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.
(see Du Val 's notation below)
first stellation B:
second stellation C:
third stellation D:
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).
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.
• Polyhedra by Peter R. Cromwell (pages 263-267), Cambridge University Press, 1997
• http://www.georgehart.com/virtual-polyhedra/stellations-info.html 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)
|February 2004 |