Saturday, 3 January 2009

Joyful Symmetry

I received some very cool gifts this Christmas.

My lovely wife got me Death From The Skies, the new book by the Bad Astronomer Phil Plait. And my friend Shannon got me Of Pandas and People, the Intelligent Design textbook at the centre of the Kitzmiller v. Dover trial.

I’ll be providing reviews of these separately once I’ve finished them. A preview: I liked one more than the other. I’ll let you guess which.

The prize for nerdiest gift goes to my sister, who made a donation to the charity Common Hope, and thereby got the right to name a mathematical symmetry group. Which she named after me.

A symmetry group is the collection of operations on a shape that preserve its symmetry: like reflections and rotations.

For example a square has symmetry under 90° rotation, and vertical and horizontal reflection:

Image1014[1](The darkened wedge is just to show the new position of the square after each movement).

Of course once you get beyond simple geometric shapes like squares and triangles it all starts getting a bit complicated. And all the simple ones have names already.

So my symmetry group corresponds to the elliptic curve described by the formula:

y2 + 769xy + 773y = x3 + 787x .


It’s called the Matthew Kippen Group.

I was toying with the idea of putting a picture of the curve here, but it’s tricky rendering a 2-dimensional picture of a 3-dimensional object embedded in a 4-dimensional space.

If anyone out there does know how to do this, please let me know. I'd be most interested.

Until then, you’ll just have to use your imagination. I’m sure it’s very pretty.


paul said...

congratulations on getting a symmetry group named after you. i reckon that's pretty damn cool.

i (of course)am not as mathemagic as you are, but the equation you posted seems to be 2dimensional to me, as it has only x and y variables. can you please explain how it is a 3 dimensional object in a 4 dimensional space?

Matt said...

Hi Paul.
The variables x and y are complex numbers (of the form a + bi, where i is the sqaure root of -1) and so have two parameters each: a and b. A 4-dimensional space is therefore required to render each (x,y) point.
The curve is 3-dimensional because it's had one dimension restricted by the equation. It's just like in high school maths when you would graph a simple linear equation on x- and y-axes. There you're putting a 1-dimensional object (a line) in a 2-dimensional space (a plane). You can also graph a 2-dimensional surface in 3-dimensional space, etc.
Hope that makes sense!

paul said...

yeah, much more sense now. maybe there's a video or 3d virtual display program somewhere on the net that could work it out.