows so many to deny its very existence as a spatially extended image in our imagination.

But although the mental image of a table can remain totally unspecified with respect to location, orientation, and scale, it is clearly possible to imagine a specific table at a specific location, orientation, and scale, and we can even select a color, and furniture style for our imagined table. The mental image remains perfectly invisible, we would still swear there is nothing there in the empty space before us. And yet at the same time we can see the imagined table right there in that empty space, with greater or lesser vividness and detail, even though its appearance in that space seems coincidental and inconsequential, like reflections in a pane of glass superimposed on the world seen through the glass. An artist or sculptor routinely sketches their mental image as if copying from a real image, demonstrating that there is some kind of information present in the mental image, and that information can be clearly spatially extended, like an actual three-dimensional image of a scene. It is even possible to locate the imagined table at a specific location in the space before us, outlining the spatial limits of its top and sides and legs with our hands, as if polishing the invisible surfaces of the imagined table. I call this exercise morphomimesis, miming the morphology of an imagined object with a wave of your palms, and thus revealing its explicit three-dimensional spatial structure. Although we can only mime two parts of the image at a time with two palms, the image itself can remain fixed and stable in space during the morphomimesis, demonstrating that it is possible to have a fully specified mental image that has the property of spatial extension across a specific region of space, even though it remains completely invisible in that space.

The fact that it is possible to form a mental image with a specific location and specific dimensions, and to mime its morphology with your palms, is proof that mental images can exist as stable three-dimensional structures, and that they can carry a specific information content. And the mental image can be formulated to have a specific location and spatial extent, even if it is not usually specified so precisely, but often remains in an indeterminate state. The fleeting evanescence and instability of many mental images should not be viewed as counter-evidence for their existence as images, but is merely evidence of a fleeting and unstable imaging system, one that is capable of representing multiple possibilities all superimposed, much like a quantum particle that can exist in multiple states simultaneously. Like a quantum particle, the mechanism or principle underlying the mental image can apparently flip or morph continuously into different forms, unless it is held to a stable state by an act of will.

So let us examine the mental image medium to see what mental images are composed of, how they present themselves to consciousness. Picture, if you will, a square, of the geometrical variety, that is, composed of Euclidean lines that span four Euclidean points marking the four corners, to define a square segment of a Euclidean plane, a surface that is perfectly flat and thin. It is possible to imagine such a square of any size, I can rotate it in my mind to any orientation in three dimensions, and I can trace out with my fingertips exactly where I am imagining the square at any moment. In other words, what I see in my mind’s eye is an image, very much like the images I see with normal vision, except that the mental image is totally and completely invisible.

Mental imagery compels us to acknowledge two different types of seeing: One is the regular type of seeing, as when viewing a colored cardboard square whose edges are defined by a visible transition in color and/or brightness across the edge, and the other is a kind of invisible seeing in which imagined objects are completely invisible, and yet we can “see” them as spatially extended structures that can occupy specific volumes of visual space. Michotte (Michotte, 1963; Kanizsa, 1979; Michotte, Thine`s & Crabbe´, 1991) has called this kind of vision “amodal” perception, because it is perception in the absence of a particular visual modality, such as color or brightness. We see a square in our imagination, but it is in a kind of invisible outline form, like a figure in a geometry text, without color or substance, just a shape.

But it is also possible to imagine a color with your mental image. I can just as easily imagine a red square, or a green one, and I can see my imagined square change color on my command, with a specific square region being painted out with the specific color that I choose to imagine, all the while remaining totally invisible in the sense that I’d be willing to swear in a court of law that I do not see a colored square before me, even though I can locate with precision the edges and corners of the colored square in my imagination that I don’t see. This ability to conjure into existence any simple geometrical structure I might choose, and to imagine it at any location and orientation and scale I might choose, and even paint it with imagined color, and yet to remain acutely aware of the distinction between reality and my imagination, is both the foundational origin of mathematical thought, and at the same time, it reveals the most basic operational principle behind human intelligence. We think primarily in pictures, the words only follow after the mental image is formed. And the words lose their meanings if they become disconnected from the images that they represent in the symbolic code of language.

Continued Chapter 3: Amodal Perception

The Perceptual Origins of Mathematics

Mathematics is a strange and wonderful mystery, that grows more strange and wonderful the more we learn about it. Our first introduction to mathematics in school is through arithmetic,  a pragmatic system of accounting and quantification,  first developed for applications such as counting sheep, for recording debts or payments, for  counting elapsed time, to quantify distances and areas for travel and surveying, and to predict the motions of the heavenly bodies, and thereby the seasons. These are of course but a tiny sample of the innumerable other applications where arithmetic comes in handy. The remarkable thing about math is that all of these varied and diverse problem domains can be addressed using a single general purpose conceptual tool. All these diverse problem domains are quantified and calculated using the same system of numbers, although the details of that system can vary from one culture to the next.

If we are to trace the origins of mathematics, we must focus on the part that is common to all number systems across all cultures and historical periods, because the common root of mathematics is likely to give a good indication as to that component of mathematics which is innate, a genetic heritage, as opposed to a cultural heritage passed down from generation to generation. Lakoff & Núñez (2000) trace the origin of mathematics to an instinctive form of counting called subitizing, in which you can instantaneously ‘see’ the number of things if the number is small, less than about five to seven items. This is the way you recognize the number of dots on dice.

You don’t have to count the individual dots, you can ‘see’ the numerical quantity immediately, in one glance, even if you had never been taught their names. Surely this is the origin of mathematics, the ability to see a difference in quantity, even if those numbers have no name, and even if that quantification only works for small numbers. Human cultural mathematics takes this innate numerical instinct and extends it to express much larger numerical quantities than can be counted by subitizing. But the cultural extension to innate math adheres to the same basic rules that govern the first few numbers, and many of the extensions to the natural numbers, such as the concept of zero, of negative numbers, fractions and rational numbers, irrational and imaginary numbers, are such a natural extension by the same rules, that they almost seem obvious after the fact, as if they had been there all along waiting to be discovered, as soon as the problems that they resolve are first encountered.

But the concept of numbers runs much deeper than a simple one, two, three. Before you can even think of counting, you have to understand the concept of countable things. Whether you are counting people, coins, or pebbles on a beach, you must decide on the set of things you want to count, and what requirements you should establish to qualify for inclusion in that set. How big can a pebble be before it counts as a rock instead of a pebble, and how small before it counts as a grain of gravel or sand? The answer depends on why you are counting them, that requires a fine mathematical judgment. Counting requires that all the members of the set can be considered equivalent, interchangable, they each count for exactly one. This is so natural and obvious an assumption (to us humans) that we don’t even realize that we are making it. But it is an artifact of the human mind, not of the natural world, that things come in identical countable units. In reality pebbles are not at all identical, each one has its own unique size and shape and color, and the distinctions between rock and pebble and gravel are overlapping and indefinite. The fact that we can make ourselves see a set of pebbles as identical units is itself a conceptual feat that is a prerequisite to learning to count them.

Lakoff explains the foundations of mathematical thinking with the concept of schemas. For example, before counting pebbles on the beach, you must first see the beach, and see the pebbles on it, and you must select which pebbles you wish to count based on some criteria that determine which pebbles belong in your countable set. If counting to merely demonstrate the concept, you might choose a dozen or so pebbles of a certain size or color, and set them apart from the rest, drawing an encompassing circle with your mind, distinguishing these pebbles as the set to be counted. For the process of counting, you must imagine another group, or imaginary circle, into which you either move the pebbles one by one, or perhaps you imagine that circle to engulf the pebbles one by one as you are counting, moving them from the set of pebbles to be counted, to the set of pebbles already counted, while keeping a tally of the total by the familiar sequence 1, 2, 3, … . All of the thought before the actual counting begins, is wordless thought that we did not have to learn in school, we knew instinctively how to judge whether this pile of pebbles is greater in number than that one. We didn’t need to know how to count for that. We don’t need to know numbers for forming a schema in our mind. However it is impossible to do any real mathematics without first forming a schema in our mind. The schema is the framework that gives meaning to the math. It is what poses the question in the first place, and what interprets the meaning of the results at the end. That is the real mathematical thinking beyond mere arithmetic, the framing of the problem to be solved, the selection of the algorithm to be used to solve it, and understanding the significance of the results after the counting is done. The counting itself is the simplest part of the problem – so simple that even a stupid computer can do it. But conceptualizing the situation and seeing the schema in it, is in fact the real mathematical part of the task. The rest is merely arithmetic.

Another schema that Lakoff cites as a guiding metaphor for numbers is the concept of pacing off a distance with steps of equal size. This relates to the idea of the number line, numerical value depicted as spatial extension in one dimension. This opens the possibility for negative numbers, when pacing backward beyond the origin, and it also presents the unitary interval as a continuum,  infinitely sub-divisible into fractional sub-intervals. Again, the actual counting of the paces, the choice of names to call the numbers, or whether to record them in binary or hexadecimal, or using Arabic or Roman numerals, is trivial compared to the real act of mathematical thought, which is seeing the land that requires surveying, knowing which dimensions to measure, making the measurements, and then understanding the significance of the paced-off quantities. The greater part of mathematical thought is not the simple arithmetic that we first learn in school as math. Real mathematical thought is an embodied process, one that cannot be meaningfully separated from our direct experience of our body located in the world.

Lakoff is right in identifying the origin of math in practical hands-on problem solving and basic conceptualization of reality. But the problem with identifying schemas as the origin of mathematics is that the notion of a schema is very vague and abstracted. We are all familiar with the experience of forming and using a schema, but we have no idea how we are actually doing it, or what a schema even is. Lakoff leaves the story of the origins of mathematics hanging at this point. This is the frontier of the terra incognita at the root of mathematics. From here on down we have no idea how we do the most primal mathematics, constructing for ourselves a schema of the world. What does that even mean?

The profound difference between schematic conceptualization and arithmetical calculation has come into sharper contrast in this era of the digital computer, a mechanism that is capable of millions of arithmetical operations per second, and of computational algorithms of fantastic complexity, with virtually perfect reliability and reproducibility, far beyond anything we can accomplish with pencil and paper. The computer is a very useful tool for the mathematician, to plot his mathematical thoughts and make them visible on the screen, And yet the digital computer is totally incapable of even the most primitive mathematical thought. The computer is completely incapable of conceptualizing a schema, or understanding the significance of the quantities that it calculates. It seems that there are two starkly contrasting aspects of math, one which is thoroughly understood, which can be performed by stupid machines even better than by humans, the other that we all do unconsciously and instinctively virtually every waking moment, but have no idea how we do it or even what it is we are doing. The focus of this book is on that other aspect of mathematical thought, what it is, and how it works, and ultimately, how we could build an artificial mind that is capable of real mathematical thought, and with it, a spatial consciousness and sense of self-existence like our own.

Continued Chapter 2: The Schema as a Mental Image

Geometric Algebra: Conformal Geometry

This is the third and final chapter in my Visual Introduction to Clifford Algebra, following on from Chapter 2 on Projective Geometry in Geometric Algebra. In Chapter 2 we showed how the perspective projection provides a wonderful means for projecting an infinite external world into a finite projection. But there is a fundamental flaw in this concept, it can only project in one direction, for example it cannot represent opposite ends of a room simultaneously.

Projective perspectives taken from opposite directions are incommensurable — they cannot be merged seamlessly without a sharp discontinuity where they join.

The problem can be traced back to the concept of projection onto a plane, which can only depict one half of reality in principle. Here we show the principle of projection again for a two-dimensional Euclidean space viewed from a third-dimensional vantage point.

We can see that the projection onto the image plane can only view the world in one direction, it cannot look in the opposite direction.

 

The solution would be to project onto a surrounding sphere, instead of a plane, so you could project in all directions. But don’t put the viewpoint at the center of the sphere. In the first place you could only ever use the lower hemisphere for projections on the Euclidean plane.

Instead, the viewpoint is located at the top pole of the sphere, and the reason for that is that it confers some extraordinary invariances to the projection, which is a conformal projection.

This is the first stage of the conformal projection in Geometric Algebra as proposed by David Hestenes.

Here is a 3-D depiction from Wikipedia of the stereographic projection of the two-dimensional Euclidean plane. The Euclidean point A projects to conformal point α = P(A) where the line PA meets the stereographic sphere. Note that a stereographic projection can be defined with the projection sphere tangent to the Euclidean origin, as shown further above, or with the center of the sphere on the Euclidean origin,  as shown here, the only difference being that in the latter case the angle APO is double the angle computed as above, but both are legitimate stereographic projections.

Here is a stereographic projection of a square grid centered on the origin. The center of the grid, point (0, 0) projects to the nadir of the sphere. Points farther from that origin project higher on the sphere, and the outer perimeter of the grid projects around the top of the sphere, surrounding the polar apex that represents all points at infinity.

The stereographic projection is closely related to the inversive function shown here from the center of a chessboard on the left, with the unit circle marked in red. To the right is a magnified view within that unit circle. Note how the nearest four squares surrounding the circle are reflected in the periphery of the circle, whereas farther ranks of squares are reflected deeper toward the center of the circle.

The stereographic and inversive functions are related by a reflection symmetry across the Euclidean plane, as shown in this Wiki Page. The stereographic projection of point P’ on the number line is point P on the unit circle. Q is a point on the unit circle that is a mirror reflection of P across the number line. The inverse of P’ is the stereographic projection of Q, which is P” or Q’. Note that the stereographic projection and the inverse function are identical at points zero and one, so the inversive function is very similar to a projection of the stereographic projection down onto the Euclidean plane.

So the stereographic projection is related to the strange and extraordinary inversive function that we encountered in Chapter 1 at the origin of our number line! This is a mapping that inverts proximal and distal, i.e. nearness flipped with farness, with the most extraordinary property that all of the distinct infinities found in every distinct direction from the center, all map to a singular central point in the conformal mapping! The bizarre inversive function found at the center of our number line is a variation on the stereographic projection.

This stereographic projection exhibits some extraordinary invariances. A circle on the Euclidean plane maps to a circle in the conformal projection.

The stereographic projection of a straight line in Euclidean space projects to a circle in conformal space, a circle that passes through infinity at the pole of the projection sphere. Two lines that cross at a point in Euclidean space project to two circles in conformal space that intersect at two points, once corresponding to where they cross in Euclidean space, and again where they cross at infinity.

Circles and straight lines are the same shape in this space!

It is a regularity that becomes apparent only in the conformal projection.

The bizarre inversion of proximal with distal, turning near and far inside-out, wraps infinity in a finite package that is more manageable for a finite mind than any kind of representation out to infinity, if such a thing is even possible for a real physical representation.

The utility of this conformal mapping has already been demonstrated in Catadioptric Photography, (Geyer and Daniilidis 2001)  i.e.  photography using a combination of mirrors and lenses. It turns out that taking a photograph through a reflection in a parabolic mirror (upper left) is mathematically equivalent to the stereographic projection (upper right). On the lower left is a photograph taken in this manner. The dark circle at the center is a mirror image of the lens that is taking the picture. Around it is a reflection of the surrounding room looking in 360 degrees all around. The camera and the mirror appear to be attached to a pegboard with an array of regularly spaced holes, evident in the reflection in the lower left quadrant. The straight lines of holes in the pegboard map to perfect circles in the conformal projection, as traced out at the lower center. And circles are particularly easy to detect due to their concentric symmetry about a center. An inverse conformal mapping applied to the conformal image (lower right) restores the linear alignment of the holes in the pegboard.

 Conformal Projection in 3-D through 4-D

The examples of conformal projection presented above involve a projection from a two-dimensional Euclidean space, for simplicity of exposition. But the conformal projection active in perception is a three-dimensional phenomenon making use of a projection through a fourth spatial dimension. It is difficult to visualize operations in a fourth dimension. However there is a peculiar feature of the conformal mapping that simplifies the visualization considerably.

In two-dimensional Euclidean space, every direction in the (x, y) plane extends outward from the origin to its own unique infinity. But in the conformal projection, all of those infinities map to the same polar point at the pole of the sphere. This works only because every one of those directions in the (x, y) plane is orthogonal to the z dimension that defines the north pole of the stereographic sphere.

To expand into three-dimensional conformal projection,  a fourth dimension is required that is simultaneously orthogonal to x, y, and z. This fourth dimension is easier to visualize because of the fact that although every distinct direction in x, y, and z, has its own unique infinity in that direction, all of those infinities map to the same single central point in the conformal projection, and that point is a finite unit distance away, so the entire squashed fourth dimension is contained within a three-dimensional shell. It is a closed dimension that mimics infinity in a finite package. So we can visualize a three-dimensional conformal mapping as a finite spherical space where infinity maps to a singular point at its center.

What we have is that same bizarre mapping that we encountered in the inversive function.

 

Here is the conformal reflection of a cylindrical rod. If the rod grows to infinite length, its conformal reflection forms a complete ring through infinity. It is possible to track this growth all the way to infinity in the conformal mapping, that is impossible in the Euclidean mapping, you cannot depict an infinite rod on a finite page no matter how small you scale it.

This is the conformal reflection of four pillars on a square base. There is a peculiar fish-eye-lens warp in the conformal projection that seems hauntingly familiar.

When I first encountered Hestenes’ conformal mapping (Hestenes et al., 2000, Perwass & Hildenbrand 2004), I was struck by the similarity to another conformally warped world that I had been studying, the perspective warp observed in visual experience (Lehar 2003, 2003, Cartoon Epistemology). You can see this conformal warp when standing on a long straight road. The sides of the road converge to a point by perspective on the horizon ahead, and if you turn around, they converge to a point in the opposite direction too. And yet at the same time the road appears straight and parallel and undistorted throughout its length. Objects in the distance appear smaller by perspective, and yet at the same time they appear undiminished in size.

In case you have never noticed this warp of perceptual experience and may even question its reality, I conducted the “Hallway experiment” in a hallway at the Schepens Eye Research Institute. Subjects placed in the hallway were given three cardboard models, and they were asked to choose which model best matched their experience of the hallway. Not the geometry of the hallway as they “knew” it to be, but the shape of their experience of the hallway. Did it appear as a rectangular prism with right angled corners and parallel sides, as in model A? Or did it appear as a flat two-dimensional projection as depicted in model C? Or did it appear somewhere in between, like a bas-relief perspective scene with lines converging toward a vanishing point as shown in model B? Subjects responses were divided between models A and B.

Then the subjects were presented with a fourth model, D, that was identical to model B except that it was etched with a perspective distorted grid and the subjects were told that the grid represented the scale of the model. In other words, this was a scale model like any scale model except that the scale varied with depth, such that the grid panels in the distance appeared smaller by perspective but they were the same objective size as the apparently larger panels in the foreground.

When offered this alternative all of the subjects chose model D. This demonstrates a profound duality in spatial perception whereby objects in the distance appear smaller by perspective, yet at the same time they are judged not to be smaller in objective size. They are perceived to be shrunken by perspective, and at the same time they are perceived undiminished in size, because they span the same number of grid lines in the warped perspective grid.

This duality in size perception is the firmest proof that the world we see around us in experience is not the world itself, but merely an internal perceptual replica of that world in an internal representation with a peculiar conformal warp that allows it to package all of external space out to infinity in all directions into a finite spherical package whose extreme conformal warping might not even be noticed by the casual observer.

There is a compelling similarity between the conformally warped world of perceptual experience, and the conformally warped inverse world of the stereographic projection. In some ways these two warped worlds bear an eerie similarity, and yet in other ways they seem like polar opposites. The Bubble World perspective is big in the middle and shrinks to nothing in the periphery, whereas the “inversive” world of the stereographic projection is big at the periphery and shrinks to nothing at the center. Besides being also turned inside-out.

It finally dawned on me how these two worlds were related. One is the radial reflection of the other, conformally reflected across the surface of the unitary sphere. The large house in the foreground in the outer Bubble World perspective corresponds to the larger house in its inversive reflection near the surface within the unitary sphere, while the smaller houses into the distance in the Bubble World perspective correspond to the smaller houses toward the center of the inversive sphere.

And in case you were wondering what happened to the percipient of this experience, there he is, nestled in between the two worlds, the bounding surface between the internal self and the external world. In my experience I do see the surrounding world warped in the Bubble World perspective. But I see no trace in my experience of that weird inside-out conformally reflected world, although I do perceive some of the regularities that are detectable only in that inversive world. I do perceive a “circularity” in an endless row of identical houses, and how progressing toward infinity is much like running on the spot in a hamster wheel.

My Own Contribution to Geometric Algebra

The conformal mapping between the Bubble World and its inversive reflection  finally made sense in my mind of the next step in the conformal mapping as proposed by David Hestenes. According to Hestenes, the conformal mapping occurs in two stages, first a conformal reflection into the unit sphere through an inversive function as described above. After that, a second outward reflection that projects the results of the symmetries detected in the inversive sphere back from the inside-out inverse world out to the right-side-in Euclidean world with an inverse of the inverse transformation.

 

 

For example a three-dimensional line in external space projects to a curved arc in conformal space, such that the radial distance of every point in that reflection from the origin is the inverse (1/r) to the radius r of the corresponding point in the line itself.

The regularity, or collinearity of the line is recognizable in the inversive reflection by the fact that the curved line is a circular arc that is part of a circle that passes through infinity at the center of the inversive space. The implications of this detected regularity are projected back out again by an inverse of the inversive function r² (every radial distance in the reflection is squared) because the square function cancels the inverse function to restore the original distance r in Euclidean space. (r² × 1/r = r).

This outward projection is not particularly useful for restoring the line itself, if the line was given in the first place.

The restoration is more useful for restoring the geometrical regularities detected in the conformal reflection, for example the fact that the given line segment is a part of an infinite line that stretches to infinity in opposite directions. That extrapolation is inverse-projected from the circle in the inverse world back out into Euclidean space, where it completes the symmetry of the line out to infinity.

Now this projection stage of the conformal mapping immediately bothered me, because points from the center of the inversive sphere would have to be projected all the way to infinity. But nothing can actually project to infinity! Infinity is fine as a mathematical abstraction, but for those of us who believe that mathematics is a physical mechanism, or analog computational projection taking place in the physical brain, the notion of projection to infinity is a physical impossibility.

The conformal distortion manifest in the Bubble World perspective suggests that the outward projection need not be a projection to actual infinity, but just to another conformal reflection bounded by a finite limit representing infinity. In this case the conformal projection chosen was the vergence measure of distance, i.e. the angle between the direction of gaze of two eyes in a binocular system that ranges from v = π for objects at zero distance to v = 0 for objects at infinity.

Actually this is none other than a one-dimensional stereographic projection from the number line to the unit circle, wrapping the infinite linear range from zero to infinity into the finite bounded angular range of 0 to π in each direction. This is also simultaneously a kind of logarithmic transformation because adding angles on the closed angular scale corresponds to multiplying the corresponding quantities on the linear scale. This also explains why Clifford multiplication of angles produces a sum of those angles, not a product of them.

 

 

 

In fact, bounding the outward projection to a finite scope solves a number of thorny problems, for example the impossibility of the reciprocal function, that suggests a one-to-one mapping between every number x from one to infinity, and its reciprocal  1/x in the bounded range between zero and one. The Bubble World conformal mapping transforms that to a one-to-one mapping between points in one finite conformal space to corresponding points in another. It is no longer an impossible mapping, but one that is perfectly possible in a real physical implementation.

A mathematics that pretends to encompass infinity, bears a permanent scar of profound paradox right at the point where it (supposedly) makes contact with infinity. A mathematics that acknowledges the profound impossibility of infinity, and thus places it in effigy at a distance that is less than infinite, marries with “infinity” as a spatial continuum, a structure that can in principle be implemented in a finite physical mechanism like the human brain.

And of course on our number line the same compression is performed in the negative direction too, the stereographic projection ranges from -π to π, to produce a number line that extends all the way from “infinity”, a pseudo-infinity, or end-point on the line in one direction to “infinity” in the opposite direction. This “infinity” represents infinity without actually being at an infinite distance from the origin. This pseudo-infinity is now just a regular number, and division by zero is no longer a forbidden operation, 1/0 is now equal exactly to “∞”, which is distinct from ∞ because nothing can actually go to infinity so we don’t even include it in our math.

A Perceptual Model

The peculiar inverse relationship between Euclidean and conformal spaces suggests a perceptual model in which the inverse conformal reflection of a sensory input serves to detect regularities hidden in a two-dimensional stimulus, and use those regularities to project a three-dimensional image of the objects most likely to have been the cause of that stimulus. In other words, the conformal model helps to solve the inverse optics problem that seeks to invert the projection of the eye to reconstruct a three-dimensional world consistent with the two-dimensional retinal image, as suggested with a simple example below.

A line in external space is projected to a two-dimensional retinal projection by the optics of the eye, represented by a patch on the surface of the conformal sphere. The information of three-dimensional depth is lost in the optical projection.

The retinal image is then inverse-projected into the inverse conformal sphere where it spreads throughout a planar probability field that represents all of the possible locations in depth that project to that same linear stimulus. This expresses the fundamental ambiguity in monocular vision whereby the depth value  is lost in the optical projection. The inverse projection represents a simultaneous reification of every possible edge at every possible orientation that all project to that linear stimulus. From that infinite set of alternative interpretations, the visual system selects one that has the greatest simplicity, or symmetry, (Gestalt prägnanz) in the conformal reflection. For example one possible interpretation of the linear stimulus is as a straight line that stretches to infinity in opposite directions, recognized by the fact that its inverse projection in the conformal sphere defines a circle through the origin.

The result of that regularity detection process is then projected back out, not into the external visual world that was the original source of the visual stimulus, but out into a conformal representation of that external space wrapped up in a finite spherical representation whose outer surface represents infinity in all directions.

Here is a diorama that I built to demonstrate the geometry of perceptual experience.

It is noteworthy that the external Euclidean world, the conformally warped “Bubble World”, and the bizarre inverse conformal world, all project radially to the same retinal image! For example a photograph of my Bubble World Diorama taken from its focal center would be identical to a photograph of the original scene represented by the model.

 

The retina is a spherical surface, and thus its fundamental coordinates are visual angles, and thus visual angle is the only component of external reality that is preserved through the retinal projection. It is this most certain and unambiguous information of the retinal image that is used to connect to the inversive conformal world, and its conformal reflection across the unit sphere in the surrounding Bubble World perspective that seeks to model the external world guided by the regularities detected in its inversive reflection.

Non-Euclidean Geometries

 

The peculiar warp observed in the Bubble World perspective and its inversive counterpart in the conformal sphere is reminiscent of non-Euclidean geometry. Three mathematicians, Karl Friedrich Gauss, Nicolai Lobaschefsky, and Janos Bolyai, each independently wondered while studying Euclid’s “Elements” why Euclid had not bothered to prove his “Fifth Postulate” with the same rigor as he had with the rest of his postulates. Essentially the “Fifth Postulate” states that if two lines that cross a third line form internal angles that sum to less than 180 degrees, then those lines must cross somewhere.

This is equivalent to the “Parallel Postulate” that parallel lines never meet, and it is also equivalent to the rule that the internal angles of a triangle sum to 180 degrees. Each of those three mathematicians set out to prove the Fifth Postulate, and all three of them failed, because although the postulate seems self-evident, it is in fact impossible to prove.

That in turn opened the possibility for non-Euclidean geometries, i.e. that it is possible to define a whole non-linear equivalent to Euclidean geometry that works in a space with positive curvature like the Bubble World perspective.

 

And likewise by symmetry, one that works in a space with negative curvature, like the inversive conformal world. For example the Pythagorean theorem can be demonstrated and proven in the distorted non-Euclidean spaces above just as easily as it can in Euclidean form. Euclidean geometry is a subset of a whole family of non-Euclidean geometries through a range of curvatures both positive and negative.

Gauss was genuinely disturbed that we cannot be sure that our world is truly Euclidean, it could just as well be of a non-Euclidean geometry. The perceptual model suggests that Gauss’ fear was well founded, that our “Bubble World” perspective that we see in the world around us does indeed exhibit a non-Euclidean geometry with positive curvature, and the inversive conformal world suggested by Geometric Algebra is a mirror image of that world in a space with negative curvature. The two nonlinear geometries are connected to each other and to external reality by the Euclidean geometry that they share in common.

Properties of the Conformal Model

The concentric inside-out mirroring between a central inversive conformal sphere of negative curvature, and a surrounding conformal sphere with positive curvature, offers an extraordinarily powerful system for representing geometrical form based on a very few foundational primitives.

The outer Bubble World conformal mapping offers a finite bounded space capable of representing an infinite unbounded space that can now represent space all the way to infinity in all directions!

The inner inversive conformal sphere at the center can be viewed as a kind of “projection mechanism” capable of projecting geometrical images into the outer projective space based on symmetries reflected in the inner inversive space.

Vectors in the conformal sphere project to points in the perceptual sphere, the radial length of the vector being inversely proportional to the radial distance of the point it represents. A vector of length zero projects to “infinity” at the periphery of the perceptual space.

Two vectors presented simultaneously or successively represents two points in perceptual space.

The wedge product between two points defines the line that joins them in perceptual space. It also defines a circular arc in conformal space.

 

The Inner Product Null Space (IPNS) of that wedge product represents the whole line of which the line segment is a part, which completes to a full circle in the inverse conformal sphere.

This is not just a single line through the center, it can represent a range of geodesics which appear curved, but represent straight lines in that curved space.

The wedge product between three points defines the circle that joins them! Circles and straight lines are the same shape in this space.

And if one of the three points moves off to “infinity”, the wedge product defines a circle of infinite radius. A circle of infinite radius is a straight line that passes through the non-infinite points.

The wedge product between four points defines the spherical shell that passes through those four points.

And if one of those points moves off to infinity, the wedge product defines a spherical shell of infinite radius, which is a plane that passes through the three non-infinite points. Spherical shells and flat planes are the “same shape” in this space.

Here are some animations using the GAViewer (Geometric Algebra Viewer) demonstrating the principles of conformal geometry showing only the objects in external Euclidean space, although the symmetries observed in this space reflect symmetries in the inverse conformal world. (The GAViewer demonstrates Hestenes’ conformal mapping that projects back out to Euclidean space, not the conformal Bubble World space)

Here are four points, a, b, c, d, disposed about an (arbitrary) origin o. In conformal geometry points are dual spheres of zero radius. The dot product between two points is a scalar that is proportional to the square of the distance between them. The dot product therefore is a distance measure in conformal geometry.

The dual of a point is a sphere, and the dual of a sphere is a point. Points and spheres are the “same shape” in this space.

The wedge product between two points defines the point-pair, or line segment that joins them. Actually a point-pair is a one-dimensional sphere, i.e. two points that are equidistant from some center.

 

The dual of a point-pair is a circle whose radius is proportional to the separation between the two points. The dual of a circle is a point-pair.

The wedge product between two points and “infinity” is the straight line containing the two points. “Infinity” is a single point that can be found in all directions.

The dual of a line is its normal plane, and the dual of a plane is its normal line.

The wedge product between three points defines the circle that joins them. This only really makes sense in the inverse conformal world where circles and straight lines are the “same shape”. It is a symmetry that shines back out to the Euclidean world.

See how the circle morphs as its defining points move. As the third point passes between the other two, the circle momentarily becomes a “circle of infinite radius”, i.e. a straight line.

The wedge product between three points and “infinity” defines the infinite plane that passes through those three points.

The wedge product between four points defines the sphere that passes through those points.

See how the sphere morphs as its defining points move. As the fourth point passes through the plane defined by the other three, the wedge product momentarily becomes an infinite plane before turning completely inside-out through “infinity”.

Closure

There are a number of curious properties of the conformal model that are a result of its circular closure.

 

 

We learned in Clifford Algebra that rotations can be expressed as double reflections. In the conformal model translation is a rotation about infinity.

Indeed all Euclidean transformations can be expressed as reflections in the conformal model! Scaling requires a reflection through a “curved mirror”, which is exactly a conformal reflection. The fact that the elements of Euclidean geometry, points, lines, planes, and spheres, are all essentially the “same shape” in conformal space, and the fact that so many Euclidean transformations can be expressed in terms of reflections is highly suggestive of a computational mechanism in the brain based fundamentally on reflections and projections between spherical elements in a conformal model.

We can now see the full implications of the principle of closure manifest at so many different levels in Geometric Algebra. Closure is not a property that makes for an accurate rendition of external reality. Closure introduces a circular warp in the representation of Euclidean space, bending straight lines into curved geodesics, and flat planes into spherical surfaces. It presents a multiplication that goes round and round instead of in and out from the origin. The purpose of closure in mathematics is the same as for the odometer, or the digital register: It provides a computational mechanism that will never “run off the edge of the page”, or roll off the end of a register, the register simply rolls back around through zero, wrapping infinity up in a finite circular reel.

There never were straight lines in the brain, the number line never extended to infinity. Like the geometry of perceptual space, the number line only approximates the linearity of the true theoretical number line, and it does so adequately only in regions relatively near the origin. It fails catastrophically at the point where it meets infinity because “infinity” falls infinitely short of true infinity at the end of the number line. Closure is what allows the mind to encompass the infinite without having to be itself infinite in extent. Closure is a practical solution to the problem of modeling an essentially infinite world in a finite bounded representation. Like the dome of the night sky, closure in mathematics is evidence of the limits and limitations of our own mind.

 Conclusion

The history of Algebra has been the history of an incremental discovery of an apparently pre-existing structure. At first, the operations of mathematics were seen as convenient tricks or techniques useful for computing quantitative results. Math seemed at first to be a human invention, as individual mathematicians were credited with inventing different tricks or techniques that simplified calculations. But as more of mathematics has been revealed, the more it has become clear that mathematics is a pre-existing structure with an inherent logic of its own, and new additions to math are not arbitrary inventions, they are incremental discoveries of the hidden logic underlying mathematics. Negative numbers and the square root of minus one, it seems, were there all along waiting to be discovered, implicit in the structure of the number line and the algebraic operations on it. But if mathematics is a pre-existing structure, what is that structure, and where is it located? The time has come to finally reject the romantic notion of mathematics as an objective external entity with independent existence in an orthogonal dimension inaccessible to scientific scrutiny. The time has come to recognize mathematics for what it truly is, an artifact of the way our brain makes sense of spatial reality. Mathematics has physical existence and embodiment in the human brain. But where in the fleshy blobby grey matter of the brain would you begin to look for geometric structures like the number line? And the crystalline perfection of points and lines and planes? The brain just seems like the wrong kind of organ to express that kind of crystalline perfection.

Mathematics is not the only artifact of the computational principles of the human mind. Many have observed the intimate relationship between mathematics and music, with its geometric harmonies and periodic rhythms, and its cyclic patterns of melody, reminiscent of the rotation of a Clifford Algebra spinor.

The periodic ticks of a metronome mark out periodic intervals of time as regular as the number line.  Could it be that music is yet another manifestation of the mechanism of mind? That would at least resolve the long-standing mystery of the origins and evolutionary purpose of music. And if music reveals the mechanism of mind, it also reveals mind to be composed fundamentally of oscillations. It is no accident that Clifford Algebra provides such an elegant model of cyclic oscillations, such as the spin of elemental particles. That is because Clifford Algebra is patterned on the oscillations in the human brain.

Yet another manifestation of the principles of perception is seen in ornamental art as seen churches, cathedrals, mosques, and palaces all around the world. Ornament is visual music, with its many symmetries and periodicities and crystalline perfection. If visual ornament is indeed an artifact of the computational principles of the mind, then visual ornament reveals a mechanism based on symmetries and periodicities, which are also properties of standing waves, as seen in Chladni figures obtained by bowing a steel plate with a violin bow.

Harmonic resonance is a self-organizing principle of physical matter expressed in terms of symmetries and periodicities.  The complexity of the pattern that emerge far exceeds the complexity of the mechanism that produces them, which is just a homogeneous steel plate.

Below are a sample of the possible Chladni figures on a square plate and a circular plate. The circular patterns can emerge at any orientation.

Chladni figures demonstrate the basic computational principle of the brain, a principle that requires no complex mechanism besides spontaneous emergence of standing waves, a principle that is consistent with the crystalline perfection of the number line, and the concepts of points and lines and planes. And if that is not yet evidence enough for harmonic resonance in the brain, Heinrich Klüver administered LSD to volunteers who then lay in a dark room and reported their visual hallucinations. The subjects reported patterns described as grating, lattice, fretwork, filigree honeycomb, chessboard, cobweb, funnel, tunnel, cone, and spiral, that spontaneously appeared.

The patterns were exceedingly rich and complex, but each pattern appeared only for a fleeting moment before morphing endlessly into other patterns. Geometrical patterns are the primitives of visual perception.

The significance of all this symmetry to the principles of perception is revealed by psychedelic art, for example this spectacular piece called Inner Sanctum by Luke Brown inspired by the Salvia experience. If the human mind spontaneously hallucinates images so rich with symmetries and periodicities, that also suggests a harmonic resonance mechanism capable of generating hundreds of such hallucinated scenes per second.

Indeed the fact that we can instantly recognize the symmetry of a kaleidescopic pattern, is itself evidence for a symmetry-based recognition system, one that must operate in parallel to solve this intractable visual problem so rapidly.

The hyperbolic geometry of the conformal mapping is in evidence in Escher’s woodcut Circle Limit IV.

If we accept the overwhelming evidence that mathematics, music, and ornamental art are all artifacts of the computational mechanism of the brain, then we can begin to study the brain based on the properties of those artifacts. Clifford Algebra in particular suggests the operational principles of the elements of perception, or how the brain constructs geometric objects out of the basic primitives of scalars, vectors, and multivectors of higher dimension. Clifford Algebra reveals that the computational operations of the brain are explicitly spatial in nature, and that spatial nature is essential to their computational function.

The concept of duality is highly suggestive of the relationship between a standing wave oscillation as on a Chladni plate, and the nodes of that wave which are the regions where there is no oscillation. The pattern of nodes determines the shape of the vibration just as well as the vibrations themselves. For example the duality between a sphere and its central point is suggestive of a spherical standing wave oscillating between the spherical surface and its central point of symmetry.

The conformal mapping of Geometric Algebra offers a clear hint as to how a finite but explicitly spatial model can encode an essentially infinite space in a manner that seems invisible to the percipient. We have the illusion of being in a practically infinite space when outdoors, although our experience is bounded by the dome of the sky. Anything farther than that horizon is usually  insignificant to our experience.

The fact that a conformal projection is itself a form of reflection, and that reflection is a most fundamental operation in Geometric Algebra suggests that the computational mechanism of the brain is built up out of simple basic elements, as are Clifford Algebra and Geometric Algebra.

The principles of phase conjugation and nonlinear optics demonstrate how a wave based system can perform computations. Whereas linear waves pass through each other transparently as if the other wave were not there, nonlinear waves warp the medium sufficient to deflect other waves, and this provides an essential control mechanism whereby one wave can modulate another, in a manner that is suggestive of a bivector formed by the wedge product between two waves, and that bivector in turn can act to deflect a third wave by Clifford multiplication. I have proposed a model of perception based on the principles of phase conjugation (Lehar).

The many phasic reversals of Clifford multiplication, and the fact that angles are represented by their projections onto the basis coordinates is suggestive of a phasic cyclic system like the Clifford Vectors Analog Electronics Analogy. I propose that the next step in investigating the computational principles of the brain would be to devise a phasic cyclic analog oscillatory model of Clifford multiplication as an image projection mechanism, capable of generating synthetic images.

The discovery of Clifford Algebra, and its subsequent elaboration into Geometric Algebra marks a profound turning point in the history of mathematics where we can finally realize what it is that we have been studying all along.  When Hamilton invented his Quaternions, and Descartes plotted numbers graphically, when Gibbs invented vectors and their dot and cross products, when Herrmann Grassman invented the wedge product, unbeknownst even to themselves, these mathematical pioneers were unwittingly studying the principles of operation of the human mind implemented in the brain. Now that we know what it is we are studying in mathematics, we can begin to reverse-engineer the operational principles of our own brain under the guidance of Clifford and Geometric Algebra.

 

 

 

 

 

Geometric Algebra: Projective Geometry

The first chapter in this series, Clifford Algebra: A Visual Introduction, presented a brief history of the incremental discovery of algebra, culminating in the discovery of Clifford Algebra, the algebra that subsumes them all, because Clifford Algebra is not just another algebra, it is the radical discovery that all of algebra is ultimately based in geometry.

In this chapter we move on to a projective geometry obtained by adding one extra dimension, and viewing the Euclidean world from that standoff perspective. If done correctly, this technique provides some extraordinary invariances to our representation, in particular by providing an elegant means to deal with the problem of “infinities”, or how to represent an infinite world within a finite representation.

A final chapter will cover the Conformal Geometry, the most spectacular addition to Clifford Algebra.

The problem of infinities can be seen when considering the number line, or the x axis. You can never depict this number line explicitly on the page the way you can depict a line segment of any length by scaling it to the page. It is the ends of the line, where it touches infinity in opposite directions, that pose the problem, because it is impossible to reduce the scale enough to depict all the way to infinity. Projective geometry offers a means to plot all the way to infinity on a finite page, and it does so using the same principle by which the lens in a camera, or in your eye, converges light to a focus to project a perspective image of the scene on a photosensitive surface.

Projective Geometry 2-D to 3-D

The idea behind projective geometry is to add an extra dimension to reality and to view the world from the perspective of that supernumerary dimension. We begin by demonstrating the principle first in two dimensions. Consider  a two-dimensional Euclidean space, or E2. We add a unit vector in a third dimension, and view the Euclidean plane from the perspective of that point. For example each integer value on the x axis becomes an angle viewed from that viewpoint, as traced in blue lines below. But angles don’t make much of an image, so we project each of those angles as a point on an image plane placed somewhere in the path of the sightline rays.

This produces a perspective projection of the two-dimensional plane that records x values in one direction all the way out to infinity, with the property that the closer you get to “infinity” (the pseudo-infinity point in the projection) the more compressed the scale becomes. If you complete the plot for all points in x and y, that produces a perspective view of the Euclidean plane with a horizon line at “infinity” and vanishing points in the directions of the x and y axes.

The same principle works just as well in three dimensions, producing full 2-D perspective views of a 3-D scene. This principle makes perfect sense in perception, where an essentially infinite world projects an image onto a 2-D sensory surface. This principle is commonly used in computer graphics to generate synthetic scenes by performing a perspective projection that is equivalent to the projection in a camera.

The computational transformation from a three-dimensional scene to its two-dimensional projection can be performed by a simple matrix multiplication of each (x,y,z) point in the model of the scene by the transformational matrix, to produce its two-dimensional projection on the image plane. The matrix encodes the entire projection including any translation, rotations, scaling, and shearing transformations that may be included in the projection. This matrix is often called “homogeneous” coordinates because it treat translation, rotation, scaling, and shearing transformations all the same way, by setting the appropriate coefficients in the matrix and performing the matrix multiplication on each point in the model. For example the coefficients along the main diagonal of the matrix, sx, sy, sz, determine the scaling of the (x,y,z) point in the projection, while the three values in the right column, dx, dy, dz, determine the translational displacement in three dimensions, etc. In  Geometric Algebra the equivalent computation is performed using a three-dimensional spatial structure representing translations, rotations, and scaling in a spatially meaningful order, including rotations, defining a transformational “structure” that resembles the transformations that it models.

We can draw an analogy between the principle of projection and a laser beam mounted on gimbals that behaves like a Clifford vector located at the origin, that can point in any direction. If the laser beam is projected onto a screen, the linear beam is transformed into a point projected on the screen. Unlike the laser beam itself, the point is no longer confined to the origin, but can now be projected anywhere on the screen. If we consider the projection itself, we see that the projection has transformed the basic elements of Clifford Algebra from linear vectors to punctate points. In projective geometry points are the most elemental feature.

 

Here we show (from Perwass & Hildenbrand 2004) Euclidean vector a in two-dimensional Euclidean space E2 transformed into vector A by the projective transformation A = P(a) where  P(a) is obtained by simply adding vectors a + e3. This is not just a vertical translation of a to A, but vector A is viewed from the origin as an angled vector, essentially transforming a displacement from the origin to an angle viewed from the origin.

This transformation is profound because it draws on the equivalence between translational dimensions and corresponding rotational ones. Translation of a in directions e1 and e2 is transformed to rotations of A in the rotational directions parallel to e1 and e2. Furthermore, the “x axis” no longer extends to infinity in opposite directions, the projective vector A is now bounded at +/- 90 degrees which now literally express +/- “infinity” (in “scare quotes” to indicate merely a “pseudo-infinity”).

The projected origin P(0,0) is no longer a special point with the polarity reversals characteristic of Clifford vectors, the projected origin is now just another point in projective space. While the scalar product of a vector with the origin in Euclidean space is always identically zero, this is not necessarily the case in projective space.

Actually the projective transformation is exactly what occurs as an analog computation in the eye, as light rays from a practically infinite world are projected through a focal point onto the spherical surface of the retina, where each point on the retina represents a visual angle to the world. The Euclidean space in the projective model, the one that extends to infinity in all directions, should therefore not be thought of as part of the representation in the brain, that part of the projective model represents the world outside the brain, which actually does extend virtually to infinity in all directions. It is only the projection itself, P(x,y) or P(x,y,z) that represents a model in the brain, and in fact projection demonstrates how a finite representational medium, whether the surface of the retina or a volume in the visual cortex, can express a virtually infinite external world by depicting that world in projection.

Interactions Between Projective Vectors

Projected vectors in Geometric Algebra interact by the same sums and products as in Clifford Algebra although they seem qualitatively different in projection.

If the laser beam “paints out” a pair of points, independently or successively alternating between them, this is equivalent to “adding” two points, presenting them “simultaneously”, although each remains independent of the other.

If we allow the laser beam to modulate between the two beams, it creates a blend of the two points represented by a bivector surface between the two vectors, as by the wedge product between two vectors, and the projection of that bivector on the screen is the line joining the two points. The basic elements of projective geometry are points, and the wedge product between two points is the line segment that joins them. Like the points themselves, the bivector line segment can be presented at any location on the screen.

Here we show(from Perwass & Hildenbrand 2004) a bivector A^B spanning projective vectors A and B, the projection of the bivector representing a line in projective space. The Outer Product Nullspace (OPNS) of the bivector A^B (i.e. NO(A^B) ) is the whole plane within which the bivector is embedded, and the projection of that plane on the projection plane represents the infinite line of which the line segment is a part. The Clifford Algebra concepts of the IPNS and OPNS are just as useful in projective space.

The wedge product between three vectors is a triangular cross-section beam, that projects to the triangular surface that joins the three projected points on the screen. Four or more vectors all projected onto the plane would fill in the polygonal area spanned by the points.

The bivector between two points is anti-commutative, i.e. A^B is different than B^A, so too is the trivector A^B^C different than trivector A^C^B, for example, as shown above. There is a directedness, or sequence between the trivector products that is preserved in the wedge product, where A^B^C takes the bivector A^B and sweeps it towards C, whereas A^C^B starts with bivector A^C and sweeps it toward B.

Projective Geometry 3-D through 4-D and back to 3-D

In the examples above for simplicity in visualization we have been dealing with projective geometry of a 2-D Euclidean space to a 3-D projective space and back to a 2-D projection. In perception the projection is from the 3-D geometry of external space viewed from a fourth-dimensional perspective, projected back into a 3-D projective model of external space. While it is difficult to visualize a fourth spatial dimension and the projection process taking place within it, it is relatively easy to visualize the effect of that projection back in the 3-D projection. We will therefore ignore the details of the projection mechanism itself and study the properties of the projection back in the three dimensional world of our experience.

The outer “room” in this figure represents a real room in external space, a space which is practically infinite in extent. The inner room is a three-dimensional projection of that outer room, “painted out” by a metaphorical laser beam that maps 3-D points from the external world to 3-D points in the projective model.

This model world may look like a three-dimensional rectangular room, but it is supposed to represent a model that is distorted by perspective like an Ames Room, with objects in the distance presented at smaller scale than objects in the foreground. In this model, “infinity” is no longer an infinite distance away, it is right there, where the perspective lines meet. The purpose of this projection is to project from an external world that is practically infinite in extent, into a model world of a finite size, small enough to fit in a human head. As with our earlier laser analogy, we can picture laser beams representing vectors projecting to points in three-dimensional space. Two points presented independently or successively represents the sum of those points in space.

The wedge product between two 3-D points defines the three-dimensional line segment that joins them.

The wedge product between three or four points in the same plane defines a triangular, or rectangular plane in space.

If we add another point outside of the plane, the wedge product between all those points defines the three-dimensional volume spanned by those points.

And if one of those points is at “infinity”, this now defines a rectangular cross-section beam that stretches all the way to “infinity” in our representation, which is only possible because “infinity” is not an infinite distance away, it is right there, where all the perspective lines meet.

The three-dimensional projective geometry through a fourth-dimensional projection provides an extraordinarily useful means to produce an explicit spatial model of an essentially infinite external world, all within a finite spatial representation that can represent points at “infinity”. This is highly suggestive of the properties of our own world of perceptual experience.

The operation of projection is very fundamental to Geometric Algebra. So much so that is is trivially easy to define a pinhole camera in Geometric Algebra. We begin with four points to define the image plane, whose surface is defined by the wedge product between them, i.e. a1^a2^a3^a4.

Next we define a focal point F on one side of the image plane. Now for any point P on the other side of the image plane, the line joining F and P is defined simply as F^P. The intersection between that line and the image plane can be computed simply as

(a1^a2^a3^a4).(F^P)

and thats it! The pinhole camera is complete! If we employ a parallel algorithm that processes every point in the scene simultaneously, we get a fully functional pinhole camera that projects an image on the screen of the world seen through the screen.

It is a fundamental property of parallel optical or wave-based systems that any computational transformation  can be reversed by simply reversing the path of each ray of light in the system, which inverts the functionality, as seen when looking through the wrong end of a telescope which makes it minify instead of magnify the image. The same system that performs a pin-hole camera projection to an image, can also perform an inverse projection of the two-dimensional image back out into the three-dimensional model world, thus performing an “inverse optics” reverse projection. Now the inverse optics problem is under-constrained, because there is an infinite range of possible inverse projections that can all correspond to a given projected image, and thus additional information must be added to resolve fundamental ambiguities. Nevertheless, the ease with which Geometric Algebra performs spatial projections, whether forward or inverse, makes possible the kind of reification by inverse-optics projection as suggested in Lehar’s (2003) Bubble World model of perceptual reification to explain how the two-dimensional retinal image is reverse-projected to form a full three-dimensional perceptual rendition of the three-dimensional stimulus most likely to have been the original cause of the given two-dimensional projection.

Limitation

The perspective projection provides a wonderful means for projecting an infinite external world into a finite projection. But there is a fundamental flaw in this concept, it cannot represent opposite ends of the room simultaneously.

Projective perspectives taken from opposite directions are incommensurable — they cannot be merged seamlessly without a sharp discontinuity where they join.

The problem can be traced back to the concept of projection onto a plane, which can only depict one half of reality in principle.

A solution to this limitation will be presented in the next chapter of my presentation on the principles of Geometric Algebra in the Conformal Projection Model of Geometric Algebra proposed by David Hestenes.

 

Clifford Algebra: A visual introduction

William Kingdon Clifford

Clifford Algebra, a.k.a. Geometric Algebra, is a most extraordinary synergistic confluence of a diverse range of specialized mathematical fields, each with its own methods and formalisms, all of which find a single unified formalism under Clifford Algebra. It is a unifying language for mathematics, and a revealing language for physics.

Unlike the standard vector analysis whose primitives are scalars and vectors for representing points and lines, Clifford Algebra has additional spatial primitives for representing plane and volume segments in two and three dimensions, and it can be extended to any number of higher dimensions by the same basic scheme, and they do, with remarkably useful properties.

Adding one extra dimension to a total of 4 produces a projective geometry, a concept not exclusive to Clifford Algebra, but very simply expressed in it, with some remarkable invariance properties. 

Adding two additional dimensions to a total of 5, enables a conformal geometry with even more extraordinary invariances which are reminiscent of certain properties of perception.

The truth, or validity of Clifford Algebra is confirmed by Occam’s Razor, it provides a simpler model of mathematical objects than does vector algebra, extending naturally from one to two, to three, and higher dimensions all under the same formalism, with a notational economy that simplifies many mathematical expressions.

For example Clifford Algebra reduces Maxwell’s equations of electromagnetism to a SINGLE EQUATION, the rest is implicit in the math!

Another feature of Clifford Algebra is that it uses a coordinate-free representation. Instead of defining motion with respect to an external coordinate system, motion is described with respect to a coordinate frame defined on the object in question, which greatly simplifies many models. Clifford Algebra reveals, for example, that the apparent chirality in electromagnetism, i.e. the right-hand rule for electric generators, and the left-hand rule for electric motors, turns out to be actually an artifact of the math used to describe the world, not a property of the world itself. It turns out that electromagnetism has no chirality, as revealed by Clifford Algebra.

Clifford Algebra fixes many of the problems inherent in linear algebra, from the arbitrary distinction between row-vectors and column-vectors, the complex process of matrix multiplication, the arbitrary arrangement of terms in the matrix, exemplified most clearly by the complexity of the formula for the determinant of a matrix, to determine whether the matrix is invertible. In Clifford Algebra the determinant is the three-dimensional volume of the parallelopiped spanned by the three row vectors of the matrix, and it becomes intuitively obvious that the volume goes to zero when the three vectors are parallel, or when they are all in the same plane. Furthermore, the whole concept of inversion becomes an explicit spatial inversion, a literal turning inside-out of a spatial structure.

Clifford Algebra provides a more intuitive model for subatomic particles that are often modeled with the Pauli matrix, by way of a spatial model that resembles the particle that it models.

And the way that Clifford Algebra achieves this extraordinary Grand Unification of mathematics is by expressing algebraic concepts in the form of spatial operations on spatial structures. Clifford Algebra, a.k.a. Geometric Algebra, is simultaneously a geometrification of algebra, and also an algebrification of geometry. We are accustomed to viewing algebra as a manipulation of abstract symbols back and forth across the equals sign. But do you remember when in geometry, you learned about a transversal crossing a pair of parallel lines, and which angles were similar and which were complementary? Your teacher never had to prove these truths, they merely had to be pointed out, they were self-evident by inspection. Clifford Algebra does that for algebraic manipulations, which become natural and intuitive spatial operations on spatial structures.

Geometry is more primal and explicit than algebra. In Clifford’s own words,

“geometry is the gate of science, and the gate is so low and small that one can only enter it as a child.”

Clifford algebra invites us to stoop low enough to re-examine some of the most basic mathematical concepts that we first learned as children, and to see new spatial and geometrical relations in them.

Why So Obscure?

So if Clifford Algebra is so great, how come you have never heard of it before? In the first place Clifford died at an early age in an era where there were several competing algebras being developed, and Clifford Algebra was eclipsed by Gibb’s Vectors that we all learned in school.

Clifford Algebra was briefly rediscovered by Dirac as a superior expression of his theory of the electron, although the wider significance was not recognized. It was rediscovered again  in the 1980’s by David Hestenes who recognized its general applicability and significance, and David has made a careeor of promoting “Geometric Algebra”, a more modern update to the original Clifford Algebra. David Hestenes prefers the term “Geometric Algebra”, which was actually Clifford’s own choice, because “Clifford Algebra” sound like “just another algebra” rather than what it really is, which is a discovery of the fundamental geometrical roots of all of algebra. Besides, Hestenes credits Hermann Grassmann at least equal to Clifford in the development of Geometric Algebra. I use the term “Clifford Algebra” for historical context and honoring a quirky dude, and “Geometric Algebra” when focusing on the modern reformulations and extensions many of which are due to David Hestenes.

My own interest in Clifford Algebra stems from my own conviction that mathematics is not a human invention, but more of a discovery of the foundational principles of thought. Mathematics is an artifact of the human mind and how it represents spatial reality. The properties of mathematics are direct evidence for the principles of operation of the human mind.

And since Clifford Algebra provides such a clear view into the fundamental nature of math, I believe Clifford Algebra to be the Rosetta Stone that reveals the basic computational and representational principles of spatial perception and algebraic cognition.

In fact the many regularities found in mathematics, from the periodicity of the number line, the alternation of even and odd, its mirror symmetry across the origin, and the polarity reversals of negation and inversion, all suggest a phasic / cyclic system of representation not unlike the trace on an oscilloscope in an analog oscillatory circuit, which in turn suggest a wave based representation in the brain underlying both spatial perception and mathematical thought.

 Historical Summary

The history of algebra has been a history of the incremental discovery of an apparently pre-existing structure. The natural numbers were discovered independently by all cultures, suggesting that the concept of number is  intuitive, or innate. The concepts of zero and the negative numbers seem to follow naturally, as if of necessity, and they too were “rediscovered” independently by different cultures over time.

The closure of circular negation was the first hint of an irregularity.

A problem arose when the solution to certain algebraic equations came out to be equal to the square root of minus one, something that was a mathematical impossibility! Rafael Bombelli, and René Descartes (who coined the term “imaginary” numbers) proposed to name this impossible number i, but that besides the fact that it squares to negative one, in every other way it was a regular number. So for example 1 times i = i. i times i (by definition) = -1. -1 times i must equal -i. And -i times i = -(-1) which brings us back to 1.

Repeated multiplication by i goes round and round in circles, making periodic excursions into an orthogonal dimension and back again. If you plot the imaginary dimension as a regular spatial dimension, this suggests that multiplication by i corresponds to a 90 degree rotation. This is an example of mathematical closure, a concept we will see again and again in Clifford Algebra.

Later it was proposed that you could rotate by less than 90 degrees, which produces a compound number composed of real and imaginary components.

But the real and imaginary dimensions operate by different rules. Multiplication of real numbers scales their magnitudes in or out from the origin. But multiplication of the imaginary component performs a rotation, it is a multiplication that goes round and round instead of in and out.

I don’t know if anyone ever pointed it out to you, but this is VERY STRANGE!

Josiah Willard Gibbs

About the same time in the late nineteenth century,  Josiah Willard Gibbs was playing around with a different kind of compound number, pairs or triplets of (x,y) or (x,y,z) which can be represented as “directed line segments”, or  vectors. The remarkable thing was that you could add or subtract these compound numbers in a manner that made a kind of intuitive spatial sense. The question arose: What happens if you multiply and divide them? Gibbs proposed two kinds of multiplication, the dot product and the cross product, as we all learned in school.

 

Oliver Heaviside used Gibbs vectors to reformulate Maxwell’s equations of electromagnetism, which did much to cement Gibbs’ vectors position as the dominant paradigm of vector manipulation. Nevertheless, Tait commented that Gibbs’ vectors were “a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassman”

The dot product was a good idea, and has been incorporated into Clifford Algebra. But it turns out in retrospect that the cross product was a big mistake! In the first place it is only really defined for three dimensions, it does not generalize to two-dimensional vectors without popping out of the page into the third dimension, and in four dimensions the concept is practically meaningless because there are so many orthogonal directions. Furthermore, Gibbs’ vectors can only express scalars and vectors, the only way it could represent a plane is by its surface normal, which is a vector.

Also the cross product can only rotate by 90 degrees, it cannot rotate through intermediate angles as do complex numbers.

Finally Gibbs’ algebra has a parity problem, the cross product is not preserved under reflection, and thus it introduces a chirality to its model of reality even where there is no chirality in the entity being modeled.

 William Rowan Hamilton

In the meantime over in another branch of mathematics, William Hamilton was trying to extend complex numbers, with their elegant rotational multiplication, into three dimensions by proposing three orthogonal imaginary dimensions, i, j,k,  each of which behaves like the imaginary dimension i, and they interact with the peculiar circular rule that i x j = k, j x k = i, and k x i = j. Here again we see another example of mathematical closure, a curious property which turns out to be very significant.

But Hamilton ran into a problem defining division for these imaginary vectors because the quotient was not unique. Imaginary vectors A and B, each composed of (i,j,k) components, multiply meaningfully as in A x B = C, but the inverse, C / B does not produce a single unique value but a whole range of possible values. This algebra lacked closure, it only computes one way, you cannot calculate backwards, as in moving terms back and forth across the equals sign.

After many years of frustration, in a flash of intuition (immortalized by a plaque on the Broom Bridge in Dublin where the intuition first struck him) Hamilton realized that he needed one more scalar value to provide a scale for his complex vectors, so he defined a compound number of the form v = a + bi + cj + dk, which could be both multiplied and divided, and the quaternion was born!

All this occurred in the era when the theory of relativity was first emerging, and Hamilton incorporated a relativistic concept in his quaternions by expressing rotations not with reference to a global x,y,z coordinate frame, but relative to the current orientation. For example if the orientation of an aircraft in flight is expressed in “Euler angles” as a triplet of roll, pitch, and yaw angles in absolute coordinates, there is a problem known as “gimbal lock” when the aircraft is heading either straight up, or straight down, where it has no defined yaw angle, or heading. Instead, quaternions employ a coordinate system that rotates with the aircraft, as perceived by the pilot in the aircraft, where concepts of “pull up”, “roll left”, and “yaw right” are meaningful whatever the current orientation of the aircraft.

Hamilton introduced another innovative departure from conventional mathematics by proposing a multiplication which was not commutative, i.e. that a x b is not equal to b x a, but neither are they unrelated, but in fact a x b = -(b x a), where negation corresponds to a reversal in the direction of rotation, just as negation of a real vector reverses its orientation. These intriguing concepts were incorporated into Clifford Algebra.

Closure

Now a word on the concept of closure, which will turn out to be significant throughout this presentation. In algebra closure refers to operations on an algebra that remain within the algebra. For example the natural numbers are closed under addition because the sum of any two natural numbers is also a natural number, but they are not closed under subtraction, because if you subtract enough you can fall right off the end of the natural number scale.

Likewise, the integers are closed under addition, subtraction, and multiplication, but they are not closed under division because certain quotients fall between the cracks of the integers. But that is not the only kind of closure in math.

0 1 2 3 4 5 6 7 8 9

There is closure in our numeral system, where when we run out of digits, we just put a zero and carry to the next column.

There is closure in an odometer where each wheel flips from 9 to 0, and the whole register flips from 99999 +1 to 00000.

There is closure in digital registers that also flip from FFFF +1 to 0000.

It’s a computational principle that prevents the calculation from “running off the edge of the page”, or the register, allowing infinite calculation within a finite representation.

And there is closure in a multiplication that goes round and round instead of in and out. It defines a continuous space like a section of the x axis, but it is a closed dimension, that is finite, but boundless. This will turn out to be very significant.

 

Hermann Günther Grassmann

Meantime Hermann  Grassmann, the inventor of linear algebra, proposed an alternative to Gibbs’ flawed cross product, in the form of the exterior product, or wedge product, whereby the wedge product of two vectors defines a patch of surface whose area is equal to the product of the two vectors, as if one vector was “swept” along the other (or the other “swept” along the first), and that surface is within the plane that contains both of the original vectors.  Just as a vector has magnitude and orientation, but it has no defined location because it is always at the “origin”, in the same way, the bivector, or wedge product of two vectors has an orientation and an area, but it has no location, and neither does it have a shape. It is an oriented area.

These bivectors can be added and subtracted just like regular vectors, and if a bivector is multiplied by a vector, its surface is “swept” along that vector in the same manner to define the volume spanned by the three component vectors, an entity known as a trivector. In other words, this is an algebra that generalizes to higher dimensions by the same exact principles that apply at the lower dimensions, and this provides an algebraic entity for scalars, vectors, bivectors, trivectors, and there is no limit to the number of dimensions it can be extended to.

Like Hamilton’s quaternions, Grassman’s wedge product is anti-commutative, whereby a^b = -(b^a). And it also incorporated a kind of closure whereby (in three dimensions) a^b^c makes a trivector, but a^b^c^d collapses back down to a scalar so as to prevent runaway expansion into ever higher dimensions.  

William Kingdon Clifford

William Kingdon Clifford was the man who took all the best ideas from previous work to achieve a single consistent formalism that generalizes to arbitrary numbers of dimensions. Clifford defined a hierarchy of compound number knows as “clifs”, or “multivectors”, that range from zero dimensions (scalar) to one dimension (vector) to two dimensions (bivector), to  three (trivector), and on upward as far as you care to go. In Clifford Algebra dimensions are referred to as “grades”, so a scalar is grade zero, a vector is grade 1, etc.

In order to attain the kind of closure seen in quaternions, you have to decide from the outset how many dimensions you want to use for your problem, so for example two-dimensional Clifford algebra, or Cl2 wraps around at two dimensions so as not to protrude out of the plane into the third, whereas Cl3 has closure in three dimensions, so as not to protrude into the fourth.

Just as a real number can be considered as a complex number with zero imaginary component, a clif can be considered to be composed of higher order clifs up to the grade of the algebra you choose, even if many of them happen to have zero coefficient for any particular problem. Clifford Algebra of zero dimensions, Cl0, is just regular scalar algebra. Clifford Algebra of one dimension, Cl1, is composed of a scalar and a vector. Cl2 is composed of a scalar, two vectors, and a bivector. Cl3 is composed of a scalar, three vectors, three bivectors, and a trivector, and that same scheme extends upward to higher dimensions by the same principle.

Here we see the hierarchy of Clifford Algebras each with its own characteristic pattern of components, that appear in numbers corresponding to Pascal’s triangle. Two-dimensional clifs, Cl2 (the even sub-algebra) are mathematically isomorphic to complex numbers. Three-dimensional clifs, Cl3 are isomorphic to Pauli matrices used to model subatomic particles. Cl4 is isomorphic to Dirac matrices that model relativistic subatomic particles.

Lets take a closer look at Cl3, Clifford Algebra in three dimensions. There is a scalar, e, then three orthogonal basis vectors e1, e2, e3, then three bivectors obtained by pairwise multiplication of the basis vectors, and finally a trivector defined by the wedge product of all three basis vectors. Each bivector has a kind of twist that represents a rotation by 90 degrees, for example bivector e1^e2 represents the right-angled turn from the direction e1 to e2, while its negation, -(e1^e2) rotates by the same angle but in the opposite direction.

Multiplication of all three basis vectors (e1^e2^e3) produces a three-dimensional volume segment or trivector, with unit cubed volume. This trivector has some peculiar properties that represent the scale of the whole space, it is often called the pseudoscalar of the space, I, it performs the function of the imaginary number i, but extended into three dimensions. Multiplication by the trivector represents a 90 degree turn in all three dimensions. Therefore two such 90 degree turns in each of the three dimensions results in a total reversal of direction, which is equivalent to a simple negation, and thus I**2 = -1. Clifford Algebra eliminates the mystery of the concept of the imaginary number, which becomes a simple rotation by 90 degrees.

For all the variety of different clifs of their various grades, there  are basically  two  ways that clifs can interact, one is addition/subtraction (add-traction?) and the other is multiplication/division (multipli-vision?). Addition represents a kind of summation of independent or successive processes, whereas multiplication represents a simultaneous modulation of one clif by another; if either clif is negative the product is negative, but if both are negative the product is positive again, another manifestation of closure. In fact analog modulation as in AM radio, or linear gain in amplification, is defined exactly as analog multiplication.

These two very basic and primal operations can be applied between clifs of any grade, and they can be applied across different grades, and in every case they produce results that seem intuitively consistent and reasonable as one might expect. Addition of scalars and vectors produces a compound scalar and vector product. Addition of vectors proceeds logically as in Gibbs’ vectors. Bivectors add to other bivectors to produce a sum bivector.

A scalar times a vector scales the vector in length, and reverses its direction if negative. A vector times a vector makes a bivector with a twist that turns from the direction of one to that of the other. A scalar times a bivector scales its area.  A vector times a bivector, or the product of three vectors produces a trivector. A vector times a trivector does not  expand to a quadrivector, but rather it collapses back down to a scalar, thus maintaining closure in three dimensions.

There is an interesting interaction that occurs across grades seen for example when a scalar, a non-spatial entity, multiplies a vector which has linear extent, it changes the spatial length by the magnitude determined by the scalar value. Likewise, taking the magnitude of a vector v as in ||v|| summarizes the spatial extent of that spatial vector with a non-spatial scalar magnitude. A bivector B can also be reduced to a scalar magnitude by taking its magnitude ||B||, which provides an abstract measure of the magnitude of the bivector which is proportional to its area. These are manifestations of inter-grade communication, whereby the scalar value can be seen as the most abstract representation, reducing clifs to a single scalar value that records their magnitude, and thus scalar algebra can be seen as an abstraction of the principles of vector algebra, preserving the corresponding magnitudes but discarding spatial extent.

Lets take a closer look at the product of two vectors. If the two vectors are parallel, the product collapses to a scalar, yet another manifestation of closure. A vector times itself collapses to a scalar value one, meaning the vector is a perfect match to itself. If the vectors are pointing in different directions, the product includes a bivector with a twist proportional to the area spanned by the two vectors.

We can see the effects of the twist by multiplying a different vector by the bivector, which rotates the vector through the same angle as that between the original vectors. Vector a times bivector c (= a x b) rotates a through the angle from a to b, which leaves it equal to b. Vector b times bivector c produces d, rotated from b through the same angle as that between a and b. As with quaternions,

Since Clifford multiplication is anti-commutative, it is important to distinguish a*b from b*a, depending on which is the pre-multiplier and which the post-multiplier. For example a*b is a rotation from a to b, when applied to a, as in a*(a*b) it rotates a into b, or b = a*(a*b). But the term can be grouped differently by moving the parentheses, i.e. (a*a)*b. Since a*a = 1, (a*a)*b = b, which produces the same result, so everything works out right in the end.

The Geometric Product

 

As with Gibbs’ vectors with its distinction between the dot and cross products, Clifford multiplication, known as the geometric product, is actually a compound multiplication composed of the sum of a dot product and a wedge product. Like Gibbs’ vectors, this product captures two aspects of multiplication. The dot product is a measure of similarity between two vectors, and is a scalar quantity, whereas the wedge product is a measure of difference, or what you’d have to do to one to transform it into the other. These are complementary operations, each one capturing what the other misses. Significantly, both are spatial projective operations, one is grade-reducing projection, the other is grade-expanding sweep. The inner product projects one vector onto the other like casting a shadow, although the shadow is then projected again to a scalar magnitude. The outer product spans the space between the vectors, a spatial sweep-like operation that is grade-increasing.

These two operations are not just complementary by design, but by definition. The dot product is computed as the (average of the) product, a*b, added to its reverse b*a. Adding a product to its reverse cancels out any component that is asymmetrical, i.e. the clockwise twist cancels its counter-clockwise reverse, leaving only the symmetrical component, which is the dot product, or what the two vectors share in common. It is a little peculiar to call this operation a “multiplication”, because multiplication is generally thought of as producing a multiplicity of its multiplicands; it is generally an expansive operation (unless the multiplicands are less than one), whereas the dot product is more like a collapse to lower grade, more akin to what we think of as division.

The wedge product is defined as the (average of the) difference between the product and its reverse, an operation that eliminates the symmetrical component, but the reversal of the reverse by negation makes them both rotate in the same direction, whose average preserves only the asymmetrical component, that represents the difference between the vectors, or how one is twisted from the other. The dot and wedge products are therefore like symmetry filters that filter the symmetrical and asymmetrical components of their multiplicands, respectively.

 A (scalar ^ a vector) makes a vector. A (vector ^ a vector) makes a bivector. A (bivector ^ a vector) makes a trivector. The wedge product is a grade-increasing operation. A trivector dot a vector collapses the dotted dimension. Trivector (P^Q^R) . R collapses the R dimension to make bivector P^Q. Bivector (P^Q) . Q collapses to P. Vector P dot P collapses to 1. The dot product therefore seems more akin to division than to multiplication.

It is interesting to note that a scalar * scalar * scalar * scalar is still a scalar! You need to have at least one spatial dimension to start expanding in grade. The basis vectors e1 e2 e3 can be thought of as pre-existing “generators” capable of generating a vector as soon as they are supplied with a non-zero coefficient.

Here is an animation of the product of the two red vectors as one rotates around the other, showing the complementarity of the dot and wedge products. (The dot product is depicted here as a vertical vector, to show its magnitude, although it is actually supposed to be a scalar quantity) Note the several polarity reversals of both the dot and wedge products as as the vectors go around, along with a reversal in the rotational orientation, or “twist” of the bivector.

Clifford Vectors Analog Electronics Analogy

This series of phase reversals, and the concept of a generator, is highly suggestive of a cyclic phasic representation not unlike the trace on an oscilloscope, as demonstrated in the above op-amp circuit. A sawtooth waveform generator at the upper-left generates the sawtooth wave plotted on the lower left, which is split into x and y streams each of which can be modulated by the potentiometers toward the right to produce X and Y traces whose amplitude can be modulated (and reversed) with the X and Y sliders to the right, plotted at the lower center of the screen. The X and Y signals are then plotted against each other at the lower right, showing a “flying spot” starting at the origin and flying cyclically toward the upper-right, tracing out the vector represented by the oscillations in the system. If you reverse the X slider the vector rotates into the second quadrant. If you then reverse the Y slider the vector rotates into the third quadrant, if you reverse the X slider back to positive the vector rotates into the fourth quadrant, and if you bring Y back to positive the vector returns to the first quadrant. The sawtooth waves that control the “flying spot” that traces out the vector correspond to the basis vectors e1 and e2, which are constantly cycling, but only produce a trace when their coefficient is non-zero.

The basic principles behind Clifford Algebra appear rather simple, which makes it all the more wonderous what complex computations they can achieve.

Clifford Algebra has been characterized as a “reflection algebra” because its most primal operation, vector multiplication, seems to mimic a simple reflection. If red vector c = b*a*b is the product of b times (a*b), it is a reflection of a through b, because the angle from c to b is the same as the angle from b to a. (We are dealing here with unit vectors so as to avoid any additional scaling, resulting in pure rotation)

This Clifford Vector Multiplication Animation shows how the reflection of a in b occurs like a reflection in a plane surface normal to vector b.

The rich and powerful operation of that rotational Clifford multiplication can be reduced to a simple reflection!

This rotational multiplication by reflection is reminiscent of a phenomenon in nonlinear optics when two laser beams cross in space, as shown for beams B1 and B2 above. An interference pattern appears through the zone of their intersection due to constructive and destructive interference. If modeled in a Reciprocal Lattice representation where each beam is represented by a vector, as shown by K1 and K2 in C above, the emerging interference pattern can itself be described as the difference vector K2 – K1 that closes the wave vector diagram.

If the crossing occurs in empty space it has no effect on anything else. However if the crossing takes place in the transparent volume of a nonlinear optical materal (and most any glass goes nonlinear with sufficient amplitude) the interference pattern warps the glass in the shape of the pattern, by the optical Kerr effect, which defines parallel planes of alternating refractive index in the glass.

This reified interference pattern in turn is able to deflect, or reflect a third beam, B3 above, producing a fourth reflected beam B4 whose direction and magnitude can also be computed as the vector that closes the wave vector diagram in the reciprocal lattice representation in B above, i.e. such that K1+K2+K3+K4 = 0. This is the principle behind phase conjugation, a mechanism by which the angle between two vectors controls the reflection of a third, reminiscent of the bivector and its effect on vectors that are multiplied with it.

Maxwell’s Equations in Clifford Algebra

The power of the geometric product is demonstrated most dramatically in the example of Maxwell’s four Equations of Electromagnetism. These describe the curl of electric and magnetic fields using cross-product terms, and the divergence of the electric and magnetic fields using dot product terms.

The magnetic field must be described by a “pseudo-vector” field because unlike a plain vector, as with the electric field, the magnetic field cannot survive an “improper rotation” i.e. a reflection without requiring an additional change of sign. (The mirror image of your right hand demonstrating the right-hand rule is your left hand, but that demonstrates a left-hand rule!)

In Clifford Algebra these four equations reduce to a single equation, the rest is implicit in the math! The magnetic field is described by a bivector, which has the sign-flipping automatically built-in. The geometric product product automatically multiplies the wedge product terms for the curl, and the dot product terms for divergence, all in a single equation.

Algebraic / Spatial Concepts

There are a number of algebraic concepts, some of which are familiar in scalar algebra, that are revealed by Clifford Algebra to be actually algebraic / spatial concepts. For example the reflection of a vector about the origin is equivalent to negation. Reflection is negation! Clifford Algebra is a reflection algebra. Rotation by 180 degrees is also a negation! Rotation by 90 degrees is equivalent to multiplication by i, the square root of minus one, because two of those multiplications reverse the direction which is also negation. This removes the mystery behind that peculiar notion of the square root of minus one, and reveals it to be a simple rotation. Furthermore, negation is also rotational reversal, thus drawing an equivalence between the open translational dimensions and the closed circular dimensions. This is very strange! Its profound significance will be discussed below.

This is simultaneously an algebrification of geometry, and the geometrification of algebra, it suggest a fundamentally spatial computational mechanism behind the principles of algebra. The similarity between numbers arrayed along the number line, and the lengths of corresponding vectors from the origin, is no mere coincidence, it reveals spatial vectors as the true basis underlying even scalar algebra.

It is a peculiar characteristic of Clifford Algebra that angles are not represented directly as angles, expressed in degrees or radians, but rather, angles are expressed by their X and Y components, a rectilinear coordinate system for expressing angles. This is significant because it makes a connection between the egocentric “dartboard” polar coordinate system where the origin is a special unique place, and the allocentric “city-block” Cartesian coordinate system in which every point is identical, and the origin is no longer special. Clifford algebra expresses one in terms of the other, thus mapping them to each other.

Another curious algebraic / spatial concept is the Dual, indicated by a superscript* asterisk. Every Clifford Algebra entity has its “orthogonal complement”, like the particle and its antiparticle, or the yin and its yang. For example the dual of a vector is a bivector, the wedge product of a pair of vectors that are orthogonal to it, and the dual of the bivector is its normal vector. The area of the bivector that is the dual of a vector is proportional to the magnitude of the vector.

The dual of a scalar, which has no spatial extent, is the pseudoscalar, which has all the spatial extent, and the dual of the pseudoscalar is a scalar.

The dual of a clif is easy to calculate, you just divide by the pseudoscalar, and this is a consequence of closure, because a vector times itself collapses to a scalar, so multiplying by the pseudoscalar collapses all vectors that are non-zero, leaving only those that were zero to begin with. It flips the whole pattern into its opposite, and the opposite of the opposite is back again to the apposite, the thing you began with in the first place.

Another algebraic / spatial concept is the reverse. For example the reverse of a*b is b*a. The reverse of a*b*c*d is d*c*b*a. Its like multiplying yourself back out again in the reverse sequence than the way you got in. We have already seen the utility of the reverse in defining the dot product and the wedge product by symmetry and anti-symmetry respectively.

Another interesting concept is the Nullspace, defined for a Clif as the set of all vectors that evaluate to zero for some function. For example the Inner Product Nullspace (IPNS) of a vector v is the set of all vectors whose dot product with v = 0, which consists of the plane that is orthogonal to v. This is in contrast to the dual of that same vector, which is a bivector normal to the vector, i.e. within the plane of the IPNS, but with an area proportional to the length of the vector, whereas the IPNS of a vector is a plane that extends to infinity. This is a generalization of the outer product to infinity. (Note that the NULL in the nullspace is a negation, thus the Inner Product NullSpace resembles an OUTER product, while the Outer Product NullSpace resembles an INNER product)

The Outer Product Nullspace (OPNS) of a vector is the set of vectors whose outer product is zero, i.e. the infinite line within which the finite vector is embedded. Again, this contrasts with the dual of a bivector which is a vector whose length is proportional to the area, whereas the OPNS of a bivector is a line that stretches to infinity in opposite directions.

These are all examples of the principle of reification, extrapolating a given local pattern outward to infinity by the same rules. A vector expands to an infinite line, a bivector surface segment expands to an infinite plane. It is easy enough to comprehend and to describe this kind of reification, but very hard to implement in any kind of finite mechanism, except by a parallel analog spatial principle like an optical mirror system.

Projections are so fundamental to Clifford Algebra that they can be expressed very simply using dot and wedge products.

For example a vector a can be decomposed into parallel and perpendicular components to a plane defined by bivector B, where the parallel component, “a par B” = a.B/B, whereas the perpendicular component “a perp B” = a^B/B. David Hestenes calls this powerful pair of complementary concepts “projection” and “rejection”.

We have seen how reflections can be performed by multiplication. Reflections can also be computed from simple projections. For example the reflection of vector a in bivector U above can be calculated as the component parallel to the plane minus the component perpendicular to it, i.e. “par – perp”.

Rotation by reflection produces a mirror-image reversal in the reflection, as does a normal mirror. This is not readily apparent in the case of vectors due to their one-dimensional symmetry, but is of significance for higher order clifs.

A more general concept of rotation is derived by two reflections through two vectors, which flips the reflection back again, resulting in a rotation without flipping. For example vector “bab” which is (b*a*b) is a reflection of vector a through b, and similarly, the reflection of bab on vector c is computed as c*bab*c, a double reflection “sandwich”

This concept extends naturally into 3-D. For example vector a is reflected through n by the rotation n*a*n, (start with n, then rotate through the angle from a to n) and then that vector “nan” is reflected through another vector m, resulting in a final product of m*nan*m, or m*n*a*n*m.

Another algebraic / spatial concept  is the concept of the “meet” and the “join” between two Clifs corresponds to the Intersection and Union of the component entities. These too have simple formulas in Clifford Algebra.

The Inverse Function

Now we come to the concept of inversion, 1/x, a concept you learned so long ago that you probably can’t even remember learning it, but in all the time since then, nobody ever pointed out to you that it is a conceptual impossibility! Supposedly for every value x that can range from one to INFINITY, there is a reciprocal value 1/x that is confined to the interval between zero and one. That suggests a one-to-one mapping from an infinite range to a finite range between zero and unity!

This is demonstrated here by plotting in red the reciprocal function between zero and one, i.e. the number whose inverse maps to it. For example ½ maps to 2, 1/3 maps to 3, ¼ maps to 4, and so forth, showing how the inverse function (the red plot line between zero and one) is like a conformal reflection of the number line (the black line plotted from one towards infinity) reflected through the unit boundary although squashed by a conformal mapping.

 

Its as if you took the number line from one to infinity, made a mirror-image reflection of it, take the reciprocal of that, Then we effectively put a bracket BEYOND INFINITY (the arrow coming in from the left) and squash the reflection back down to a finite range so that the entire reflection is squashed into the unit interval. This is a completely impossible operation because it pretends that it is possible to place a bracket beyond infinity and to squash infinity back to “finity”, a finite range. And on our familiar number line the same operation is done in the negative direction too. This creates a double mirror image of the whole number line from negative to positive infinity excluding the unit radius, all packed within the finite range of the unit radius from -1 to +1. This is an extraordinary paradoxical impossibility, but it is the impossibility that brings closure to multiplication, that offers a reciprocal to multiplication as if every positive and negative number had a reciprocal value that falls between positive and negative unity, and the fact that every number has its reciprocal is what makes it possible to multiply and divide both sides across the equals sign, to calculate forward to compute implications, as well as backward to impute original causes.

This incredible “impossible” reciprocal function was already present in scalar algebra that we learned in grade school. What Clifford Algebra adds to this concept is a circular symmetry, like a set of number lines rotated through all orientations, tracing a unit-radius circle at the center, within which is a conformal reflection of the infinite surrounding space. For every (x,y) point in two-dimensional space that falls on some radial ray from the origin, there is a reciprocal point that falls on the same ray in the same direction from the origin, but with radial distance from the origin that is the reciprocal of the distance to the point, just as occurs on the regular number line itself. If the original point is greater than unity, then its reciprocal is always less than unity, and vice-versa, because by symmetry, the reciprocal of a reciprocal gets back to itself. Clifford Algebra generalizes the algebra on the familiar number line to an algebra that occurs symmetrically at all orientations equally, and thus merges algebra with its parent geometry to define an algebra of space. And this same concept extends naturally into three dimensions and higher, where the central unit-radius sphere, or hyper-sphere, contains within it a reciprocal inverse conformal reflection of the entire space outside the sphere squashed into a finite spherical volume.

Play around with MathForum.org‘s Inversion of A Point Demo and observe that when point x and its inverse point 1/x are close to the circumference of the unit circle, they are mirror-reflections of each other across the unit circle, showing how multiplication is reflection near the unit boundary. When you slide point x  off outward towards infinity, its reciprocal point 1/x begins to approach zero, but never gets there unless you could slide x all the way out to infinity! Now if you slide x back from infinity and make it cross the unit circle, it changes place with its reciprocal doppelgänger like Alice changing places with her own reflection in a mirror, and now as  x approaches the central origin, its reciprocal shoots off towards infinity. The difference this time is that you can move point x exactly to zero, which means that its reciprocal must have arrived at actual infinity, although no computer display could possibly show it there, we just stop accounting for it when it runs off the edge of our screen. 

Alexandre Duret -Stereographic projection

Alexandre Duret’s photo art Stereographic Projection gives an impression of how this idea would look extend into three dimensions, with a spherical boundary at a unit distance from some origin, containing an inverse conformal reflection of the surrounding world.  See the tall tower at 2 o’clock, blocking the sun, and its inverse reflection within the unit radius in the 2 o’clock direction, with a peculiar inversion: the tower itself extends outward from the center above its base (“up” is “outward”). In the reflection the tower extends inward from the base toward the center, (i.e. “up” is now “inward”) because in the inverse world, the direction towards the circumference of the circle is proximal, whereas the direction towards the center of the circle is distal, with the bizarre peculiarity that the singular point at the very center of the picture represents infinity in all directions! This is a very strange singular structure to be found at the very core of our number system! It is not there by accident. It provides a lever balance across the unit boundary between a value x and its reciprocal 1/x.

The inverse function embodies an assumption that all of infinity in every direction can be packaged as its reciprocal within a finite unitary range, the finite range from zero to one reflects the entire world from one to infinity.  And even more outrageous to common sense, is the fact that although every direction in space points outward towards its own unique infinity in that direction, in the reciprocal reflection they all map to a single central point, which represents not only infinity, but all of the infinities in all directions simultaneously!

In my next post I will show how the impossibility of the inverse function suggests that the number line should not be conceived as infinite, but merely a pseudo-infinity that represents the entire range from one to infinity, all squashed into a finite range, a mirror-image of its conformal reflection, as suggested in non-euclidean geometries.

And of course the same thing would be required in the negative direction. That would render the point-for-point mapping between x and its inverse 1/x  no longer an impossible paradox, but now perfectly possible, because it is now a point-for-point mapping from a finite range to another finite range. The existence of the paradox of the inverse function is evidence that our true representation of mathematics in our mind does not use a representation that extends to infinity (how could it?) but it simulates infinity within a finite representation that does not incorporate a paradoxical impossibility. 

In fact this much should have been obvious by inspection of our world of perceptual experience, which itself seems to be trapped in some kind of egocentric warp. The sides of a long straight road converge to a point on the horizon before us, and if we turn around, they converge to a point back there too! And yet they appear to be straight and parallel and equidistant throughout their length. This reveals a conformal warp in our very perception of space, such that lines that curve along with the warp are by definition “straight” in that space, and they terminate at (pseudo-) “infinity” at either end. The dome of the sky marks the outer boundary of our perceptual bubble, the maximal extent of our spatial experience, no matter how large a space we are exposed to.

The reason for the warp in our perceptual bubble is clearly to allow a finite spatial representation to depict a practically infinite surrounding space. This is yet another manifestation of closure, with remarkable similarities to the true nature of the number line as revealed by Clifford Algebra.

In my next posting I will show how the similarity with perception does not end there, and that the conformal mapping in Geometric Algebra suggests a similar inversion as a part of perception, except extended out into three dimensions, and the purpose of that inverse representation is the same purpose it serves in mathematics, it reveals certain regularities and symmetries that are not readily apparent in the non-inverse world.

Conclusion

The history of algebra has been the history of an incremental discovery of an evidently pre-existing structure.  Clifford Algebra has turned out to be the lynchpin that shows how all the previous discoveries are  interrelated within a single self-consistent scheme whose laws operate the same in all directions.  It is the very simplicity and generality of Clifford Algebra that confirms its “truth”. But truth to what? What is it that mathematics represents? What must it remain true to? This gets to the question of, as Lakoff and Núñez (2000) called it in the title of their book, “Where Mathematics Comes From“. If algebra is the discovery of a pre-existing structure, then what is that structure and where is it located? Plato and his many modern adherents believe mathematics to be an objective eternal Truth that exists independently of human minds, although it is only accessible through the human mind. It is true that mathematics is not a human invention, but more of a discovery, and thus has objective existence in that sense. But to declare the magnificent edifice of Mathematics to inhabit an orthogonal dimension that is inaccessible to scientific scrutiny, is a hypothesis that cannot in principle be falsified, and thus, it is not a scientific hypothesis but more of a belief, for those who are inclined to believe it.  I demand a more scientifically sound falsifiable hypothesis for the ultimate nature of mathematics that gives it objective existence in the universe known to science. I agree with Lakoff and Núñez that mathematics is an artifact of how our mind makes sense of reality, and that therefore mathematics has physical existence or instantiation within the human brain. Our view of the world is not direct, we see the world indirectly, through a representation, or model of the world constructed in our brain, and that model is painted out in the geometric primitives of points and lines and planes, which are the elemental features of perception, and those geometric primitives correspond closely to the algebra which was ultimately derived from them. Since Clifford Algebra offers such a clear view of the essential principles of mathematics, it thereby also offers a clear view of the principles of visual perception and the way we conceptualize shapes in space, revealing algebra to be at the root of it a spatial process of spatial operations that operate on spatial structures. The scalar algebra that we learn first in school is an abstraction of the vector algebra that underlies it.

The periodicity of the number line, the alternation between even and odd, the concepts of negation and inversion, are all strongly suggestive of a phasic cyclic oscillatory phenomenon which must ultimately reside in the brain. The fact that Clifford Algebra reveals all of algebra to be a spatial phenomenon at the root of it, suggests that the computational processes of the human brain are themselves spatial in nature, like the computational operations of nonlinear optics and phase conjugation which they resemble, where spatial waves interact with other waves to produce modulated waves. The fact that Clifford algebra is particularly good at modeling phasic / cyclic phenomena such as the “spin” of elementary particles, all suggests that the human brain operates with a spatial representation based on cyclic oscillations and standing waves in the brain, which just happens to be similar to the cyclic resonances of subatomic particles. The equivalence found in Clifford Algebra between the linear and rotational domains, how negation maps also to rotational reversal, and translation along a linear axis maps to angular rotation about the origin, is surely further evidence for a wave based  principle of computation in the brain. Light waves come in the form of transverse waves that oscillate vertically or horizontally, but it also manifests in the form of rotational oscillations whereby the photon spirals either clockwise or counter-clockwise as it travels through space. It is the essential equivalence between linear and rotational oscillations that reflects their equivalence in Clifford Algebra.

The many manifestations of closure seen in different forms throughout algebra suggests a computational principle that makes optimal use of a finite representational resource, like the odometer whose individual dials each form closed loops, as does the odometer as a whole, and thus the mechanism never “runs off the end”, it just wraps around again. This is not a strategy for an accurate representation of an effectively infinite universe, as we discover whenever our odometer “rolls over” to zero, but rather it is a strategy for a wise design of a limited representational resource. It is an artifact of the computational mechanism of our numerical mind with which we represent the world, not a property of the world itself. But there is also a certain invariance to a rotational representation that resolves the problem of the “infinities” found at both ends of every number line. The equivalence between linear and rotational dimensions suggests that our mind never actually uses linear dimensions like the X axis, but approximates them in a circular dimension to avoid the problem of “infinities”.

This entire analysis of Clifford Algebra was based on my own foundational assumption that mathematics is not a human invention, but more of a discovery of the essential principles of computation in the brain. If this hypothesis is correct, then it should be possible to reverse engineer the principles of operation of the brain with guidance from the properties of mathematics as revealed through Clifford Algebra. I propose the next step in investigating the computational principles of perception would be to devise oscillatory circuits such as this Clifford Vectors Analog Electronics Analogy, to see what is possible using waves as a principle of spatial representation and computation. When we discover what kinds of things can be computed using a cyclic / phasic analog system built to emulate Clifford Algebra, only then will we be ready to hypothesize the computational function of the oscillations found in the brain.

This concludes my introduction to Clifford Algebra, but this is by no means the end of the story! This narrative continues with the story of Geometric Algebra, the modern reformulations and some extraordinary extensions to Clifford’s and Grassman’s work, largely due to David Hestenes, that reveal the true potential of Clifford Algebra and its intimate connection with perception. David Hestenes saw fit to rename Clifford Algebra to Geometric Algebra, which it turns out was Clifford’s own choice, because Hestenes wanted to emphasize the fact that Clifford Algebra is not just another algebra, but a radical discovery of the true roots of all algebras, and those roots are geometric in nature.

 

The story continues with Geometric Algebra: Projective Geometry.

The final chapter is Geometric Algebra: Conformal Geometry.