Message #1255

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Parity in 12 Colors
Date: Fri, 05 Nov 2010 23:44:51 -0500

On Thu, Nov 4, 2010 at 4:01 PM, Melinda Green <melinda@superliminal.com>wrote:

>
> Regarding your math, I was rather surprised to learn that you use
> Complex arithmetic. I’ve never really understood why they are so useful
> in a number of sciences. What do you use them for?
>

I had hoped to see more on this, because it is thoroughly magical how
complex arithmetic is behind all this geometry (and how it fits into other
areas of science). I’ve mentioned the first 6 chapters of Visual Complex
Analysis. The book is about complex numbers, yet those chapters are
simultaneously an introduction to non-Euclidean geometry, which shows how
intertwined they are.

Complex arithmetic also made MagicTile possible, and it uses the same kinds
of calculations as in MHT633 (described below) to do its twisting in all
three of the geometries. Ironically, complex numbers simplify the scenarios
immensely. Though I don’t have experience applying them one dimension up, I
have the impression things are similar. After all, you can take an H2 slice
of H3, an R2 slice of R3, or an S2 slice of S3, and the 2D behaviors are
connected to the 3D behaviors.

The core of things are the "linear fractional transformations" or "Mobius
Transformations
<http://en.wikipedia.org/wiki/Mobius_transformations>". A Mobius
Transformation
is a simple looking function of a complex variable z.

f(z) = (az+b)/(cz+d)

The a,b,c,d coefficients are complex numbers as well, and you get
differently behaving functions depending on how they are chosen. (To be
complete for those who might not know, a complex number has two components
and so can be interpreted as a point in a plane, so these functions
transform point locations in a plane.)

To get a sense of how they can transform points in a plane and some cool
geometric insight, watch this short YouTube
video<http://www.youtube.com/watch?v=JX3VmDgiFnY>.
In the video, you see how the functions can represent the "isometry"
movements (distance preserving) of a stereographically projected sphere, but
they can represent more. Selecting certain coefficients allows you to
describe all the movements of H2 in the Poincare disk model as well! Don
used these functions for his tessellation applet. In fact, the simple
formula above can be used to describe isometries in all three standard
geometries (stereographically projected S2, R2, and the Poincare disk model
of H2), which is crazy to me.

As if that’s not enough, Tristan Needham writes in Visual Complex Analysis:

> …we explain how we may set up a one-to-one correspondence between these
> light rays and complex numbers. Thus each Lorentz transformation of
> space-time induces a definite mapping of the complex plane. What kinds of
> complex mappings do we obtain in this way? The miraculous answer turns out
> to be this: The complex mappings that correspond to the Lorentz
> transformations are the Mobius transformations! Conversely, every Mobius
> transformation of *C* yields a unique Lorentz transformation of
> space-time. Even among professional physicists, this "miracle" is not as
> well known as it should be.

So there you have it, these transforms are also intimately related to
special relativity!

If anyone wants to dig deeper, all the details of the possibilities for
the a,b,c,d coefficients and the classes of behavior they lead to are in
Visual Complex Analysis. It’s safe to say the Mobius Transformations are
all over the place in the models of H3 geometry (I know Andrey used them in
his implementation, though he could provide more insight about what it takes
to fully jump to the 3D geometry from 2D).

All the best,
Roice