Explorations of the Minkowski Sphere in 3D
by Rob Halliday
The goal of the Spacesphere project is to turn the animations and illustration that the Alexander group of the Illinois Geometry Laboratory, has done into 3D animations. The Alexander group has been operating since the beginning of the IGL and has produced numerous animations about the geometry of the Minkowski sphere, in addition to new research about the extended Poincare disk for Hyperbolic-deSitter Space. All of this work has been done using Mathematica, and so, in order properly visual these animations, they must be made into C++/OpenGL programs.
The fundamental goal this project stands to accomplish is to create a system of points which trace out the sphere. In doing so, the framework for all other parts of the project is laid out. The second step being creating the interactions in the Cube.
The first functionality that must be implemented is a way of showing what parts of the space get what kinds of angles, given a certain point. The convention of the animation as of now is to call this point the measuring sphere. The measuring sphere will be moved around, hopefully by clicking a button while pointing at the sphere, and the points will be color coded according to their distance type. In addition to being a beautiful piece of mathematical art, this animation is used to show how incredibly fundamental causality is as a guideline to gaining intuition in this space.
From here, I propose to finish at least 2 of the next 4 objectives:
Another important and interesting facet of this space is its geodesics. One resemblance that it bears to the Euclidean sphere is that its geodesics can be thought of as intersections of the sphere with planes through the origin (except the x-y plane). One functionality that should be implemented in Spacesphere is the drawing of the geodesics and segments of them, as well as a way of conveniently displaying the complex arclength along these geodesic segments.
It would be a great tool to be able to actually color a surface of the sphere based on a schema similiar to the Admissible Observers. The actual animation of this is slightly less useful than the Admissible Observer formulation because you cannot see the whole space at once without the point array technique. This has also not been perfected in Mathematica so it is unclear if it will be easer in C++/OGL because this requires a much more base control of the graphic, or if not having a Mathematica template as a starting point will lead to trouble.
So far the Alexander Group has not touched on any possible Topological features this space could have, however it would be simple to make a homotopy-esque animation out of geodesics.
Space Time Topology
This last option would be a forray into pure mathematics to talk about, and make rigorous, some possible ways to think about topology on this sphere.