# 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