A 3-cube with the property that each facet is perpendicular to its opposite.

Micha Sharir asked the question whether such a cube exists. This construction solves the problem in the affirmative.

It is fairly obvious that no such cube (quadrangle) exists in 2 dimensions. On the other the present example in dimension 3 can be used to lift the construction to higher dimensions: consider a prism over an n-dimensional Sharir-Cube and cut it suitably by a pair of halfspaces whose bounding hyperplanes are perpendicular in order to obtain an (n+1)-dimensional Sharir-Cube.

Model produced with: polymake 1.3.1

Keywords
| cube | |

MSC-2000 Classification
| 52B10 (52B12) | |

Zentralblatt No.
| 01682989 |

- Master File: SharirCube.poly
- Applet File: SharirCube.jvx
- Preview: SharirCube.gif

Submitted: Sun Sep 10 07:10:24 CET 2000.

Accepted: Mon Nov 20 17:06:57 CET 2000.

Technische Universität Berlin

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10623 Berlin

Germany

ziegler@math.tu-berlin.de

http://www.math.tu-berlin.de/~ziegler