Geometric Algebra, Geometric Calculus

[Roquen]

I'm not following.  But I babble a little anyway.  Remember than complex numbers in the standard model of 2D in GA are the scalar and the bivector parts and can be formulated that way.  So some 'z' point is x + 0 e₁ +0 e₂ + y e₁₂.

If you wanted to play around with variants in some extended model you'd have to move the x and y into the vector and the equation is reformed in the same way as in my quaternion thread.  Humm...you could probably do the two other configurations as well...but I don't think that would be of any interest.

Of course I may not be understanding what you mean by visualize...I've been assuming you mean "thinking about the elements" instead of literally "make a picture".  If the latter, then it depends on the equation.  In many cases you'll have zeros at fixed positions...otherwise you need to play around with projections.

[kram1032]

I just had the weirdest idea:
What would happen if you took a movie (2+1+3D) and just applied typical transformations on it?
I've seen switching space and time axes before. That can cause some interesting effects. But what about rotating the color axes into the spacetime axes? (remapped such that there is no color clipping)
I bet that would be pretty trippy.

Datavisualization deals with the issue of dimensional reduction all the time. There are algorithms which filter the data, automatically decide which parts are the "most significant" (e.g. their structure is the most complex/ they contribute the most to the used data) and, from there, produce a low-dimensional image which can be understood and interpreted by us more easily.
It's not perfect, but it's the best we can do.

[hermann]

May be I have used the word dimension for my purposes. But geometric algebra is relative new for me, being classicly educated without geometric algebra a
3-Dimensional space was constructed by .

Now in geometric algebra I have named the following structur 3-dimensional. May be not the correct name in mathematics, but that is it what I mean:




The following is a thing I call 4 dimensional


for 5 dimensions my LaTex code is not accepted. But the html-version works:
http://www.wackerart.de/mathematik/geometric_algebra.html

's form a vector space.
form a bivector space.
form a trivector space
one can continue to construc n-vector spaces.
From this Items one can construct a multivector. Which builds a multivector space.
All the vector spaces defined above are of cause vector spaces in the abstract sence!

Each of these vector spaces has of cause his own individual dimensionality.

Hermann

[kram1032]

those objects are the descriptions of an orthonormal system (this is not actually a requirement, you can do non-orthonormal ones too) of all possible subspaces of a given dimensionality, added together to a larger vector space.

In practice you'll rarely need more than just a couple of elements to be presented at the same time.
It's not clear at all what it means to have a scalar AND a directed stretch AND a directed area AND a directed volume all at the same time.

A dimension, in mathematics, is no more than a number. It need not be visualized per se. It's not of great importance.
Intuition is greater in low-dimensional spaces we can actually grasp mentally, but that's precisely the power of geometric algebra: To intuitively extend this to higher-dimensional spaces, making the notion of reflection almost primitive. (It's just a double-multiplication with an element you want to reflect in, and its inversion, e.g. a sandwich operation).

You can't actually visualize those higher-dimensional spaces, but through geometric algebra, the operations on them become more intuitive. The notions of rotation or reflection become really clear.

[hermann]

I just had the weirdest idea:
What would happen if you took a movie (2+1+3D) and just applied typical transformations on it?
I've seen switching space and time axes before. That can cause some interesting effects. But what about rotating the color axes into the spacetime axes? (remapped such that there is no color clipping)
I bet that would be pretty trippy.

May be one day I have a complete version of geometric algebra software. The next step is then how to define fractals and how visualise them. Sounds like a aweful lot of programming.

[kram1032]

eh, once you have the basic architecture down, defining any kind of geometric process should be easy enough.
In particular, already in this thread we found how to represent the MSet in Geometric Algebra. Applying that to higher dimensions unfortunately (but expectedly) just reproduces the quaternion-style rotated M-Set, but it works nicely for 2D.

What you gotta do is pick a base direction (the direction which will point in the positive real direction of the classical M-Set) and then just multiply it both sides with a (not reversed) arbitrary vector to get the "z²" part. Then, just add another vector c, and you have your usual situation.

Example:
Input vectors:

chosen base direction:

Process:

[hermann]

I am still working on V9 and V18!

Maybe you can check my V9. Attention: This page will challange your browser!
http://www.wackerart.de/mathematik/geometric_algebra/multiplication_table_short_9.html

  How did you note V18 with paper and pencil? 

Hermann

[kram1032]

it's not technically hard to write all this down, even if resulting tables are absolutely massive. Though if you multiply in a non-systematic way, by brute-forcing each possible combination of elements one by one to obtain the full multiplication tables, it's a very mind-numbing if technically easy task.

[hermann]

eh, once you have the basic architecture down, defining any kind of geometric process should be easy enough.
In particular, already in this thread we found how to represent the MSet in Geometric Algebra. Applying that to higher dimensions unfortunately (but expectedly) just reproduces the quaternion-style rotated M-Set, but it works nicely for 2D.

Hallo Kram,

I have no problem in programming M-Sets with geometric algebra. Its more a technical question how to get a graphic interface to my software.
At the moment I am thinking to implement a 3D-JavaScript intefrace on my homepage. With severel windows in each window running a 3D animation.
The only problem is, I have to program it.

Hermann
P.S Something like this: http://www.wackerart.de/windows.html (use the scroll bars!)

[kram1032]

hah that's a neat demo

[hermann]

I have produced the following page with tables of the canonical base vectors for the construction of multi vectors in different dimensions.
http://www.wackerart.de/mathematik/geometric_algebra/several_base_vector_tables.html

Hermann

[hermann]

With geometic algebra it is possible to build complex numbers and quaternions.
But is it also possible to build Octonions and Sedenions with Geometric Algebra?

They also seems to have very interesting properties.
http://en.wikipedia.org/wiki/Octonion
http://en.wikipedia.org/wiki/Sedenion

Hermann

[kram1032]

I'm pretty sure yes, however the correspondence might not be quite as pretty.
Quaternions are directly encoded by taking a 3D Clifford algebra together with the scalar element.
Trying to do this with Octonions and Sedenions is not quite as straight forward.
a 6D Clifford algebra happens to have 15 different bivectors, however, this sub-algebra would also generate quadvectors and the hexvector for this space. So you'd end up with 32 distinct elements in that space.

However, a 4D Clifford Algebra has 6 bivectors and its one pseudo-scalar quadvector. This, together with its scalar, should generate Octonions.
And a 5D Clifford Algebra generates 10 bivectors and 5 quadvectors. Taking those plus scalar should be able to generate the Sedenions.

I did not actually test if those are equivalent, but that's the easiest way of how I could see those algebras.

I did test something quickly though: The subalgebra of even-dimensional subspaces of an n-D Clifford Algebra actually has dimension for any dimension n. So my guess is that all those even-dimensional subspace subalgebras are precisely the, err, -ions.

Note that this trend also fits with the complex numbers. You have your elements {1, x, y, xy=I} of which the even sub-algebra is {1,I} and this happens to precisely model the complex numbers with the dimension .

This would interestingly mean that, technically, octonions are made up not just of planar rotations (or double-reflections) but also of 4-space rotations (4x-reflections) and sedenions even have a full-space rotation (6 consecutive reflections in different directions) of sorts.

[kram1032]

In this (rather large, 268 pages) journal, there is a paper (starting page 101) showing how to properly do calculus with arbitrary non-commutative variables (as you might find them in geometric algebra, for instance)
http://ejtp.com/articles/ejtpv11i31.pdf#page=101
I haven't looked into all the other papers in this thing, so I can't say if anything else interesting is in there. Feel free to dig through it all

[hermann]

Hallo Kram,

thanks for the information.

Some where it was stated that it was not possible to express octonions and sedonions but I can not remember the link.
The paper you posted looks very challanging. So I have my usual time constraints.
Also my holidays are know over and I am back in the company with challanging and time consuming jobs.

Normaly I prefere not to make software public before it is complete. But I discovered, that I can generate HTML-Documentation from GPS (Gnat Programing Studio).
So I have generated the documentation of my geometric algebra software to give a snapshot of the actual state of the work.
http://www.wackerart.de/mathematik/geometric_algebra.html#rainy_days
It is still very incomplete, but the tables generated with this software can be seen on my geometric algebra page:
http://www.wackerart.de/mathematik/geometric_algebra.html and in this thread.

I also made this incomplete work public to give you, jehovac and all in fractal forums a positive feedback for all the inspiration I have received.

Hermann