Let S be a rotation through angle 2*Pi/p , and let A0 be any point not on the axis of S . Then the points Ai = A0^(S^i), i = ...,-2,-1,0,1,2,... are the vertices of a regular polygon {p} , whose sides are the segments A0A1, A1A2, A2A3,... But the polygon can be closed without p being integral; it is merely necessary that the period of S be finite, i.e., that p be rational.

When the rational number p is expressed as a fraction in its lowest terms, we denote its numerator and denominator by np and dp . Thus p =np/dp where np and dp are coprime. The regular polygon {p} is traced out by a moving point which continuously describes equal chords of a fixed circle and returns to its original position after describing np chords and making dp revolutions about the center. The following exhibites some instances of {p} 's:



The general regular polygon {p} can be derived from the convex polygon {np} by either one of two reciprocal processes: stellating and faceting.

In order to stellate a polyhedron, we have to extend its faces symmetrically until they again form a polyhedron. To investigate all possibilities, we consider the set of lines in which the plane of a particular face would be cut by all the other faces (sufficiently extended), and try to select regular polygons bounded by set of these lines.

For a given star-polyhedron or compound, the core is the largest convex solid that can be drawn inside it. And the compound or star-polyhedron may be constructed by stellating its core (which has the same face planes).

Maple currently supports stellation of the five Platonic solids, and the two quasi-regular solids: the cuboctahedron and the icosidodecahedron . To stellate a given polyhedron, use the Maple command stellate(gon,core,n); where gon is the name of the stellated polyhedron to be created, core the core polyhedron, and n a non-negative integer.

For the tetrahedron {3,3} and the cube {4,3} , the only lines are the sides of the face itself. Hence, there is only one possible value of n , namely 0.




The Core: Tetrahedron
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The Core: Hexahedron
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For the octahedron {3,4} , the eight facial planes enclose not only the original octahedron, bug also other portions of space exterior to this octahedron. For a particular face ABC , the faces opposite to those which immediately surround ABC meet the plane in a larger triangle MNP . The eight large triangles so derived form all the faces form the stella octangula. Hence, possible values of n are 0, 1.



The Core: Octahedron
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The First Stellated Octahedron
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Let us now stellate the dodecahedron {5,3} , and denote one face as 11111 . By stellating this pentagon we obtain the pentagram 22222 , which is a face of {5/2,5} . The large pentagon that has the same vertices 22222 is a face of {5,5/2} . By stellating this pentagon we obtain the large pentagram 33333 , which is a face of {5/2,3} . The process now terminates, since the ten lines account for all the other faces of {5,3} , the twelfth face being parallel to 11111 . Hence, possible values of n are 0,1,2,3.




Pattern: #0


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Pattern: #1


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Pattern: #2


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Pattern: #3


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For the case of the icosahedron {3,5} , its stellations are so numerous and complicated that some care is needed in stateting precisely what varieties shall be considered properly significant and distinct.

In The Fifty-Nine Icosahedra , Coxeter, Du Val, Flather,and Petrie developed a complete enumeration of stellated icosahedra by considering the possible faces (first method), and by considering solid cells, and were able to prove that there exists 59 icosahedra based on the following set of five restricted rules suggested by J.C.P. Miller:

  1. The faces must lie in twenty planes, viz, the bouding planes of the regular icosahedron.
  2. All parts composing the faces must be the same in each plane, although they might be quite disconnected.
  3. The parts included in any one plane must have trigonal symmetry, with or without reflection. This secures icosahedral symmetry for the whole solid.
  4. The parts included in any plane must all be "accessible" in the complete solid (i.e., they must be on the "outside").
  5. We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure. But we allow the combination of an enantimorphous pair having no common part.

Of the 59 icosahedra, thirty two with full icosahedral symmetry and twenty-seven enantimorphous forms with twisted apprearance.

The following shows a subset of the fifty-nine stellated icosahedra. Each stellation pattern and its stellated polyhedron are shown. For a complete list, see the 59 icosahedra.



Pattern: #2


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Pattern: #9


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Pattern: #13


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Pattern: #18


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Pattern: #20


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Pattern: #26


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Pattern: #34


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Pattern: #46


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For a complete set of the fifty-nine icosahedra, see the fifty-nine icosahedra (VRML)

It can be proved that only three of the stellated dodecahedra {5/2,5} , {5/5/2}, {5/2,3} and one stellated icosahedron {3,5/2} are regular polyhedra. They are called Kepler-Poinsot polyhedra. The two with star faces - the two stellated dodecahedra - were found by Kepler (1571-1630); the others with regular faces and star vertices - the great icosahedron and the great dodecahedron - by Poinsot (1777-1859). These four polyhedra, together with the five Platonic solids, known to the ancient world form the set of nine regular polyhedra.

The other stellated polyhedra are compound polyhedra.

The same process can be applied to the Archimedean solids; each facial plane must be extended indefinitely to generate cells exterior to the original solid. Using these cells as building blocks, one can form many new solids. A complete enumeration of all the possible stellation is a mathematical question that has yet to be investigated.

The following shows the stellation process being applied to the two Archimedean solids: the cuboctahedron and the icosidodecahedron. The set of polyhedra shown below are five stellated cuboctahedra and five stellated icosidodecahedra. For the set of nineteen stellate icosidodecahedra as described in Polyhedron Models by Magnus J. Wenninger, see stellated icosidodecahedra




Pattern: #0 a


Pattern: #0 b


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Pattern: #1 a


Pattern: #1 b


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Pattern: #2 a


Pattern: #2 b


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Pattern: #3 a


Pattern: #3 b


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Pattern: #4 a


Pattern: #4 b


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Stellations of the cuboctahedron




Pattern: #4 a


Pattern: #4 b


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Pattern: #6 a


Pattern: #6 b


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Pattern: #9 a


Pattern: #9 b


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Pattern: #10 a


Pattern: #10 b


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Pattern: #15 a


Pattern: #15 b


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Stellations of the icosidodecahedron


Author: Ha Le (hle@cecm.sfu.ca)
Create: April 6, 1996. Last Modified: Nov 17, 1998.