Geodesic
Pairs of mutually complementary $2$-dimensional simplicial polyhedra: Interesting examples
Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 42-55

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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. 
@article{CHEB_2023_24_3_a2,
     author = {S. Lawrence and A. S. Lao and M. E. Lao and O. I. Chelyapina},
     title = {Pairs of mutually complementary $2$-dimensional simplicial polyhedra: {Interesting} examples},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {42--55},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_3_a2/}
}
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S. Lawrence; A. S. Lao; M. E. Lao; O. I. Chelyapina. Pairs of mutually complementary $2$-dimensional simplicial polyhedra: Interesting examples. Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 42-55. http://geodesic.mathdoc.fr/item/CHEB_2023_24_3_a2/