`Drivin Me Crazy'
was partly inspired by
the theory of moduli spaces of polygons with fixed angles.
I first learned about this theory
from Bill Thurston, and there is paper
by Ghys and Bavard on the topic. I'm not sure
who originated the theory. The example
relevant to the game is the space of
convex hexagons whose interior angles are all 120 degrees.
Call these ``pseudoregular'' hexagons.
(The regular hexagon is the archtypical example.)
Given the right
coordinates, the space of pseudoregular hexagons
is just a convex cone C of 4-dimensional Euclidean space.
On C there is a
quadratic form of type (3,1) whose diagonal part is the area
function. Equipped with this form C becomes a cone
in Minkowski space, the space familiar from special
relativity. The set of unit length vectors
- that is, unit area pseudoregular hexagons - in C has a natural interpretation
as a subset of 3-dimensional hyperbolic space.
`Drivin Me Crazy' is set up in such a way that the inner
part of the expanding hexagon is pseudoregular and
has unit area. So, when you
play ``Drivin Me Crazy'', you are
steering around a room in hyperbolic
space, trying not to hit
the walls.