Symmetry in Motion: Stellae Octangulae, and Equifaced Polyhedra

Abstract

We discuss three seemingly independent topics concerning polyhedra, which finally show some relations. The first topic concerns generalisations of the classical Stella Octangula, whereby the pair of indirect congruent tetrahedra, which generates such a Stella, allows forced motions of one tetrahedron along the other. Symmetry arguments are used to identify pairs of right triangular pyramids and pairs of indirect congruent tetrahedra as candidates for such “movable” Stellae Oc- tangulae. The second topic discusses equifaced octahedra, as they occur as the common body of equifaced Stellae Octangulae. It turns out that, in general, there exist four octahedra with the same acute face triangles. They differ in the way their diagonals intersect and in the number of symme- tries and are, in this paper, distinguished “type A-octahedra” and “type B-octahedra”. For obtuse face triangles the is no type A-octahedron. For general right triangles there are two type B- octahedra, which for isosceles right triangles coincide and become the single Rodrigues- octahedron. The third topic concerns polyhedra with congruent isosceles right triangular faces, thus generalising the Rodrigues-octahedron. This chapter also the aims at providing material for educational purposes.