inside the icosahedron

The sequences of images in this webpage shows how the convex regular polyhedra (aka platonic solids), three concave regular polyhedra and polyhedron compounds (of the convex regular) are obtained from the regular icosahedron. The three Kepler-Poinsot polyhedra are also obtainable through stellation, but here we obtained them through faceting.

Regular facets of the icosahedron

From the vertices of the icosahedron, we may obtain three types of regular facets, that are shown below: regular triangles, pentagons and pentagrams. From these, we may facet the icosahedron to obtain three concave regular polyhedra (the last three items of this webpage).

Tetrahedra inside the dodecahedron

Up to 10 tetrahedra are inscribable inside the dodecahedron, if we take certain sets of the face centroids as their vertices. We may thus obtain five different stella octangulas, two compounds of five tetrahedra, and the compound of ten tetrahedra:

Cubes inside the icosahedron

Specific sets of four face centroids of the icosahedron allows us to obtain a square and six of these, a cube. Thus, we may obtain up to five cubes inside the icosahedron, as well as the compound of five cubes, shown below:

Octahedra inside the icosahedron

Octahedra are also inscribable in the icosahedron, if the midpoints of given edges are taken as their vertices. There are five possible ways to inscribe an octahedron inside the icosahedron, as well as the compound of five octahedra, all of which are shown in the images below. This regular compound is also obtainable from the dodecahedron (as shown here) and its convex hull is the icosidodecahedron, shown at the end of the sequence of images.

Dodecahedron inside the icosahedron

The icosahedron and the dodecahedron are duals of each other, so the vertices of the icosahedron are the face centroids of the dodecahedron (and vice-versa). The second image below highlights the relation of the dodecahedron with the golden rectangles, whose vertices outline a regular icosahedron.

small stellated dodecahedron inside the icosahedron

Twelve pentagramatic facets allows us to obtain the small stellated dodecahedron:

great dodecahedron inside the icosahedron

Twelve pentagonal facets allows us to obtain the great dodecahedron:

great icosahedron inside the icosahedron

Twenty triangular facets allows us to obtain the great icosahedron:

References:

  1. Coxeter, H. (1973). Regular Polytopes. New York: Dover Publications.
  2. Cromwell, P. 1997. Polyhedra. Cambridge, U.K.: Cambridge University Press.
  3. Pugh, A. 1976. Polyhedra: A visual approach. Berkeley: University of California Press.
  4. Weisstein, E. n/d. “Polyhedron Compound.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedronCompound.html
  5. Wenninger, M. (1975). Polyhedron models. Cambridge: University Press.
  6. 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.