inside the octahedron

The sequences of images in this webpage shows how the convex regular polyhedra (aka platonic solids) and polyhedron compounds (of the convex regular) are obtained from the regular octahedron.

Regular facets of the octahedron

From the vertices of the octahedron, it is only possible to obtain one type of regular facet: the square.

Tetrahedra inside the octahedron

Two tetrahedra are inscribable inside the octahedron, if we take the face centroids of the octahedron as vertices. In the images below, a nested cube is shown to better understand their relations, but the cube itself is unnecessary to obtain the tetrahedron.

Cube inside the octahedron 01

The cube and the octahedron are duals of each other, so the face centroids of the octahedron are the vertices of the cube (and vice-versa). In this situation, they do not share the same reciprocation sphere.

Cubes inside the octahedron 02

It is also possible to draw cubes inside the octahedron in three different positions (and an interesting compound), in which the vertices of the former belong to the edges of the latter. Each of the cubes shares one 4-fold symmetry with the octahedron:

Piero della Francesca illustrated this relation between the cube and the octahedron ca. 1460-1480, in Libellus De Quinque Corporibus Regularibus:

Piero della Francesca (ca. 1460) Libellus De Quinque Corporibus Regularibus, p. 41b [3]

In the following images, we will consider only one of the cubes to better understand the situation. According to Coxeter [3] the vertices of the cube are the intersection of the edges of the compound of three octahedra (that Escher depicts in the wood engraving Stars, dated 1948) and are located in the edges of each octahedron in the ratio [square root of 2]:1, as the images below shows:

Dodecahedron inside the octahedron

We begin with the cube whose vertices are the face centroids of the octahedron and obtain a dodecahedron from it, as explained here. Eight vertices of the dodecahedron belong to the faces of the octahedron.

Icosahedron inside the octahedron

The vertices of the icosahedron inside the octahedron are determined through the division of its edges in the golden ratio, here determined by interior division of an edge:

References:

  1. Pugh, A. (1976). Polyhedra: A visual approach. Berkeley: University of California Press.
  2. Piero della Francesca circa 1460 – Libellus De Quinque Corporibus Regularibus – Bibliothèque municipale de Bordeaux – Libellus De Quinque Corporibus Regularibus de Piero della Francesco cote G.F. 9702/3 Rés, Pubblico dominio, https://commons.wikimedia.org/w/index.php?curid=74662295
  3. Coxeter, H. (1985). Book Review. The Mathematical Intelligencer, 7(1), 59-71. doi:10.1007/bf03023010
  4. Viana, V. 2020. Aplicações didácticas sobre poliedros para o ensino da geometria / Didactic Applications on Polyhedra for the teaching of Geometry (http://hdl.handle.net/10348/10337) [Tese de Doutoramento, Universidade de Trás-os-Montes e Alto Douro]. Repositório Científico da UTAD.