Pairs of mutually complementary 2-dimensional simplicial polyhedra: Interesting examples
https://doi.org/10.22405/2226-8383-2023-24-3-42-55
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 CH4 “linked” by a pair of internally disjoint spanning polyhedral M¨obius strips.
Keywords
About the Authors
Serge Alexandrovich Lawrence
Russian State University of Tourism and Service; Institute of Service Technologies
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate
professor
Alex Sergeevich Lao
IT Company “Kometa Games”
Russian Federation
Russian Federation
Maria Evgenievna Lao
IT Company “Kometa Games”
Russian Federation
Russian Federation
Olga Ivanovna Chelyapina
Russian State University of Tourism and Service; Institute of Service Technologies
Russian Federation
Russian Federation
candidate of technical sciences, associate professor
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Review
For citations:
Lawrence S.A., Lao A.S., Lao M.E., Chelyapina O.I. Pairs of mutually complementary 2-dimensional simplicial polyhedra: Interesting examples. Chebyshevskii Sbornik. 2023;24(3):42-55. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-3-42-55
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