Geodesic
On the Symmetry of a Convex Polyhedron in a Translational Point Multilattice
Informatics and Automation, Geometry, Topology, and Mathematical Physics, Tome 325 (2024), pp. 322-332 Cet article a éte moissonné depuis la source Math-Net.Ru

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In geometric crystallography, there are 32 well-known point crystallographic groups, or A. V. Gadolin's 32 crystal classes, which make up a complete list of symmetry groups of crystal shapes whose internal structure is subordinate to one of the 230 Fedorov groups existing in $\mathbb R^3$. In 2022, the author constructed two point crystal structures located in $\mathbb R^3$ whose possible external shapes have the symmetry groups $D_{8\textup {h}}$ and $D_{12\textup {h}}$, respectively. However, the internal structure of the crystal was not taken into account in the considerations of these groups. The central result of the author's 2022 paper is as follows: if a possible external shape of an ideal crystal has an ordinary rotation of non-crystallographic order $n$, then either $n=8$ or $n=12$ and in this case the external shape is a right prism of finite height. But only after the paper was published did the author notice that the proof of this result was incomplete, although the result itself is correct. The present paper provides a complete proof of this result without relying on the 2022 text.
Keywords: Fedorov group, crystal structure, lattice, cut (faceting), symmetry group.
Mots-clés : net 
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M. I. Shtogrin. On the Symmetry of a Convex Polyhedron in a Translational Point Multilattice. Informatics and Automation, Geometry, Topology, and Mathematical Physics, Tome 325 (2024), pp. 322-332. http://geodesic.mathdoc.fr/item/TRSPY_2024_325_a17/

[1] Diamond cut—Wikipedia, The Free Encyclopedia, 2024 https://en.wikipedia.org/w/index.php?title=Diamond\textunderscore cut&oldid=1229617863

[2] Dolbilin N., Garber A., Leopold U., Schulte E., Senechal M., “On the regularity radius of Delone sets in $\mathbb R^3$”, Discrete Comput. Geom., 66 (2021), 996–1024 | DOI | MR | Zbl

[3] Gadolin A.V., “Vyvod vsekh kristallograficheskikh sistem i ikh podrazdelenii iz odnogo obschego nachala”, Zap. Imp. S.-Peterb. mineralog. o-va. Ser. 2. Ch. IV, 1869, 112–200; 1954, Изд-во АН СССР, Л.; Gadolin A., “Mémoire sur la déduction d'un seul principe de tous les systèmes cristallographiques avec leurs subdivisions”, Acta Soc. Sci. Fenn., 9 (1871), 1–73

[4] Shtogrin M.I., “O ratsionalnykh napravleniyakh v ploskoi reshetke”, Chebyshev. sb., 16:2 (2015), 273–281 | MR | Zbl

[5] M. I. Shtogrin, “On a convex polyhedron in a regular point system”, Izv. Math., 86:3 (2022), 586–619 | DOI | DOI | MR | Zbl

[6] Vavilov V.V., Ustinov A.V., Mnogougolniki na reshetkakh, MTsNMO, M., 2006