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List of Wenninger polyhedron models

  (Redirected from Stellations of the dodecahedron)

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.

Contents

Platonic solids (regular) W1 to W5Edit

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 W18Edit

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   2 3|3  
3.6.6
Td U02 K07 12 18 8 4{3} + 4{6}
7 Truncated octahedron   tetrakis hexahedron   2 4|3  
4.6.6
Oh U08 K13 24 36 24 6{4} + 8{6}
8 Truncated hexahedron   triakis octahedron   2 3|4  
3.8.8
Oh U09 K14 24 36 14 8{3} + 6{8}
9 Truncated icosahedron   pentakis dodecahedron   2 5|3  
5.6.6
Ih U25 K30 60 90 32 12{5} + 20{6}
10 Truncated dodecahedron   triakis icosahedron   2 3|5  
3.10.10
Ih U26 K31 60 90 32 20{3} + 12{10}
11 Cuboctahedron   rhombic dodecahedron   2|3 4  
3.4.3.4
Oh U07 K12 12 24 14 8{3} + 6{4}
12 Icosidodecahedron   rhombic triacontahedron   2|3 5  
3.5.3.5
Ih U24 K29 30 60 32 20{3} + 12{5}
13 Small rhombicuboctahedron   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   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   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   2 3 5|  
4.6.10
Ih U28 K33 120 180 62 30{4} + 20{6} + 12{10}
17 Snub cube   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   |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 W41Edit

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 W66Edit

Stellations of octahedronEdit

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

Stellations of dodecahedronEdit

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 icosahedronEdit

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 cuboctahedronEdit

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 icosidodecahedronEdit

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 W119Edit

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}

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See alsoEdit

ReferencesEdit

  • 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.

External linksEdit