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Zbl 1133.52302
Ziegler, Günter M.
Sharir's cube. (English)
[J] Electronic Geometry Model No. 2000.09.028, no pag., electronic only (2000).

Summary: 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 hand, 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 half-spaces whose bounding hyperplanes are perpendicular in order to obtain an $(n+1)$-dimensional Sharir-cube.
MSC 2000:
*52B10 Three-dimensional polytopes
52B12 Special polytopes (linear programming, centrally symmetric, etc.)

Keywords: cube

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Highlights
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