Cubes
8. The SharirZiegler 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 3dimensional 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 3cube 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 3dimensional version of a SharirZiegler cube, Ziegler also showed that what could happen in 3space also happened in all higher dimensional spaces as well. To help visualize the SharirZiegler cube, Michael Joswig provided a net of the SharirZiegler cube.The discovery of the SharirZiegler 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.

Introduction

Some history

The 3dimensional cube

Combinatorial perspectives on cubes

A recursive way of constructing cubes

Cube puzzles

Symmetries of the cube

The SharirZiegler cube

References

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