Chiral 4-polytopes in ordinary space (Q1702739)

Let \(\mathcal{P}\) be a polytope (finite or infinite) in \(3\)-space, let \(G(\mathcal{P})\) be its symmetry group (of euclidean isometries) and let \(G^+(\mathcal{P})\) be the even subgroup of \(G(\mathcal{P})\) that permutes flags that differ by an even number of adjacent steps. A polytope \(\mathcal{P}\) is ranked, so that a \(4\)-polytope has vertices, edges, \(2\)-faces and facets. Then \(\mathcal{P}\) is (geometrically) regular if \(G(\mathcal{P})\) is transitive on all the flags of \(\mathcal{P}\), and chiral if \(G(\mathcal{P})\) induces two orbits on the flags so that adjacent flags belong to different orbits; in the latter case, \(G(\mathcal{P}) = G^+(\mathcal{P})\). Observe that \(G(\mathcal{P})\) can differ from the automorphism group \(\Gamma(\mathcal{P})\), consisting of the combinatorial symmetries; thus \(\mathcal{P}\) can be combinatorially regular, but only chiral as a geometric polytope. The author begins by introducing several Euclidean nets (connected graphs with discrete vertex sets admitting a full lattice of translational symmetries), and giving a survey of the basic theory of polytopes and their symmetries. He then describes three chiral (infinite) \(4\)-polytopes, giving details of their symmetry groups (in particular their generators) and geometric structure; the euclidean nets play a part in the latter. (One of these was the subject of the author's earlier paper [Ars Math. Contemp. 12, No. 2, 315--327 (2017; Zbl 1382.52011)].) He then proves that these polytopes are combinatorially chiral; in fact, they have chiral facets. Since finite polyhedra in \(3\)-space cannot be chiral, nor can polygons or tessellations in the plane, these polytopes provide the strongest possible refutation of a claim by the reviewer in [Discrete Comput. Geom. 32, No. 1, 1--35 (2004; Zbl 1059.52019)] that polytopes of full rank could not be chiral; a finite chiral \(4\)-polytope of full rank was found by \textit{J. Bracho} et al. [Discrete Comput. Geom. 52, No. 4, 799--805 (2014; Zbl 1306.05260)].