Stellating the Icosahedron and Facetting the Dodecahedron
Updated 10 May 2022
The regular icosahedron and dodecahedron are two of the five Platonic solids. The stellated icosahedra and facetted dodecahedra are related by polar reciprocation - for every stellated icosahedron there is a twin facetted dodecahedron. The stellated icosahedra were thought to have been understood over sixty years ago, but I have been taking a fresh look. The facetted dodecahedra appear not to have been studied systematically until Bridge in the early 1970's.
For more about stellation and facetting in general, see my stellation and facetting pages.
A series of papers tackling the subject, broadly in logical order.
- In search of the lost icosahedra
Substantially as in The Mathematical Gazette 86, July 2002, p.p. 208-215.
There probably aren't exactly 59 stellations after all. Starts with a critique of The fifty-nine icosahedra and especially Miller's rules, presents some counter-examples including four lost icosahedra, examines some new but unsatisfactory rules, and grinds to a halt over problems with holes. First in a continuing series of investigations. The text is reproduced by kind permission, with some minor updates.
- Towards stellating the icosahedron and facetting the dodecahedron
Substantially as in Symmetry: culture and science Vol. 11, 1-4 (2000) pp. 269-291.
Consideration of the rules for stellation leads to investigation of the duality of stellations and facettings. The facetting diagram of the regular dodecahedron and a new complete stellation diagram of the icosahedron are presented. Cell sets are inappropriate for analysis, but useful for determining visual appearance; a new notation for stellations is described. We find 'untidy' polyhedra which extend to infinity, have holes in, or have n-methoric or n-synaptic edges.
- Tidy dodecahedra and icosahedra
Bridge's 1974 paper is revisited. Tidy facets of the regular dodecahedron are listed. Tidy facettings, and their reciprocal stellations of the icosahedron, are enumerated, including a rediscovered lost stellation and some infinite ones. The stellations are compared with the fully suppported stellations. Du Val's notation for the stellated icosahedra is extended.
- Icosahedral precursors – Unfinished
Precursors provide a way to describe and identify the stellations of the regular icosahedron and their reciprocal facettings of the dodecahedron. Some theoretical foundations are presented. Some problems with untidy polyhedra are discussed, and yet another lost stellation noted. Revised rules for identifying stellations and facettings based on precursors are proposed. A good idea, but subject to radical revision.
Images of some lost icosahedra
Here are some assorted images that don't fit the more organised format of the main list of lost stellations.
EpdEplEnGn is a very elegant icosahedron with its twenty dodecagonal faces. It was noted by N. J. Bridge in 1974 (as D+f2). For more about this, see Tidy dodecahedra and icosahedra. I believe that I am the first to image it; as with all stellations of the regular icosahedron the golden ratio of Classical art is everywhere. In the background of my idealised landscape hovers the ghostly dual – a facetting of the regular dodecahedron – and serene over all is the Platonic icosahedron itself. The artwork appears on Page 202 of Symmetry: culture and science, 13 Nos. 1-2 (2002).
This is one of the lost icosahedra from In search of the lost icosahedra.
The 3D image was created on a RISC OS computer using
!Stellate and saved
as a RISC OS drawfile. It was imported to Xara X on a Windows PC for prettying-up and conversion to jpeg (I could have done all this on my Acorn, but I wanted to try out Xara X. It's great). Dave Crennell of
produced the ray-traced pictures on a modern RISC OS computer using POVray.
RISC OS (Acorn) PolyData files
To view and manipulate these 3D data files you will need !PolyDraw from Fortran Friends running on either an Acorn compatible PC with RISC OS 3.1 or above, or on a standard Windows PC with VirtualAcorn.
Other stellation and facetting resources