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Equivelar toroids with few flag-orbits. (English) Zbl 07308717
Toroids are generalisations of maps on the torus to higher dimensions. To be more specific, an \((n+1)\)-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group.
The classification of regular and chiral toroids of any rank was completed by P. McMullen and E. Schulte [Adv. Math. 117, No. 1, 17–51 (1996; Zbl 0847.52010)]. In particular, they showed that there exist no chiral toroids of rank greater than 3. Furthermore, M. I. Hartley et al. showed for \(n\ge 2\), the only Euclidean space form that admits chiral tessellations is the \(n\)-torus and proved this is only possible when \(n = 2\) (see [Can. J. Math. 51, No. 6, 1230–1239 (1999; Zbl 0965.51008)]). In [Discrete Comput. Geom. 48, No. 4, 1110–1136 (2012; Zbl 1263.51016)], I. Hubard et al. studied properties of the automorphism groups of equivelar toroids and classified all equivelar toroids of ranks 3 and 4 into families according to their symmetry types.
The present nice paper gives a classification of equivelar \((n+1)\)-toroids with at most \(n\) orbits. In particular, a classification of 2-orbit \((n+1)\)-toroids is discussed. Symmetry types of families of equivelar \((n+1)\)-toroids with at most \(n\) orbits are also determined.
51B15 Laguerre geometries
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
51F15 Reflection groups, reflection geometries
51M20 Polyhedra and polytopes; regular figures, division of spaces
Full Text: DOI
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