theory of the stellation: example of the dodecahedron

The set of planes containing the faces of a convex polyhedron partitions the space into a set of cells. The polyhedron itself is one of these cells (the central core), and there are unbounded cells (ie "infinite") without interest. If the angle between two faces is greater than 90° there are several layers of bounded cells which may be assembled to build new polyhedra.

The regular dodecahedron is surrounded by three layers of bounded cells : 12 golden pentagonal pyramids, then 30 wedges (tetrahedra) which insert themselves between the pyramids, and finally 20 spikes (triangular bipyramids) which fit between the wedges. The animations below show how each layer covers the preceding one and builds a new stellation.

dodecahedron A
(see Du Val 's notation below)

first stellation B:
dodecahedron + 12 pyramids
      = small stellated dodecahedron

second stellation C:
small stellated dodecahedron + 30 wedges
      = great dodecahedron

third stellation D:
great dodecahedron + 20 spikes
      = great stellated dodecahedron

These three animations are drawn at the same scale.

The three stellations of the dodecahedron are non convex regular polyhedra (the great icosahedron is a stellation of the icosahedron).

With the icosahedron things become more complicated : 473 bounded cells of 12 types permit a great number of combinations among which we have to decide which ones are "acceptable".
Miller's rules define a "stellation":
 - the faces must lie in the face-planes of the original polyhedron,
 - the regions composing the faces must be the same in each plane,
 - the regions included in a plane must have the same rotational symmetry as a face of the original polyhedron,
 - the regions included in a plane must be accessible in the completed stellation,
 - compounds of simpler stellations are excluded.
Stellation thus preserves the rotational symmetry of the original polyhedron.
According to these criteria the icosahedron has 59 stellations (see examples on an other page), among them 31 with symmetry planes.

Du Val's notation allows to classify the cells and to describe which ones are used in a given stellation: the successive layers are a, b, c, d...  The upper case letters mean that all the preceding layers are used (example: C=abc for the three first layers).
For the dodecahedron we thus have a (the dodecahedron), b (the 12 pyramids), c (the 30 wedges), and d (the 20 pikes).
With the icosahedron we have eight layers of cells in twelve sets: a, b, c, d, e=e1+e2, f=f1+f2=(f11+f12)+f2, g=g1+g2 and h.
C is the compound of five octahedra, G is the great icosahedron, and H, composed of all the cells, is the complete icosahedron.

references: •  Polyhedra by Peter R. Cromwell (pages 263-267), Cambridge University Press, 1997
• by George W. Hart
•  Stella: Polyhedron Navigator by Robert Webb (published in Symmetry: Culture and Science, vol.11 n°1, 2000)
•  polyhedra stellation applet by Vladimir Bulatov (the program may be downloaded for free)

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updated 04-07-2005