Abstract
To some extent, we can claim to“understand” 3-dimensional polytopes. in fact, Steinitz’ Theorem
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“the combinatorial types of 3-polytopes are given by the simple, 3-connected planar graphs” (Steinitz, see Steinitz & Rademacher [12])
reduces much of the geometry of 3-polytopes to entirely combinational questions. Its powerful extensions answer basic questions about representing combinatorial types by actual 3-dimensional polytopes:
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“every 3-polytope can be realized with rational vertex coordiantes”(a trivial consequence of the inductive proof for Steinitz’ theorem),
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“every combinatorial type of 3-polytopes can be realized with the shape of one facet (2-face)arbitrarily prescribed” (a theorem obtained by subtle adaption of the proof, by BArnette & Grýbaum [2]),
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“the space of all realizations of a convex 3-polytope, up to affine equivalence, is contractible, and thus in particular connected” (this is what Steinitz actually proved, see[12]).
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Ziegler, G.M. (1994). Three Problems About 4-Polytopes. In: Bisztriczky, T., McMullen, P., Schneider, R., Weiss, A.I. (eds) Polytopes: Abstract, Convex and Computational. NATO ASI Series, vol 440. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0924-6_20
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