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List of Wenninger polyhedron models

From Wikipedia, the free encyclopedia

This is 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 and quasiregular polyhedra.

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

The polyhedra are grouped 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.

Platonic solids (regular convex polyhedra) W1 to W5

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

Archimedean solids (Semiregular) W6 to W18

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

Kepler–Poinsot polyhedra (Regular star polyhedra) W20, W21, W22 and W41

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

Stellations: models W19 to W66

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Stellations of octahedron

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

Stellations of dodecahedron

[edit]
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

Stellations of icosahedron

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Index Name Symmetry group Picture Facets
4 Icosahedron (regular) 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
26 Small triambic icosahedron
(First stellation of icosahedron)
(Triakis icosahedron) 		Ih
27 Second stellation of icosahedron Ih
28 Excavated dodecahedron
(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
Great triambic 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

Stellations of cuboctahedron

[edit]
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

Stellations of icosidodecahedron

[edit]
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

Uniform nonconvex solids W67 to W119

[edit]
Index Name Picture Dual name Dual picture Wythoff symbol Vertex figure Symmetry group U# K# V E F Faces by type
67 Tetrahemihexahedron Tetrahemihexacron 3/23|2
4.3/2.4.3		 Td U04 K09 6 12 7 4{3}+3{4}
68 Octahemioctahedron Octahemioctacron 3/23|3
6.3/2.6.3		 Oh U03 K08 12 24 12 8{3}+4{6}
69 Small cubicuboctahedron Small hexacronic icositetrahedron 3/24|4
8.3/2.8.4		 Oh U13 K18 24 48 20 8{3}+6{4}+6{8}
70 Small ditrigonal icosidodecahedron Small triambic icosahedron 3|5/23
(5/2.3)3		 Ih U30 K35 20 60 32 20{3}+12{5/2}
71 Small icosicosidodecahedron Small icosacronic hexecontahedron 5/23|3
6.5/2.6.3		 Ih U31 K36 60 120 52 20{3}+12{5/2}+20{6}
72 Small dodecicosidodecahedron Small dodecacronic hexecontahedron 3/25|5
10.3/2.10.5		 Ih U33 K38 60 120 44 20{3}+12{5}+12{10}
73 Dodecadodecahedron Medial rhombic triacontahedron 2|5/25
(5/2.5)2		 Ih U36 K41 30 60 24 12{5}+12{5/2}
74 Small rhombidodecahedron Small rhombidodecacron 25/25|
10.4.10/9.4/3		 Ih U39 K44 60 120 42 30{4}+12{10}
75 Truncated great dodecahedron Small stellapentakis dodecahedron 25/2|5
10.10.5/2		 Ih U37 K42 60 90 24 12{5/2}+12{10}
76 Rhombidodecadodecahedron Medial deltoidal hexecontahedron 5/25|2
4.5/2.4.5		 Ih U38 K43 60 120 54 30{4}+12{5}+12{5/2}
77 Great cubicuboctahedron Great hexacronic icositetrahedron 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 Hexahemioctacron 4/34|3
6.4/3.6.4		 Oh U15 K20 12 24 10 6{4}+4{6}
79 Cubitruncated cuboctahedron
(Cuboctatruncated cuboctahedron)		 Tetradyakis hexahedron 4/33 4|
8/3.6.8		 Oh U16 K21 48 72 20 8{6}+6{8}+6{8/3}
80 Ditrigonal dodecadodecahedron Medial triambic icosahedron 3|5/35
(5/3.5)3		 Ih U41 K46 20 60 24 12{5}+12{5/2}
81 Great ditrigonal dodecicosidodecahedron Great ditrigonal dodecacronic hexecontahedron 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 Small ditrigonal dodecacronic hexecontahedron 5/33|5
10.5/3.10.3		 Ih U43 K48 60 120 44 20{3}+12{5/2}+12{10}
83 Icosidodecadodecahedron Medial icosacronic hexecontahedron 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)		 Tridyakis icosahedron 5/33 5|
10/3.6.10		 Ih U45 K50 120 180 44 20{6}+12{10}+12{10/3}
85 Nonconvex great rhombicuboctahedron
(Quasirhombicuboctahedron)		 Great deltoidal icositetrahedron 3/24|2
4.3/2.4.4		 Oh U17 K22 24 48 26 8{3}+(6+12){4}
86 Small rhombihexahedron Small rhombihexacron 3/22 4|
4.8.4/3.8		 Oh U18 K23 24 48 18 12{4}+6{8}
87 Great ditrigonal icosidodecahedron Great triambic icosahedron 3/2|3 5
(5.3.5.3.5.3)/2		 Ih U47 K52 20 60 32 20{3}+12{5}
88 Great icosicosidodecahedron Great icosacronic hexecontahedron 3/25|3
6.3/2.6.5		 Ih U48 K53 60 120 52 20{3}+12{5}+20{6}
89 Small icosihemidodecahedron Small icosihemidodecacron 3/23|5
10.3/2.10.3		 Ih U49 K54 30 60 26 20{3}+6{10}
90 Small dodecicosahedron Small dodecicosacron 3/23 5|
10.6.10/9.6/5		 Ih U50 K55 60 120 32 20{6}+12{10}
91 Small dodecahemidodecahedron Small dodecahemidodecacron 5/45|5
10.5/4.10.5		 Ih U51 K56 30 60 18 12{5}+6{10}
92 Stellated truncated hexahedron
(Quasitruncated hexahedron)		 Great triakis octahedron 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)		 Great disdyakis dodecahedron 4/32 3|
8/3.4.6		 Oh U20 K25 48 72 26 12{4}+8{6}+6{8/3}
94 Great icosidodecahedron Great rhombic triacontahedron 2|5/23
(5/2.3)2		 Ih U54 K59 30 60 32 20{3}+12{5/2}
95 Truncated great icosahedron Great stellapentakis dodecahedron 25/2|3
6.6.5/2		 Ih U55 K60 60 90 32 12{5/2}+20{6}
96 Rhombicosahedron Rhombicosacron 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)		 Great pentakis 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)		 Medial disdyakis triacontahedron 5/32 5|
10/3.4.10		 Ih U59 K64 120 180 54 30{4}+12{10}+12{10/3}
99 Great dodecicosidodecahedron Great dodecacronic hexecontahedron 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 Small dodecahemicosacron 5/35/2|3
6.5/3.6.5/2		 Ih U62 K67 30 60 22 12{5/2}+10{6}
101 Great dodecicosahedron Great dodecicosacron 5/35/23|
6.10/3.6/5.10/7		 Ih U63 K68 60 120 32 20{6}+12{10/3}
102 Great dodecahemicosahedron Great dodecahemicosacron 5/45|3
6.5/4.6.5		 Ih U65 K70 30 60 22 12{5}+10{6}
103 Great rhombihexahedron Great rhombihexacron 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)		 Great triakis icosahedron 2 3|5/3
10/3.10/3.3		 Ih U66 K71 60 90 32 20{3}+12{10/3}
105 Nonconvex great rhombicosidodecahedron
(Quasirhombicosidodecahedron)		 Great deltoidal hexecontahedron 5/33|2
4.5/3.4.3		 Ih U67 K72 60 120 62 20{3}+30{4}+12{5/2}
106 Great icosihemidodecahedron Great icosihemidodecacron 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 Great dodecahemidodecacron 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)		 Great disdyakis triacontahedron 5/32 3|
10/3.4.6		 Ih U68 K73 120 180 62 30{4}+20{6}+12{10/3}
109 Great rhombidodecahedron Great rhombidodecacron 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 Small hexagonal hexecontahedron |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 Medial pentagonal hexecontahedron |25/25
3.3.5/2.3.5		 I U40 K45 60 150 84 60{3}+12{5}+12{5/2}
112 Snub icosidodecadodecahedron Medial hexagonal hexecontahedron |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 Great inverted pentagonal hexecontahedron |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 Medial inverted pentagonal hexecontahedron |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 Great hexagonal hexecontahedron |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 Great pentagonal hexecontahedron |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 Great pentagrammic hexecontahedron |3/25/32
(3.3.3.3.5/2)/2		 I U74 K79 60 150 92 (20+60){3}+12{5/2}
118 Small retrosnub icosicosidodecahedron Small hexagrammic hexecontahedron |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 Great dirhombicosidodecacron |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}

See also

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References

[edit]
  • 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 0-521-29432-0.
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