Cubes

8. The Sharir-Ziegler Cube

In June of 1999 at the Monte Verita Conference on Discrete and Computational Geometry, the computer scientist and mathematician Micha Sharir asked what appeared to be a strange question. Could one construct a polyhedron in 3-dimensional space which was combinatorially a cube and where the opposite faces of the cube were perpendicular to each other? When we think of a cube we think of it as having three pairs of faces which are opposite to each other and which are parallel. Sharir raised the possibility of having a 3-cube at the other extreme; its opposite pairs of faces would be perpendicular. Its hard to imagine this can be done!

Not long after Sharir posed his question (July 1999), Günter Ziegler solved the problem using the software tool Polymake to assist with the process. With the computer's assistance in finding a 3-dimensional version of a Sharir-Ziegler cube, Ziegler also showed that what could happen in 3-space also happened in all higher dimensional spaces as well. To help visualize the Sharir-Ziegler cube, Michael Joswig provided a net of the Sharir-Ziegler cube.The discovery of the Sharir-Ziegler cube demonstrates the way that mathematical scientists interact and the role that software is having as a tool for helping get insights into mathematical questions.

1. Introduction
2. Some history
3. The 3-dimensional cube
4. Combinatorial perspectives on cubes
5. A recursive way of constructing cubes
6. Cube puzzles
7. Symmetries of the cube
8. The Sharir-Ziegler cube
9. References