List of Wenninger polyhedron models

From Wikipedia, the free encyclopedia

You have new messages (last change).

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 WN 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

[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] Stellation 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

[edit] External links

Retrieved from "http://en.wikipedia.org../../../l/i/s/List_of_Wenninger_polyhedron_models_c02f.html"
In other languages