List of Wenninger polyhedron models

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This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.

The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes.

It contains the 75 nonprismatic uniform polyhedra, as well as 44 stellated forms of the convex regular polyhedra.

This list was written to honor this early polyhedral work from Wenninger, and to provide a detailed reference to the 119 numbered models in his book.

Models listed here can be cited as "Wenninger Model Number N", or W_N for brevity.

The polyhedra are grouped below in 5 tables: Regular (1-5), Semiregular (6-18), regular star polyhedra (20-22,41), Stellations and compounds (19-66), and uniform star polyhedra (67-119). The four regular star polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings.

Contents 1 Platonic solids (Regular) W1 to W5 2 Archimedean solids (Semiregular) W6 to W18 3 Kepler-Poinsot solids (Regular star polyhedra) W20,W21,W22 and W41 4 Stellations: models W19 to W66 4.1 Stellations of octahedron 4.2 Stellations of dodecahedron 4.3 Stellations of icosahedron 4.4 Stellations of cuboctahedron 4.5 Stellations of icosidodecahedron 5 Uniform nonconvex solids W67 to W119 6 References 7 External links

[edit] Platonic solids (Regular) W1 to W5

Index	Name	Picture	Wythoff symbol	Vertex figure and Schläfli symbol	Symmetry group	U#	K#	V	E	F	Faces by type
1	Tetrahedron		3|2 3	{3,3}	Td	U01	K06	4	6	4	4{3}
2	Octahedron		4|2 3	{3,4}	Oh	U05	K10	6	12	8	8{3}
3	Hexahedron (Cube)		3|2 4	{4,3}	Oh	U06	K11	8	12	6	6{4}
4	Icosahedron		5|2 3	{3,5}	Ih	U22	K27	12	30	20	20{3}
5	Dodecahedron		3|2 5	{5,3}	Ih	U23	K28	20	30	12	12{5}

[edit] Archimedean solids (Semiregular) W6 to W18

Index	Name	Picture	Wythoff symbol	Vertex figure	Symmetry group	U#	K#	V	E	F	Faces by type
6	Truncated tetrahedron		2 3|3	3.6.6	Td	U02	K07	12	18	8	4{3}+4{6}
7	Truncated octahedron		2 4|3	4.6.6	Oh	U08	K13	24	36	14	6{4}+8{6}
8	Truncated hexahedron		2 3|4	3.8.8	Oh	U09	K14	24	36	14	8{3}+6{8}
9	Truncated icosahedron		2 5|3	5.6.6	Ih	U25	K30	60	90	32	12{5}+20{6}
10	Truncated dodecahedron		2 3|5	3.10.10	Ih	U26	K31	60	90	32	20{3}+12{10}
11	Cuboctahedron		2|3 4	3.4.3.4	Oh	U07	K12	12	24	14	8{3}+6{4}
12	Icosidodecahedron		2|3 5	3.5.3.5	Ih	U24	K29	30	60	32	20{3}+12{5}
13	Small rhombicuboctahedron		3 4|2	3.4.4.4	Oh	U10	K15	24	48	26	8{3}+(6+12){4}
14	Small rhombicosidodecahedron		3 5|2	3.4.5.4	Ih	U27	K32	60	120	62	20{3}+30{4}+12{5}
15	Great rhombicuboctahedron (Rhombitruncated cuboctahedron) (Truncated cuboctahedron)		2 3 4|	4.6.8	Oh	U11	K16	48	72	26	12{4}+8{6}+6{8}
16	Great rhombicosidodecahedron (Rhombitruncated icosidodecahedron) (Truncated icosidodecahedron)		2 3 5|	4.6.10	Ih	U28	K33	120	180	62	30{4}+20{6}+12{10}
17	Snub cube		|2 3 4	3.3.3.3.4	O	U12	K17	24	60	38	(8+24){3}+6{4}
18	Snub dodecahedron		|2 3 5	3.3.3.3.5	I	U29	K34	60	150	92	(20+60){3}+12{5}

[edit] Kepler-Poinsot solids (Regular star polyhedra) W20,W21,W22 and W41

Index	Name	Picture	Wythoff symbol	Vertex figure and Schläfli symbol	Symmetry group	U#	K#	V	E	F	Faces by type
20	Small stellated dodecahedron		5|25/2	{5/2,5}	Ih	U34	K39	12	30	12	12{5/2}
21	Great dodecahedron		5/2|2 5	{5,5/2}	Ih	U35	K40	12	30	12	12{5}
22	Great stellated dodecahedron		3|25/2	{5/2,3}	Ih	U52	K57	20	30	12	12{5/2}
41	Great icosahedron (16th stellation of icosahedron)		5/2|2 3	{3,5/2}	Ih	U53	K58	12	30	20	20{3}

[edit] Stellations: models W19 to W66

[edit] Stellations of octahedron

Index	Name	Symmetry group	Picture	Facets
2	Octahedron (regular)	Oh
19	Stellated octahedron (Compound of two tetrahedra)	Oh

[edit] Stellations of dodecahedron

Index	Name	Symmetry group	Picture	Facets
5	Dodecahedron (regular)	Ih
20	Small stellated dodecahedron (regular) (First stellation of dodecahedron)	Ih
21	Great dodecahedron (regular) (Second stellation of dodecahedron)	Ih
22	Great stellated dodecahedron (regular) (Third stellation of dodecahedron)	Ih

[edit] Stellations of icosahedron

Index	Name	Symmetry group	Picture	Facets
4	Icosahedron (regular)	Ih
26	First stellation of icosahedron (Triakis icosahedron)	Ih
23	Compound of five octahedra (First compound stellation of icosahedron)	Ih
24	Compound of five tetrahedra (Second compound stellation of icosahedron)	I
25	Compound of ten tetrahedra (Third compound stellation of icosahedron)	Ih
27	Second stellation of icosahedron	Ih
28	Third stellation of icosahedron	Ih
29	Fourth stellation of icosahedron	Ih
30	Fifth stellation of icosahedron	Ih
31	Sixth stellation of icosahedron	Ih
32	Seventh stellation of icosahedron	Ih
33	Eighth stellation of icosahedron	Ih
34	Ninth stellation of icosahedron	Ih
35	Tenth stellation of icosahedron	I
36	Eleventh stellation of icosahedron	I
37	Twelfth stellation of icosahedron	Ih
38	Thirteenth stellation of icosahedron	I
39	Fourteenth stellation of icosahedron	I
40	Fifteenth stellation of icosahedron	I
41	Great icosahedron (regular) (Sixteenth stellation of icosahedron)	Ih
42	Final stellation of the icosahedron	Ih

[edit] Stellations of cuboctahedron

Index	Name	Symmetry group	Picture	Facets (octahedral planes)	Facets (cube planes)
11	Cuboctahedron (regular)	Oh
43	Compound of cube and octahedron (First stellation of cuboctahedron)	Oh
44	Second stellation of cuboctahedron	Oh
45	Third stellation of cuboctahedron	Oh
46	Fourth stellation of cuboctahedron	Oh

[edit] Stellations of icosidodecahedron

Index	Name	Symmetry group	Picture	Facets (icosahedral planes)	Facets (dodecahedral planes)
12	Icosidodecahedron (regular)	Ih
47	(First stellation of icosidodecahedron) Compound of dodecahedron and icosahedron	Ih
48	Second stellation of icosidodecahedron	Ih
49	Third stellation of icosidodecahedron	Ih
50	Fourth stellation of icosidodecahedron (Compound of small stellated dodecahedron and triakis icosahedron)	Ih
51	Fifth stellation of icosidodecahedron (Compound of small stellated dodecahedron and five octahedra)	Ih
52	Sixth stellation of icosidodecahedron	Ih
53	Seventh stellation of icosidodecahedron	Ih
54	Eighth stellation of icosidodecahedron (Compound of five tetrahedra and great dodecahedron)	I
55	Ninth stellation of icosidodecahedron	Ih
56	Tenth stellation of icosidodecahedron	Ih
57	Eleventh stellation of icosidodecahedron	Ih
58	Twelfth stellation of icosidodecahedron	Ih
59	Thirteenth stellation of icosidodecahedron	Ih
60	Fourteenth stellation of icosidodecahedron	Ih
61	Compound of great stellated dodecahedron and great icosahedron	Ih
62	Fifteenth stellation of icosidodecahedron	Ih
63	Sixteenth stellation of icosidodecahedron	Ih
64	Seventeenth stellation of icosidodecahedron	Ih
65	Eighteenth stellation of icosidodecahedron	Ih
66	Nineteenth stellation of icosidodecahedron	Ih

[edit] Uniform nonconvex solids W67 to W119

Index	Name	Picture	Wythoff symbol	Vertex figure	Symmetry group	U#	K#	V	E	F	Faces by type
67	Tetrahemihexahedron		3/23|2	4.3/2.4.3	Td	U04	K09	6	12	7	4{3}+3{4}
68	Octahemioctahedron		3/23|3	6.3/2.6.3	Oh	U03	K08	12	24	12	8{3}+4{6}
69	Small cubicuboctahedron		3/24|4	8.3/2.8.4	Oh	U13	K18	24	48	20	8{3}+6{4}+6{8}
70	Small ditrigonal icosidodecahedron		3|5/23	(5/2.3)3	Ih	U30	K35	20	60	32	20{3}+12{5/2}
71	Small icosicosidodecahedron		5/23|3	6.5/2.6.3	Ih	U31	K36	60	120	52	20{3}+12{5/2}+20{6}
72	Small dodecicosidodecahedron		3/25|5	10.3/2.10.5	Ih	U33	K38	60	120	44	20{3}+12{5}+12{10}
73	Dodecadodecahedron		2|5/25	(5/2.5)2	Ih	U36	K41	30	60	24	12{5}+12{5/2}
74	Small rhombidodecahedron		25/25|	10.4.10/9.4/3	Ih	U39	K44	60	120	42	30{4}+12{10}
75	Truncated great dodecahedron		25/2|5	10.10.5/2	Ih	U37	K42	60	90	24	12{5/2}+12{10}
76	Rhombidodecadodecahedron		5/25|2	4.5/2.4.5	Ih	U38	K43	60	120	54	30{4}+12{5}+12{5/2}
77	Great cubicuboctahedron		3 4|4/3	8/3.3.8/3.4	Oh	U14	K19	24	48	20	8{3}+6{4}+6{8/3}
78	Cubohemioctahedron		4/34|3	6.4/3.6.4	Oh	U15	K20	12	24	10	6{4}+4{6}
79	Cubitruncated cuboctahedron (Cuboctatruncated cuboctahedron)		4/33 4|	8/3.6.8	Oh	U16	K21	48	72	20	8{6}+6{8}+6{8/3}
80	Ditrigonal dodecadodecahedron		3|5/35	(5/3.5)3	Ih	U41	K46	20	60	24	12{5}+12{5/2}
81	Great ditrigonal dodecicosidodecahedron		3 5|5/3	10/3.3.10/3.5	Ih	U42	K47	60	120	44	20{3}+12{5}+12{10/3}
82	Small ditrigonal dodecicosidodecahedron		5/33|5	10.5/3.10.3	Ih	U43	K48	60	120	44	20{3}+12{5/2}+12{10}
83	Icosidodecadodecahedron		5/35|3	6.5/3.6.5	Ih	U44	K49	60	120	44	12{5}+12{5/2}+20{6}
84	Icositruncated dodecadodecahedron (Icosidodecatruncated icosidodecahedron)		5/33 5|	10/3.6.10	Ih	U45	K50	120	180	44	20{6}+12{10}+12{10/3}
85	Uniform great rhombicuboctahedron (Quasirhombicuboctahedron)		3/24|2	4.3/2.4.4	Oh	U17	K22	24	48	26	8{3}+(6+12){4}
86	Small rhombihexahedron		3/22 4|	4.8.4/3.8	Oh	U18	K23	24	48	18	12{4}+6{8}
87	Great ditrigonal icosidodecahedron		3/2|3 5	(5.3.5.3.5.3)/2	Ih	U47	K52	20	60	32	20{3}+12{5}
88	Great icosicosidodecahedron		3/25|3	6.3/2.6.5	Ih	U48	K53	60	120	52	20{3}+12{5}+20{6}
89	Small icosihemidodecahedron		3/23|5	10.3/2.10.3	Ih	U49	K54	30	60	26	20{3}+6{10}
90	Small dodecicosahedron		3/23 5|	10.6.10/9.6/5	Ih	U50	K55	60	120	32	20{6}+12{10}
91	Small dodecahemidodecahedron		5/45|5	10.5/4.10.5	Ih	U51	K56	30	60	18	12{5}+6{10}
92	Stellated truncated hexahedron (Quasitruncated hexahedron)		2 3|4/3	8/3.8/3.3	Oh	U19	K24	24	36	14	8{3}+6{8/3}
93	Great truncated cuboctahedron (Quasitruncated cuboctahedron)		4/32 3|	8/3.4.6	Oh	U20	K25	48	72	26	12{4}+8{6}+6{8/3}
94	Great icosidodecahedron		2|5/23	(5/2.3)2	Ih	U54	K59	30	60	32	20{3}+12{5/2}
95	Truncated great icosahedron		25/2|3	6.6.5/2	Ih	U55	K60	60	90	32	12{5/2}+20{6}
96	Rhombicosahedron		25/23|	6.4.6/5.4/3	Ih	U56	K61	60	120	50	30{4}+20{6}
97	Small stellated truncated dodecahedron (Quasitruncated small stellated dodecahedron)		2 5|5/3	10/3.10/3.5	Ih	U58	K63	60	90	24	12{5}+12{10/3}
98	Truncated dodecadodecahedron (Quasitruncated dodecahedron)		5/32 5|	10/3.4.10	Ih	U59	K64	120	180	54	30{4}+12{10}+12{10/3}
99	Great dodecicosidodecahedron		5/23|5/3	10/3.5/2.10/3.3	Ih	U61	K66	60	120	44	20{3}+12{5/2}+12{10/3 }
100	Small dodecahemicosahedron		5/35/2|3	6.5/3.6.5/2	Ih	U62	K67	30	60	22	12{5/2}+10{6}
101	Great dodecicosahedron		5/35/23|	6.10/3.6/5.10/7	Ih	U63	K68	60	120	32	20{6}+12{10/3}
102	Great dodecahemicosahedron		5/45|3	6.5/4.6.5	Ih	U65	K70	30	60	22	12{5}+10{6}
103	Great rhombihexahedron		4/33/22|	4.8/3.4/3.8/5	Oh	U21	K26	24	48	18	12{4}+6{8/3}
104	Great stellated truncated dodecahedron (Quasitruncated great stellated dodecahedron)		2 3|5/3	10/3.10/3.3	Ih	U66	K71	60	90	32	20{3}+12{10/3}
105	Uniform great rhombicosidodecahedron (Quasirhombicosidodecahedron)		5/33|2	4.5/3.4.3	Ih	U67	K72	60	120	62	20{3}+30{4}+12{5/2}
106	Great icosihemidodecahedron		3 3|5/3	10/3.3/2.10/3.3	Ih	U71	K76	30	60	26	20{3}+6{10/3}
107	Great dodecahemidodecahedron		5/35/2|5/3	10/3.5/3.10/3.5/2	Ih	U70	K75	30	60	18	12{5/2}+6{10/3}
108	Great truncated icosidodecahedron (Great quasitruncated icosidodecahedron)		5/32 3|	10/3.4.6	Ih	U68	K73	120	180	62	30{4}+20{6}+12{10/3}
109	Great rhombidodecahedron		3/25/32|	4.10/3.4/3.10/7	Ih	U73	K78	60	120	42	30{4}+12{10/3}
110	Small snub icosicosidodecahedron		|5/23 3	3.3.3.3.3.5/2	Ih	U32	K37	60	180	112	(40+60){3}+12{5/2}
111	Snub dodecadodecahedron		|25/25	3.3.5/2.3.5	I	U40	K45	60	150	84	60{3}+12{5}+12{5/2}
112	Snub icosidodecadodecahedron		|5/33 5	3.3.3.3.5.5/3	I	U46	K51	60	180	104	(20+6){3}+12{5}+12{5/2}
113	Great inverted snub icosidodecahedron		|5/32 3	3.3.3.3.5/3	I	U69	K74	60	150	92	(20+60){3}+12{5/2}
114	Inverted snub dodecadodecahedron		|5/32 5	3.5/3.3.3.5	I	U60	K65	60	150	84	60{3}+12{5}+12{5/2}
115	Great snub dodecicosidodecahedron		|5/35/23	3.5/3.3.5/2.3.3	I	U64	K69	60	180	104	(20+60){3}+(12+12){5/2}
116	Great snub icosidodecahedron		|25/25/2	3.3.3.3.5/2	I	U57	K62	60	150	92	(20+60){3}+12{5/2}
117	Great retrosnub icosidodecahedron		|3/25/32	(3.3.3.3.5/3)/2	I	U74	K79	60	150	92	(20+60){3}+12{5/2}
118	Small retrosnub icosicosidodecahedron		|3/23/25/2	(3.3.3.3.3.5/2)/2	Ih	U72	K77	180	60	112	(40+60){3}+12{5/2}
119	Great dirhombicosidodecahedron		|3/25/335/2	(4.5/3.4.3.4.5/2.4.3/2)/2	Ih	U75	K80	60	240	124	40{3}+60{4}+24{5/2}

[edit] References

• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. 
  • Errata
    • In Wenninger, the vertex figure for W90 is incorrectly shown as having parallel edges.
• Wenninger, Magnus (1979). Spherical Models. Cambridge University Press. ISBN 0521294320. 

[edit] External links

• Magnus J. Wenninger
• Software used to generate images in this article:
  • Stella: Polyhedron Navigator Stella (software) - Can create and print nets for all of Wenninger's polyhedron models.
  • Vladimir Bulatov's Polyhedra Stellations Applet
• Gallery of Polyhedron Models
• M. Wenninger, Polyhedron Models, Errata: known errors in the various editions.