Pairs of mutually complementary 2-dimensional simplicial polyhedra: Interesting examples
Serge Alexandrovich Lawrence, Alex Sergeevich Lao, Maria Evgenievna Lao, Olga Ivanovna Chelyapina · Чебышевский сборник · 2023
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.