inside the dodecahedron

The sequences of images in this webpage shows how the convex regular polyhedra (aka platonic solids), one concave regular polyhedra and polyhedron compounds (of the convex regular) are obtained from the pentagonal dodecahedron. The great stellated dodecahedron is also obtainable through stellation, but here we obtained it through faceting.

Regular facets of the dodecahedron

From the vertices of the dodecahedron, it is possible to obtain six types of regular facets, that are shown below: smaller triangles and larger triangles, squares, smaller pentagrams and larger pentagrams, and pentagons:

Tetrahedra inside the dodecahedron

Up to 10 tetrahedra are inscribable inside the dodecahedron, if we take the larger triangular facets shown above as their faces. Inside the pentagonal dodecahedron, we also find: five stella octangulas, two compounds of five tetrahedra (the second, with the enantiomers of the first) and the compound of ten tetrahedra. All of these are equally obtainable from the icosahedron, as shown here:

Cubes inside the dodecahedron

From the squared facets of the dodecahedron, five different cubes are obtainable, all of which are shown below, as well as the compound of five cubes that is shown below and also here, obtained from the icosahedron.

Octahedra inside the dodecahedron

Octahedra are also inscribable within the dodecahedron, if the midpoints of given edges are taken as their vertices. There are five possible ways to inscribe an octahedron inside the dodecahedron, as well as the compound of five octahedra, all of which are shown in the images below. The same compound is obtainable from the icosahedron, as shown here.

Icosahedron inside the dodecahedron

The icosahedron and the dodecahedron are diuals 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 the regular icosahedron.

great stellated dodecahedron inside the dodecahedron

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

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.