Draft for a proposal to use ALICE to enter Non-Euclidean Space.
We all know what a surface is. To a topologist, a surface is locally
like a patch in the Euclidean plane; say a square-mile section of
East Central Illinois. A complete surface can be continued in all
directions forever. But, unlike the plane, a topologists surface
can come back on itself, like the surface of our spherical earth.
It can be one-sided (like a Moebius band) so that a left hand can
become a right hand merely by taking a trip around the surface.
A surface in the shape of an innertube or donut is called a torus.
One can connect two tori by a cylindrical tube, obtaining a 2-holed
torus. To a geometer, a surface also comes with a way of measuring angles,
distances and areas. Even people driving in the flat midwestern
countryside know that the geometry on a sphere is not flat; every
so often a north/south road makes a jog to accommodate the curvature
of the earth.
As video-game generation people, we are also familiar with the notion
of a wrap-around screen. When a game token flies off the top or the
left edge of the screen, it reappears at the bottom or right edge and
proceeds at the appropriate angle. This establishes a Euclidean
geometry on a topological torus. The most typical geometry on
arbitrary surfaces is, however, the non-Euclidean, non-spherical
geometry of Lobachevski, Bolyai and Gauss, also called hyperbolic geometry.
It is an old story that all surfaces can be can be assembled by connecting
tori and sewing Moebius-bands into an initial sphere. It is a profound
theorem of Poincare and Klein, from the last century, that surfaces can
have only one of three geometries: Euclidean, hyperbolic or spherical.
What about 3D worlds, called 3-manifolds? Locally, a 3-manifold looks
like a block of Euclidean 3-space. Our own world is part of some
3-manifold. But which one? If we could explore our world geometrically,
unconstrained by physical limitations such as the speed of light, would
we find it closed on itself like a sphere, a torus, a Moebius band?
We known from Einstein's Theory of Relativity that our world is curved,
albeit so gently that we have not yet discovered the 3D analog of the
jog in the road.
It has been know for a long time that 3-manifolds can also be decomposed
into elementary constituents and these, in turn, admit the impositions
of only a few different kinds of geometries. The eminent geometer,
William Thurston, conjectured in 1980 that there are only eight of these
for 3-manifolds, as there are only three for surfaces. He and his
collaborators proved the conjecture for almost all manifolds. And, just
as for surfaces, for almost all of them their geometry is hyperbolic.
Besides 3D spherical and Euclidean geometry, there are only two more
simple ones, corresponding to the Cartesian product of a 2-sphere or
a hyperbolic plane with a line. The remaining three are the exotic
geometries of 3D Lie groups, one of which, the Heisenberg group, is
important in relativity theory.
Although the geometry engines of the Silicon Graphics Inc (SGI)
workstations were designed with only Euclidean space in mind, it
turns out that hyperbolic and spherical geometries can be handled
very efficiently by the same design. We demonstrated this in the
4-sided CAVES at SIGGRAPH'94 with the very popular "Post Euclidean
Walkabout". The exotic 3-D geometries present a challenge of
considerable mathematical importance to optimize for real-time
exploration of their fascinating worlds.
Our nine-year experience from frequent public demonstrations of hyperbolic
and spherical geometry in a conventional (4-walled) CAVE suggests
that the illusion is not adequate to "experience" non-Euclidean geometry.
Too much has to be explained; too few questions are asked. The
exotic geometries are even more subtle, they are anisotropic. That means
physical laws do depend on orientation. Some directions are more equal
than others, to misquote George Orwell. The sensation of navigating
these anisotropic worlds will be totally different than any experience
ever had.
Our experience in the conventional CAVE has, however, proved that
the gravity based sense of an absolute "down" can be, fleetingly,
overcome by the visual suggestion to the contrary. Still, out of
the corners of the eye people see the empty space above, and the
dimly lit room to the rear of the convential CAVE. In an illusion
without "above" and "rear" to confirm the gravitational "down" we
expect to overcome the Euclidean reference frame.
George Francis
Introduction to the geometric portion
of the original grant proposal for Alice.