Geometric Algebra, Geometric Calculus

[flexiverse]

Very interesting this.  Having just learnt how important grassmanian algebra is.

What is the best resource to Learn clifford algebra for absolute beginners ?

I always thought there should be a more unified algebra that works in all dimensions! I'm completely mind blown grassmann was doing this so long ago !!!

Just answered my own question !!!


Geometric Algebra for computer graphics book :


http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/%5BJohn_A._Vince%5D_Geometric_algebra_for_computer_gra%28BookFi.org%29.pdf

[kram1032]

flexiverse, it may be a tall task by now (considering this thread is 12 pages long already!) but there are a LOT of resources of varying difficulty strewn out all across it. Tendentially the more beginner-level ones are earlier in the thread (although I think I remember there being a couple easier ones in the more recent past as well.)

Though even most of the more technical papers tend to recap the same ideas over and over. They do this because Geometric Algebra isn't in the mainstream of maths and thus, to expand the audience, it needs to always be re-explained.

Thanks for that link though! I may be wrong but I think that one's new to this thread.
In fact this whole thing has some nice things http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/

[eiffie]

Sweet! Thanks for the link(s).

[flexiverse]

flexiverse, it may be a tall task by now (considering this thread is 12 pages long already!) but there are a LOT of resources of varying difficulty strewn out all across it. Tendentially the more beginner-level ones are earlier in the thread (although I think I remember there being a couple easier ones in the more recent past as well.)

Though even most of the more technical papers tend to recap the same ideas over and over. They do this because Geometric Algebra isn't in the mainstream of maths and thus, to expand the audience, it needs to always be re-explained.

Thanks for that link though! I may be wrong but I think that one's new to this thread.
In fact this whole thing has some nice things http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/

Mind you I was specifically referring to the book "geometric algebra for computer graphics" by John vince.

But it turns out the root folder is accessible and contains EVERY book and paper about this subject on this entire planet !!! Really !

Fantastic find I must say so my self. 

[hermann]

My Starting point was the paper of Richard Alan Miller. You can find a link in this thread or you can find the link at my Geometric Algebra page, which is inspired by this thread here in fractal forums.

Hermann

http://www.wackerart.de/mathematik/geometric_algebra.html

[flexiverse]

My Starting point was the paper of Richard Alan Miller. You can find a link in this thread or you can find the link at my Geometric Algebra page, which is inspired by this thread here in fractal forums.

Hermann

http://www.wackerart.de/mathematik/geometric_algebra.html

Thanks. Actually David Hestenes who is mentioned in the first sentence in the introduction in that paper, has produced what looks like the best book on the subject.  Namely geometric Algebra for physics.

[jehovajah]

Er, I can't find qquazxxsw on you tube ?

Thanks for the correction flexiverse . I have corrected my original post. Here is a link to qqazxxsw for you.

http://m.youtube.com/watch?v=rz8A5l_yn34

[jehovajah]

Mind you I was specifically referring to the book "geometric algebra for computer graphics" by John vince.

But it turns out the root folder is accessible and contains EVERY book and paper about this subject on this entire planet !!! Really !

Fantastic find I must say so my self. 

Really?
It is a great resource though, but worth downloading what you want now before the access is controlled more vigorously!

[flexiverse]

The john vince books seem great !

Geometric Algebra: An Algebraic System for Computer Games and Animation. By  John vince


_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf]http://dl.lux.bookfi.org/genesis/122000/bac7a55b79a54365ad875863c1225bc4/_as/[John_A._Vince]_Geometric_Algebra_An_Algebraic_Sy(BookFi.org).pdf

Honestly guy, geometric algebra is the most fascinating thing I've ever come across!

I've been looking for something like this for a long time, it's the true mathematics, I always knew geometry was the basis for everything.....
I honestly can't believe I've never heard of clifford / grassmann algebra only until recently, it really does unify everything.

[kram1032]

Geometric Algebra can do a lot of awesome things and geometry is very prevalent all throughout both math and life.  However,  I wouldn't quite go as far as to say it's the basis of everything. A lot of stuff is better described with mere topology which has less structure than geometry.  (It's like geometry but without a metric or rather,  geometries are topologies equipped with a metric.)

Meanwhile,  Topology, Type Theory and Logic can all be combined into a beautifully structured theory called homotopy type theory. It and extensions of it to more general categories seem to be much more natural for a full foundation of mathematics. Geometric Algebra already has too much, too specific structure for that kind of thing.
Though if your sole intention is to manipulate geometric spaces, geometric algebra really is where it's at.

[flexiverse]

Geometric Algebra can do a lot of awesome things and geometry is very prevalent all throughout both math and life.  However,  I wouldn't quite go as far as to say it's the basis of everything. A lot of stuff is better described with mere topology which has less structure than geometry.  (It's like geometry but without a metric or rather,  geometries are topologies equipped with a metric.)

Meanwhile,  Topology, Type Theory and Logic can all be combined into a beautifully structured theory called homotopy type theory. It and extensions of it to more general categories seem to be much more natural for a full foundation of mathematics. Geometric Algebra already has too much, too specific structure for that kind of thing.
Though if your sole intention is to manipulate geometric spaces, geometric algebra really is where it's at.

I' m talking about a much deeper level here.  The concept of "sacred geometry" is quite common in the spiritualist realm as the true underlying structure of the universe.

It's only now that physics dudes are seeing how higher dimensional geometry is the real hidden structure if the universe.

E.g.  http://www.wired.com/2013/12/amplituhedron-jewel-quantum-physics/

E.g. https://www.ted.com/talks/garrett_lisi_on_his_theory_of_everything?language=en

Also the books the approach of geometric algebra in physics clearly state it's obviously the real underlining structure and it should be used because it unifies so much.

Topology is just geometry it's working with higher dimensional objects.

Don't get bogged down with early mathematical thinking that didn't seem to grasp complex numbers and grassmanian alegra,

So that's basically the hidden truth of the universe it's all about higher multi-dimensional objects.

That's why fractals hold a fascination it's working with higher dimensional objects.
In order to explain the universe we need to describe, manipulate higher dimensional objects.
Geometric algebra is just the start of this process, and because of it's clear success in physics just shows that's truly the way the universe works - it's sacred geometry.

[kram1032]

Early mathematical thinking? On the contrary, Homotopy Type Theory is on the very cutting edge of mathematics and newer than Geometric Algebra.
Both of those things unify a lot. Though GA unifies all of geometric manipulation while HoTT unifies three pillars of maths foundation.
You can express GA in HoTT but the opposite will be impossible.

And topology isn't geometry in higher dimensions. Geometry in higher dimensions is geometry in higher dimensions. It's also topology in higher dimensions plus a distance metric.

[jehovajah]

There is a lot to be said for both of your statements, but in fact the more i study Grassmann the more it becomes crucial not to mix up ideas and terminologies and labels.

The Hegelian Philosophy underpins how Grassmann resolved these issues allowing a much broader but precise apprehension of the human experience.

You must "get bogged down" terminologically in each subject division, but you must recognise that you are free to soar above all of them by Analogical thought patterns. What wisdom you as an individual draw from those precise 2 activities is your reward, but personally i cannot impose my apprehension on anyone else. in the same vein i will not accept anyone elses as "absolute".

Yin and Yang are the best approximation i have personal experience of, and of course i am waiting to see how far that pans out! 

[kram1032]

A method to use GA for "holographic reduced representations" or HRRs which are one method to store associations between different terms in vector spaces:
http://arxiv.org/pdf/0710.2611v2.pdf

An example for a HRR would be:

Code:

France = [Capital*Paris + Location*WesternEurope + Currency*Euro]

So "France" is defined to be a sum of the "Capital" being "Paris", the "Location" being "WesternEurope" and the "Currency" being "Euro".

Then, if you want to know a fact about France, all you have to do is to multiply with that fact:

Code:

WhatCity = Capital *France
= Capital * [Capital*Paris + Location*WesternEurope + Currency*Euro]
= Capital*Capital*Paris + Capital*Location*WesternEurope + Capital*Currency*Euro
= Paris + (noise terms).

This noise is (ideally) relatively small so you can manipulate this result so that you end up with purely Paris.

In the GA variation, this last denoising step becomes especially simple because all the noise ends up in a direction orthogonal to the actual result, so all you have to do is to project the result onto (take the scalar product with) all your plausible terms: For instance, you might know Paris, London and Oslo are Capitals. So you try them all. The largest number resulting from this scalar product will be your answer. A sort of vector based memory.

Also interesting about this paper is how they start explaining GA. I was a little confused at first but they actually chose this interpretation well and it also is an interesting demonstration that the non-commutativity of GA is a direct consequence of it being able to talk about direction. In an undirection version of GA, where , you'll actually end up with something commutative. (Though also something way less useful: It could represent lines and surfaces and the like but it couldn't tell you what direction you go along a given path or what way you turn, etc.)

[kram1032]

Geometric Algebra in Quantum Information Processing http://arxiv.org/pdf/quant-ph/0004031v3.pdf and
A Geometric Algebra Perspective On Quantum Computational Gates And Universality In Quantum Computing http://arxiv.org/pdf/1006.2071.pdf