Pairs of mutually complementary $2$-dimensional simplicial polyhedra: Interesting examples

S. Lawrence ; A. S. Lao ; M. E. Lao ; O. I. Chelyapina

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 42-55.

Résumé

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.

Détail

Keywords: polyhedron, triangulation, torus, projective plane, Möbius strip, Schlegel diagram, GeoGebra.

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