Icosahedron Stellations

Applying the stellation process to a regular icosahedron gives

cells of 10 different shapes and sizes (Wenninger 1989, p. 41).

After application of five restrictions known as Miller's rules to define which forms should be considered significant and distinct (Coxeter et al. 1999, pp. 15-16), 59 stellations (including the original regular icosahedron itself) are possible (Coxeter et al. 1999). These stellations are illustrated above are given in the original ordering of Maeder (1994).

18 of the 59 stellations are fully supported. Of these, 16 are reflexible and 2 are chiral (Webb).

Of the 59 icosahedron stellations, 32 have full icosahedral symmetry and 27 are enantiomeric forms (Coxeter et al. 1999, pp. 64-65). One is a Platonic solid (the icosahedron itself), one is a Kepler-Poinsot polyhedron, four are polyhedron compounds, and one is the dual polyhedron of an Archimedean solid. Note that the first real stellation (stellation #2 in Coxeter's counting) is that obtained by cumulating the icosahedron until the faces of each triangular pyramid lie parallel to the surrounding original faces. This gives fairly small spikes, and results in a solid known as the small triambic icosahedron. Note also that the great stellated dodecahedron is not an icosahedron stellation since the faces of its groups of five triangular pyramids do not lie in the same plane (even though they appear very close to it).

The original icosahedron, its 20 facial planes, and the intersections of those planes with the facial plane of the "top" face are illustrated above.

The stellation diagram showing the 18 lines of intersections of one face with the 18 other nonparallel faces is shown above, together with the regions into which these intersections divide the plane (Coxeter et al. 1999, p. 17).

The numbering of the regions can be simplified by writing for both 2 and , for 4 and , for 11 and , and for all of 13, , and (Coxeter et al. 1999, pp. 18-19).

The following table (extending Coxeter et al. 1999, pp. 13 and 64-65) summarizes different orderings and notations used by a number of authors. The newer Maeder numbering ("Stellated Icosahedra" web page, "Fifty-Nine Icosahedra" Demonstration) orders by increasing circumradius . Rogers ("Playing with Stellations of the Icosahedron" Demonstration) uses the ordering and §3 notation of Coxeter et al. (1999). Stellations with Coxeter index 1-32 are amphichiral, while those with index 33-59 are chiral stellations listed in dextro form. To obtain the laevo form, change Roman to italic and vice versa in both "§2" and "§3" notations (Coxeter et al. 1999, pp. 64-65).

MathWorld name	Coxeter	Coxeter §2	Coxeter §3	Coxeter plate	Coxeter name	Wenninger	Wenninger name	Wheeler	Brückner	Maeder	Maeder
regular icosahedron	1			I	icosahedron	4		1		1	1
small triambic icosahedron	2			I	triakisicosahedron	26	triakis icosahedron	2	Fig. 2, Taf. VIII	2	2
octahedron 5-compound	3			I	five octahedra	23		3	Fig. 6, Taf. IX	3	3
	4			I				4	Fig. 17, Taf. IX	5	4
	5			II						16	9
	6			II		27	2nd stellation	19		21	35
great icosahedron	7			II	great icosahedron	41	great icosahedron	11	Fig. 24, Taf. XI	11	54
echidnahedron	8			III	complete stellation	42	final stellation	12	Fig. 14, Taf. XI	4	59
	9			IV		37	12th stellation			6	7
	10			IV						30	14
	11			IV		29	4th stellation	21		10	24
	12			V						24	10
	13			V				20		26	20
	14			V						32	23
	15			VI						14	6
	16			VI				22		8	33
	17			VI						22	55
	18			VII						15	32
	19			VII						28	44
	20			VII		30	5th stellation			19	45
	21			VIII		32	7th stellation	10		7	8
tetrahedron 10-compound	22			VIII		25	compound of ten tetrahedra	8	Fig. 3, Taf. IX	18	12
	23			VIII		31	6th stellation	17	Fig. 3, Taf. X	23	39
	24			IX						27	11
	25			IX						29	21
inwardly augmented dodecahedron	26			IX				9	Fig. 26, Taf. VIII	20	22
	27			X		28	3rd stellation	5		12	5
	28			X				18	Fig. 20, Taf. IX	17	34
	29			X		33	8th stellation	14		9	47
	30			XI		34	9th stellation	13		13	31
	31			XI						25	43
	32			XI						31	46
	33			XII		35	10th stellation			33	13
	34			XII		36	11th stellation			34	15
	35			XII						35	16
	36			XIII						39	25
	37			XIII		39	14th stellation			45	26
	38			XIII						47	27
	39			XIV						50	56
	40			XIV						54	57
	41			XIV						58	58
	42			XV						48	51
	43			XV						52	52
	44			XV						56	53
	45			XVI		40	15th stellation			42	19
	46			XVI						40	18
tetrahedron 5-compound (dextro)	47			XVI	five tetrahedra	24	compound of five tetrahedra	7		36	17
tetrahedron 5-compound (laevo)								6	Fig. 11, Taf. IX
	48			XVII						57	30
	49			XVII						53	29
	50			XVII						49	28
	51			XVIII		38	13th stellation			43	38
	52			XVIII						41	37
(dextro)	53			XVIII				15		37	36
(laevo)								16
	54			XIX						59	42
	55			XIX						55	41
	56			XIX						51	40
	57			XX						46	50
	58			XX						44	49
	59			XX						38	48

Coxeter stellation number 30 () corresponds to the hull of the medial triambic icosahedron and great triambic icosahedron (Wenninger 1983, pp. 45-50).