(AxB)xC does not, in general, equal Ax(BxC). The lack of associativity can be seen from the connection of the wedge product (which is associative) and the cross product:

AxB = *(A^B) and so you get into trouble with triple cross products

(AxB)xC = *( *[A^B] ^ C)

Ax(BxC) = *( A ^ *[B^C])

and since the right-hand-sides are not, in general, equal the cross product is not an associative product.

That seems rather unnatural to me.

Yours, John Smith

]]>What I may guarrantee you though is that whenever I talk to someone who studies cognitive/neuroscience I’ll come up with your blog and ideas, I’m already a GA advocate, and am gonna stimulate its use in this field too.

Hopefully in a near future I’ll be in condition to fully understanding your work, as many others I think this language will help me with. Please, keep developing and publishing it! Me from the future certainly thanks you.

]]>The Perceptual Origins of Mathematics

(work in progress)

I had been bothered for a long time about i, the square root of -1. The concept of “complex” numbers was shrouded in the deepest mystery. Then I ran into BetterExplained and his Visual Intuitive Guide to Imaginary Numbers I was blown away that such an abstruse abstraction could become intuitively obvious in the right spatial context was quite a revelation to me.

But in my Google searches I kept getting hits “Clifford Algebra”, “Clifford Algebra”, “Clifford Algebra”… and I kept thinking I’m too old to be learning a whole new algebra! But one Sunday lying in bed smoking pot and surfing the internet I thought “What the hell!” and clicked on a Clifford Algebra link. I have never been the same since!

I have also made a connection between mind and harmonic resonance

Harmonic Resonance In the Brain

and provided an

Intuitive Explanation of Phase Conjugation

which I see as a model of resonances acting like physical gears, pushing and pulling on each other like genuine physical objects. The theory of computation was revolutionized by concepts like the transistor, the op-amp, amplification, that involve an input, output, and “gate” signal that controls the flow from input to output.

I see phase conjugation as a similar concept except, instead of single one-bit inputs, outputs, and controls, there is an input IMAGE, output IMAGE, and control IMAGE (in 2 or 3 dimensions!) where the control image modulates the input image to produce the output image.

Do you see where I am going with this stuff? Have you seen my

Constructive Aspect of Visual Perception

to understand how spatial waves can interact as a computational process? If you begin to see the huge potential of this direction of investigation please help me! I can’t do it all on my own! There is a huge HUGE payoff when we ultimately discover the (analog analogical spatial) computational principle of the human brain as a direct counterpoise to the world of digital logic. There is so much here to be discovered once you understand the potential significance, I’m an old man, I won’t be around for too many more years. Don’t let these ideas die with me! Help me think this through if you can see the potential that I see. I think it is the funnest, most intriguing problem that science has ever faced. The theory of our own minds.

]]>1. Yes, it’s a great video, but it’s [unfortunately] not me! It’s from 3Blue1Brown.com

2. You can make neat computer vision algorithm with tauquernions.

-m

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