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arXiv · 0710.4453

Non-rational configurations, polytopes, and surfaces

Günter M. Ziegler
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Abstract

It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes'': combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates. We describe a simple construction of non-rational polytopes that does not need duality (Perles' ``Gale diagrams''): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions. We also show that there are non-rational polyhedral surfaces in 3-space, a discovery by Ulrich Brehm from 1997. His construction also starts from any non-rational point configuration in the plane, and then performs what one should call Brehm extensions, in order to obtain non-rational partial surfaces. These examples and objects are first mile stones on the way to the remarkable "universality theorems'' for polytopes and for polyhedral surfaces by Mnëv (1986), Richter-Gebert (1994), and Brehm (1997).

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Günter M. Ziegler. 2007-11-16. Non-rational configurations, polytopes, and surfaces. https://arxiv.org/abs/0710.4453

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