اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNon-rational configurations, polytopes, and surfaces
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Gunter M. Ziegler
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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 Mnev (1986), Richter-Gebert (1994), and Brehm (1997).
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We will study the angle sums of polytopes, listed in the $alpha$-vector, working to exploit the analogy between the f-vector of faces in each dimension and the alpha-vector of angle sums. The Gram and Perles relations on the $alpha$-vector are analog
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