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Pairs of mutually complementary $2$-dimensional simplicial polyhedra: Interesting examples
S. Lawrencea, A. S. Laob, M. E. Laob, O. I. Chelyapinaa a Russian State University of Tourism and Service; Institute of Service Technologies
(Moscow)
b IT Company “Kometa Games” (Moscow)
Abstract:
We construct an example of a pair of ($2$-dimensional) $8$-vertex simplicial toroidal polyhedra (each polyhedron without self-intersection) with same $1$-dimensional skeleton in (Euclidean) $3$-space, which do not have a single common $2$-face, and the union of the $2$-skeletons of these two polyhedra gives a geometric realization of the $2$-skeleton of the $4$-dimensional hyperoctahedron in $3$-space. Also, we construct an example of a pair of $6$-vertex simplicial polyhedral projective planes with the same $1$-skeleton in $4$-space, which do not have a single common $2$-face, and the union of these projective planes gives a geometric realization of the $2$-skeleton of the $5$-hypertetrahedron in $4$-space. Finally, it is shown how to imagine, figuratively, the atoms in the molecule of methane ${\rm{CH}}_4$ “linked” by a pair of internally disjoint spanning polyhedral Möbius strips.
Keywords:
polyhedron, triangulation, torus, projective plane, Möbius strip, Schlegel diagram, GeoGebra.
Received: 21.08.2022 Accepted: 12.09.2023
Citation:
S. Lawrence, A. S. Lao, M. E. Lao, O. I. Chelyapina, “Pairs of mutually complementary $2$-dimensional simplicial polyhedra: Interesting examples”, Chebyshevskii Sb., 24:3 (2023), 42–55
Linking options:
https://www.mathnet.ru/eng/cheb1324 https://www.mathnet.ru/eng/cheb/v24/i3/p42
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Abstract page: | 57 | Full-text PDF : | 13 | References: | 11 |
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