inside the tetrahedron

The sequences of images in this webpage shows how each of the convex regular polyhedra (aka platonic solids) are obtained from the regular tetrahedron.

Tetrahedron inside the tetrahedron

The tetrahedron is self-dual and, as such, the only convex regular polyhedron that can be obtained from itself. The vertices of the inner tetrahedron are the face centroids of the outer tetrahedron.

Cube inside the tetrahedron

The face centroids of the outer tetrahedron outline two skew diagonals of the cube. If we translate a copy of each diagonal in order that a pair of diagonals intersect in their midpoint, we obtain the remaining vertices of the cube.

Octahedron inside the tetrahedron

The vertices of the octahedron are the midpoints of the edges of the tetrahedron:

Icosahedron inside the tetrahedron

To obtain the icosahedron from the tetrahedron, its edges must be divided in the golden ratio and the resulting points smmetrically placed form each vertex. From these, 3 cevians in each face of the tetrahedron are drawn. Their intersection give us each face of the icosahedron, as the sequence of images shows:

Dodecahedron inside the tetrahedron

To obtain the dodecahedron, we must depart from the previous icosahedron, whose face centroids are the vertices of the dodecahedron. Thus, four vertices of the dodecahedron belong to the faces of the tetrahedron:

the Archimedeans inside the regular tetrahedron

Pugh [1: 25] denotes that each of the Archimedeans is inscribable inside a regular tetrahedron and that this property is precisely what allows us to distinguish them, not only from the semiregular prisms and antiprisms, but also the rhombicuboctahedron from Johnson Solid J37, the elongated square gyrobicupola that once, Kepler recognized as the 14th Archimedean but later disregarded.

When I was writing my doctoral thesis, I did not find any image of the Archimedeans inside the tetrahedron, so I decided to model them myself. The following are images similar to those found in [2: 55-57]. The faces shown in red are coplanar with the tetrahedron and the sequence is the following: cuboctahedron, icosidodecahedron, truncated tetrahedron (first, with the hexagonal faces coplanar with the tetrahedron; and, second, with the triangular faces), truncated octahedron, truncated icosahedron, truncated cube, truncated dodecahedron, rhombicuboctahedron, rhombicosidodecahedron, rhombitruncated cuboctahedron, rhombitruncated icosidodecahedron, snub cube and the snub dodecahedron.

References:

  1. Pugh, A. (1976). Polyhedra: A visual approach. Berkeley: University of California Press.
  2. 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.