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arXiv · 2002.00355

$f$-vectors of $3$-polytopes symmetric under rotations and rotary reflections

Maren H. Ring·Robert Schüler
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Abstract

The $f$-vector of a polytope consists of the numbers of its $i$-dimensional faces. An open field of study is the characterization of all possible $f$-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e. to characterize $f$-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer for all three dimensional polytopes that are symmetric with respect to a finite rotation or rotary reflection group. We solve these cases constructively by developing tools that generalize Steinitz's approach.

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Maren H. Ring, Robert Schüler. 2020-02-02. $f$-vectors of $3$-polytopes symmetric under rotations and rotary reflections. https://arxiv.org/abs/2002.00355

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