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Encyclopedia > Johnson solid
The elongated square gyrobicupola (J37), a Johnson solid
The elongated square gyrobicupola (J37), a Johnson solid
This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles.)
This 24 square example is not a Johnson solid because it is not strictly convex (has zero-angled dihedral angles.)
This 24 equilateral triangle example is not a Johnson solid because it is not convex.
This 24 equilateral triangle example is not a Johnson solid because it is not convex.

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1); it has one square face and four triangular faces. Johnson solid #37 Another attempt at using user:Cyps image:Poly. ... Johnson solid #37 Another attempt at using user:Cyps image:Poly. ... In geometry, the elongated square gyrobicupola is one of the Johnson solids (J37). ... Image File history File links Download high-resolution version (873x866, 24 KB) I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (873x866, 24 KB) I, the creator of this work, hereby release it into the public domain. ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... Image File history File links Stella_octangula. ... Image File history File links Stella_octangula. ... Stella octangula The stella octangula (eight-pointed star), also known as the stellated octahedron, is the polyhedral compound of two tetrahedra. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Look up Convex set in Wiktionary, the free dictionary. ... A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In geometry, a Platonic solid is a convex regular polyhedron. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... In geometry, an equilateral polygon has all sides of the same length. ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ...


As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J2) is an example that actually has a degree-5 vertex. In geometry, the Pentagonal pyramid is one of the Johnson solids (J2). ...


Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.


In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete. Year 1966 (MCMLXVI) was a common year starting on Saturday (link will display full calendar) of the 1966 Gregorian calendar. ... Norman W. Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. ... Victor (Viktor) Abramovich Zalgaller (Russian: ) (born on December 25, 1920 in Parfino, Novgorod oblast) is a mathematician in the fields of geometry and optimization. ... For the Stargate SG-1 episode, see 1969 (Stargate SG-1). ...


Of the Johnson solids, the elongated square gyrobicupola (J37) is unique in being vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle. In geometry, the elongated square gyrobicupola is one of the Johnson solids (J37). ...

Contents

Names

The names are listed below and are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundae), together with the Platonic and Archimedean solids, prisms, and antiprisms. This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... The square cupola with regular faces In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of triangles and rectangles. ... In Mosta, Malta, the Rotunda of Santa Marija Assunta is covered by a saucer dome. ... In geometry, a Platonic solid is a convex regular polyhedron. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...

  • Bi- means that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, they can be joined so that like faces (ortho-) or unlike faces (gyro-) meet. In this nomenclature, an octahedron would be a square bipyramid, a cuboctahedron would be a triangular gyrobicupola, and an icosidodecahedron would be a pentagonal gyrobirotunda.
  • Elongated means that a prism has been joined to the base of the solid in question or between the bases of the solids in question. A rhombicuboctahedron would be an elongated square orthobicupola.
  • Gyroelongated means that an antiprism has been joined to the base of the solid in question or between the bases of the solids in question. An icosahedron would be a gyroelongated pentagonal bipyramid.
  • Augmented means that a pyramid or cupola has been joined to a face of the solid in question.
  • Diminished means that a pyramid or cupola has been removed from the solid in question.
  • Gyrate means that a cupola on the solid in question has been rotated so that different edges match up, as in the difference between ortho- and gyrobicupolae.

The last three operations — augmentation, diminution, and gyration — can be performed more than once on a large enough solid. We add bi- to the name of the operation to indicate that it has been performed twice. (A bigyrate solid has had two of its cupolae rotated.) We add tri- to indicate that it has been performed three times. (A tridiminished solid has had three of its pyramids or cupolae removed.) An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Gyration is a company that produces an optical mouse known also as the Gyration. ...


Sometimes, bi- alone is not specific enough. We must distinguish between a solid that has had two parallel faces altered and one that has had two oblique faces altered. When the faces altered are parallel, we add para- to the name of the operation. (A parabiaugmented solid has had two parallel faces augmented.) When they are not, we add meta- to the name of the operation. (A metabiaugmented solid has had two oblique faces augmented.)


Enumeration

Prismatoids and rotundae

Jn Solid name Image V E F F3 F4 F5 F6 F8 F10 Symmetry
1 Square pyramid 5 8 5 4 1 C4v
2 Pentagonal pyramid 6 10 6 5 1 C5v
3 Triangular cupola 9 15 8 4 3 1 C3v
4 Square cupola 12 20 10 4 5 1 C4v
5 Pentagonal cupola 15 25 12 5 5 1 1 C5v
6 Pentagonal rotunda 20 35 17 10 6 1 C5v

A prismatoid is a polyhedron where all vertices lie in two parallel planes. ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... The square cupola with regular faces In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of triangles and rectangles. ... In geometry, the Pentagonal rotunda is one of the Johnson solids (J6). ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ... Image File history File links Johnson solid #1 Another attempt at using user:Cyps image:Poly. ... In geometry, the Pentagonal pyramid is one of the Johnson solids (J2). ... Image File history File links Pentagonal_pyramid. ... In geometry, the triangular cupola is one of the Johnson solids (J3). ... Image File history File links Triangular_cupola. ... In geometry, the Square cupola is one of the Johnson solids (J4). ... Johnson solid #4 Another attempt at using user:Cyps image:Poly. ... In geometry, the Pentagonal cupola is one of the Johnson solids (J5). ... Image File history File links Pentagonal_cupola. ... In geometry, the Pentagonal rotunda is one of the Johnson solids (J6). ... Image File history File links Pentagonal_rotunda. ...

Modified pyramids and dipyramids

  • elongated pyramid
  • gyroelongated pyramid
  • bipyramid
  • elongated dipyramid
  • gyroelongated dipyramid
Jn Solid name Image V E F F3 F4 F5 F6 F8 F10 Symmetry
7 Elongated triangular pyramid 7 12 7 4 3 C3v
8 Elongated square pyramid
or (augmented cube)
or (augmented square prism)		 9 16 9 4 5 C4v
9 Elongated pentagonal pyramid 11 20 11 5 5 1 C5v
10 Gyroelongated square pyramid 9 20 13 12 1 C4v
11 Gyroelongated pentagonal pyramid
or (diminished icosahedron)		 11 25 16 15 1 C5v
12 Triangular dipyramid 5 9 6 6 D3h
13 Pentagonal dipyramid 7 15 10 10 D5h
14 Elongated triangular dipyramid 8 15 9 6 3 D3h
15 Elongated square dipyramid
or (biaugmented cube)
or (biaugmented square prism)		 10 20 12 8 4 D4h
16 Elongated pentagonal dipyramid 12 25 15 10 5 D5h
17 Gyroelongated square dipyramid 10 24 16 16 D4d

A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... In geometry, the elongated dipyramids are an infinite set of polyhedra, constructed by elongating an n-agonal bipyramid (by inserting an n-agonal prism between its congruent halves. ... In geometry, the gyroelongated dipyramids are an infinite set of polyhedra, constructed by elongating an n-agonal bipyramid (by inserting an n-agonal antiprism between its congruent halves. ... Hi. ... Image File history File links Elongated_triangular_pyramid. ... In geometry, the elongated square pyramid is one of the Johnson solids (J8). ... Image File history File links Elongated_square_pyramid. ... In geometry, the elongated pentagonal pyramid is one of the Johnson solids (J9). ... Image File history File links Elongated_pentagonal_pyramid. ... In geometry, the gyroelongated square pyramid is one of the Johnson solids (J10) The 92 Johnson solids were named and described by Norman Johnson in 1966. ... Image File history File links Gyroelongated_square_pyramid. ... In geometry, the Gyroelongated pentagonal pyramid is one of the Johnson solids (J11). ... Image File history File links Gyroelongated_pentagonal_pyramid. ... In geometry, the triangular dipyramid is a polyhedron made entirely out of 6 faces, which are all equilateral triangles, 9 edges, and 5 vertexes. ... Image File history File links Triangular_dipyramid. ... In geometry, the pentagonal dipyramid is one of the Johnson solids (J13). ... Image File history File links Pentagonal_dipyramid. ... In geometry, the elongated triangular dipyramid is one of the Johnson solids (J14). ... Image File history File links Elongated_triangular_dipyramid. ... In geometry, the elongated square dipyramid is one of the Johnson solids (J15). ... Image File history File links Elongated_square_dipyramid. ... In geometry, the elongated pentagonal dipyramid is one of the Johnson solids (J16). ... Image File history File links Elongated_pentagonal_dipyramid. ... In geometry, the gyroelongated square dipyramid is one of the Johnson solids (J19). ... Image File history File links Gyroelongated_square_dipyramid. ...

Modified cupolas and rotunda

  • elongated cupola
  • elongated rotunda
  • elongated birotunda
  • elongated cupolarotunda
  • elongated bicupola
  • gyroelongated cupola
  • gyroelongated rotunda
  • bicupola
  • cupolarotunda
  • gyroelongated bicupola
  • gyroelongated birotunda
  • gyroelongated cupolarotunda
Jn Solid name Image V E F F3 F4 F5 F6 F8 F10 Symmetry
18 Elongated triangular cupola 15 27 14 4 9 1 C3v
19 Elongated square cupola
(diminished rhombicuboctahedron)		 20 36 18 4 13 1 C4v
20 Elongated pentagonal cupola 25 45 22 5 15 1 1 C5v
21 Elongated pentagonal rotunda 30 55 27 10 10 6 1 C5v
22 Gyroelongated triangular cupola 15 33 20 16 3 1 C3v
23 Gyroelongated square cupola 20 44 26 20 5 1 C4v
24 Gyroelongated pentagonal cupola 25 55 32 25 5 1 1 C5v
25 Gyroelongated pentagonal rotunda 30 65 37 30 6 1 C5v
26 Gyrobifastigium 8 14 8 4 4 D2d
27 Triangular orthobicupola
(gyrate cuboctahedron)		 12 24 14 8 6 D3h
28 Square orthobicupola 16 32 18 8 10 D4h
29 Square gyrobicupola 16 32 18 8 10 D4d
30 Pentagonal orthobicupola 20 40 22 10 10 2 D5h
31 Pentagonal gyrobicupola 20 40 22 10 10 2 D5d
32 Pentagonal orthocupolarotunda 25 50 27 15 5 7 C5v
33 Pentagonal gyrocupolarotunda 25 50 27 15 5 7 C5v
34 Pentagonal orthobirotunda
(gyrate icosidodecahedron)		 30 60 32 20 12 D5h
35 Elongated triangular orthobicupola 18 36 20 8 12 D3h
36 Elongated triangular gyrobicupola 18 36 20 8 12 D3d
37 Elongated square gyrobicupola
(gyrate rhombicuboctahedron)		 24 48 26 8 18 D4d
38 Elongated pentagonal orthobicupola 30 60 32 10 20 2 D5h
39 Elongated pentagonal gyrobicupola 30 60 32 10 20 2 D5d
40 Elongated pentagonal orthocupolarotunda 35 70 37 15 15 7 C5v
41 Elongated pentagonal gyrocupolarotunda 35 70 37 15 15 7 C5v
42 Elongated pentagonal orthobirotunda 40 80 42 20 10 12 D5h
43 Elongated pentagonal gyrobirotunda 40 80 42 20 10 12 D5d
44 Gyroelongated triangular bicupola
(2 chiral forms)		 18 42 26 20 6 D3
45 Gyroelongated square bicupola
(2 chiral forms)		 24 56 34 24 10 D4
46 Gyroelongated pentagonal bicupola
(2 chiral forms)		 30 70 42 30 10 2 D5
47 Gyroelongated pentagonal cupolarotunda
(2 chiral forms)		 35 80 47 35 5 7 C5
48 Gyroelongated pentagonal birotunda
(2 chiral forms)		 40 90 52 40 12 D5

The gyrobifastigium (J26) can be considered a linear gyrobicupola. ... In geometry, the elongated triangular cupola is one of the Johnson solids (J18) Categories: Stub | Johnson solids ... Image File history File links Elongated_triangular_cupola. ... In geometry, the Elongated square cupola is one of the Johnson solids (J19). ... Image File history File links Johnson solid #19 Another attempt at using user:Cyps image:Poly. ... In geometry, the elongated pentagonal cupola is one of the Johnson solids (J20). ... Image File history File links Elongated_pentagonal_cupola. ... In geometry, the Elongated pentagonal rotunda is one of the Johnson solids (J21). ... Image File history File links Elongated_pentagonal_rotunda. ... In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J22). ... Image File history File links Gyroelongated_triangular_cupola. ... In geometry, the Gyroelongated square cupola is one of the Johnson solids (J23). ... Image File history File links Gyroelongated_square_cupola. ... In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J24). ... Image File history File links Gyroelongated_pentagonal_cupola. ... In geometry, the Gyroelongated pentagonal rotunda is one of the Johnson solids (J25). ... Image File history File links Gyroelongated_pentagonal_rotunda. ... In geometry, the gyrobifastigium is one of the Johnson solids (J26). ... Image File history File links Gyrobifastigium. ... In geometry, the triangular orthobicupola is one of the Johnson solids (J27). ... Image File history File links Triangular_orthobicupola. ... In geometry, the Square orthobicupola is one of the Johnson solids (J28). ... Image File history File links Square_orthobicupola. ... In geometry, the Square gyrobicupola is one of the Johnson solids (J29). ... Image File history File links Square_gyrobicupola. ... In geometry, the pentagonal orthobicupola is one of the Johnson solids (J30). ... Image File history File links Pentagonal_orthobicupola. ... In geometry, the pentagonal gyrobicupola is one of the Johnson solids (J31). ... Image File history File links Pentagonal_gyrobicupola. ... In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (J32). ... Image File history File links Pentagonal_orthocupolarotunda. ... In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids (J33). ... Image File history File links Pentagonal_gyrocupolarotunda. ... In geometry, the Pentagonal orthobirotunda is one of the Johnson solids (J34). ... Image File history File links Pentagonal_orthobirotunda. ... In geometry, the elongated triangular orthobicupola is one of the Johnson solids (J35). ... Image File history File links Elongated_triangular_orthobicupola. ... In geometry, the elongated triangular gyrobicupola is one of the Johnson solids (J36) Categories: Stub | Johnson solids ... Image File history File links Elongated_triangular_gyrobicupola. ... In geometry, the elongated square gyrobicupola is one of the Johnson solids (J37). ... Johnson solid #37 Another attempt at using user:Cyps image:Poly. ... In geometry, the elongated pentagonal orthobicupola is one of the Johnson solids (J38). ... Image File history File links Elongated_pentagonal_orthobicupola. ... In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (J39). ... Image File history File links Elongated_pentagonal_gyrobicupola. ... In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids (J40). ... Image File history File links Elongated_pentagonal_orthocupolarotunda. ... In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids (J41). ... Image File history File links Elongated_pentagonal_gyrocupolarotunda. ... In geometry, the Elongated pentagonal orthobirotunda is one of the Johnson solids (J42). ... Image File history File links Elongated_pentagonal_orthobirotunda. ... In geometry, the Elongated pentagonal gyrobirotunda is one of the Johnson solids (J43). ... Image File history File links Elongated_pentagonal_gyrobirotunda. ... In geometry, the gyroelongated triangular bicupola is one of the Johnson solids (J44) Categories: Math stubs | Johnson solids | Chiral polyhedra ... Image File history File links Gyroelongated_triangular_bicupola. ... In geometry, the gyroelongated square bicupola is one of the Johnson solids (J45). ... Image File history File links Gyroelongated_square_bicupola. ... In geometry, the gyroelongated pentagonal bicupola is one of the Johnson solids (J46). ... Image File history File links Gyroelongated_pentagonal_bicupola. ... In geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids (J47). ... Image File history File links Gyroelongated_pentagonal_cupolarotunda. ... In geometry, the Gyroelongated pentagonal birotunda is one of the Johnson solids (J48). ... Image File history File links Gyroelongated_pentagonal_birotunda. ...

Augmented prisms

Jn Solid name Image V E F F3 F4 F5 F6 F8 F10 Symmetry
49 Augmented triangular prism 7 13 8 6 2 C2v
50 Biaugmented triangular prism 8 17 11 10 1 C2v
51 Triaugmented triangular prism 9 21 14 14 D3h
52 Augmented pentagonal prism 11 19 10 4 4 2 C2v
53 Biaugmented pentagonal prism 12 23 13 8 3 2 C2v
54 Augmented hexagonal prism 13 22 11 4 5 2 C2v
55 Parabiaugmented hexagonal prism 14 26 14 8 4 2 D2h
56 Metabiaugmented hexagonal prism 14 26 14 8 4 2 C2v
57 Triaugmented hexagonal prism 15 30 17 12 3 2 D3h

In geometry, the augmented triangular prism is one of the Johnson solids (J49). ... Image File history File links Augmented_triangular_prism. ... In geometry, the biaugmented triangular prism is one of the Johnson solids (J50). ... Image File history File links Biaugmented_triangular_prism. ... In geometry, the triaugmented triangular prism is one of the Johnson solids (J51). ... Image File history File links Triaugmented_triangular_prism. ... In geometry, the augmented pentagonal prism is one of the Johnson solids (J52). ... Image File history File links Augmented_pentagonal_prism. ... In geometry, the biaugmented pentagonal prism is one of the Johnson solids (J53). ... Image File history File links Biaugmented_pentagonal_prism. ... In geometry, the augmented hexagonal prism is one of the Johnson solids (J54). ... Image File history File links Augmented_hexagonal_prism. ... In geometry, the parabiaugmented hexagonal prism is one of the Johnson solids (J55). ... Image File history File links Parabiaugmented_hexagonal_prism. ... In geometry, the metabiaugmented hexagonal prism is one of the Johnson solids (J56). ... Image File history File links Metabiaugmented_hexagonal_prism. ... In geometry, the triaugmented hexagonal prism is one of the Johnson solids (J57). ... Image File history File links Triaugmented_hexagonal_prism. ...

Modified Platonic solids

  • Augmented dodecahedrons
  • Diminished icosahedrons
Jn Solid name Image V E F F3 F4 F5 F6 F8 F10 Symmetry
58 Augmented dodecahedron 21 35 16 5 11 C5v
59 Parabiaugmented dodecahedron 22 40 20 10 10 D5d
60 Metabiaugmented dodecahedron 22 40 20 10 10 C2v
61 Triaugmented dodecahedron 23 45 24 15 9 C3v
62 Metabidiminished icosahedron 10 20 12 10 2 C2v
63 Tridiminished icosahedron 9 15 8 5 3 C3v
64 Augmented tridiminished icosahedron 10 18 10 7 3 C3v

In geometry, the augmented dodecahedron is one of the Johnson solids (J58). ... Image File history File links Augmented_dodecahedron. ... In geometry, the parabiaugmented dodecahedron is one of the Johnson solids (J59). ... Image File history File links Parabiaugmented_dodecahedron. ... In geometry, the metabiaugmented dodecahedron is one of the Johnson solids (J60). ... Image File history File links Metabiaugmented_dodecahedron. ... In geometry, the triaugmented dodecahedron is one of the Johnson solids (J61). ... Image File history File links Triaugmented_dodecahedron. ... In geometry, the Metabidiminished icosahedron is one of the Johnson solids (J62). ... Image File history File links Metabidiminished_icosahedron. ... In geometry, the Tridiminished icosahedron is one of the Johnson solids (J63). ... Image File history File links Tridiminished_icosahedron. ... In geometry, the Augmented tridiminished icosahedron is one of the Johnson solids (J64). ... Image File history File links Augmented_tridiminished_icosahedron. ...

Modified Archimedean solids

  • augmented truncated tetrahedron
  • augmented truncated cube
  • augmented truncated dodecahedron
  • gyrate rhombicosadodecahedron
  • diminished rhombicosadodecahedron
  • gyrate diminished rhombicosadodecahedron
  • diminished rhombicosadodecahedron
  • gyrate diminished rhombicosadodecahedron
  • diminished rhombicosadodecahedron
Jn Solid name Image V E F F3 F4 F5 F6 F8 F10 Symmetry
65 Augmented truncated tetrahedron 15 27 14 8 3 3 C3v
66 Augmented truncated cube 28 48 22 12 5 5 C4v
67 Biaugmented truncated cube 32 60 30 16 10 4 D4h
68 Augmented truncated dodecahedron 65 105 42 25 5 1 11 C5v
69 Parabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 D5d
70 Metabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 C2v
71 Triaugmented truncated dodecahedron 75 135 62 35 15 3 9 C3v
72 Gyrate rhombicosidodecahedron 60 120 62 20 30 12 C5v
73 Parabigyrate rhombicosidodecahedron 60 120 62 20 30 12 D5d
74 Metabigyrate rhombicosidodecahedron 60 120 62 20 30 12 C2v
75 Trigyrate rhombicosidodecahedron 60 120 62 20 30 12 C3v
76 Diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v
77 Paragyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v
78 Metagyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs
79 Bigyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs
80 Parabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 D5d
81 Metabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 C2v
82 Gyrate bidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 Cs
83 Tridiminished rhombicosidodecahedron 45 75 32 5 15 9 3 C3v

In geometry, the augmented truncated tetrahedron is one of the Johnson solids (J65). ... Johnson solid #65 Part of a series using user:Cyps image:Poly. ... In geometry, the augmented truncated cube is one of the Johnson solids (J66). ... Image File history File links Augmented_truncated_cube. ... In geometry, the biaugmented truncated cube is one of the Johnson solids (J67). ... Image File history File links Biaugmented_truncated_cube. ... In geometry, the augmented truncated dodecahedron is one of the Johnson solids (J68). ... Image File history File links Johnson solid #68 Part of a series using user:Cyps image:Poly. ... In geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids (J69). ... Image File history File links Johnson solid #69 Part of a series using user:Cyps image:Poly. ... In geometry, the metabiaugmented truncated dodecahedron is one of the Johnson solids (J70). ... Image File history File links Johnson solid #70 Part of a series using user:Cyps image:Poly. ... In geometry, the triaugmented truncated dodecahedron is one of the Johnson solids (J71). ... Image File history File links Triaugmented_truncated_dodecahedron. ... In geometry, the Gyrate rhombicosidodecahedron is one of the Johnson solids (J72). ... Image File history File links Gyrate_rhombicosidodecahedron. ... In geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids (J73). ... Image File history File links Parabigyrate_rhombicosidodecahedron. ... In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids (J74). ... Image File history File links Metabigyrate_rhombicosidodecahedron. ... In geometry, the Trigyrate rhombicosidodecahedron is one of the Johnson solids (J75). ... Image File history File links Trigyrate_rhombicosidodecahedron. ... In geometry, the Diminished rhombicosidodecahedron is one of the Johnson solids (J76). ... Image File history File links Diminished_rhombicosidodecahedron. ... In geometry, the paragyrate diminished rhombicosidodecahedron is one of the Johnson solids (J77). ... Image File history File links Paragyrate_diminished_rhombicosidodecahedron. ... In geometry, the metagyrate diminished rhombicosidodecahedron is one of the Johnson solids (J78). ... Image File history File links Metagyrate_diminished_rhombicosidodecahedron. ... In geometry, the bigyrate diminished rhombicosidodecahedron is one of the Johnson solids (J79). ... Image File history File links Bigyrate_diminished_rhombicosidodecahedron. ... In geometry, the Parabidiminished rhombicosidodecahedron is one of the Johnson solids (J80). ... Image File history File links Parabidiminished_rhombicosidodecahedron. ... In geometry, the Metabidiminished rhombicosidodecahedron is one of the Johnson solids (J81). ... Image File history File links Metabidiminished_rhombicosidodecahedron. ... In geometry, the gyrate bidiminished rhombicosidodecahedron is one of the Johnson solids (J82). ... Image File history File links Gyrate_bidiminished_rhombicosidodecahedron. ... In geometry, the Tridiminished rhombicosidodecahedron is one of the Johnson solids (J83). ... Image File history File links Tridiminished_rhombicosidodecahedron. ...

Miscellaneous

Jn Solid name Image V E F F3 F4 F5 F6 F8 F10 Symmetry
84 Snub disphenoid
(Siamese dodecahedron)		 8 18 12 12 D2d
85 Snub square antiprism 16 40 26 24 2 D4d
86 Sphenocorona 10 22 14 12 2 C2v
87 Augmented sphenocorona 11 26 17 16 1 Cs
88 Sphenomegacorona 12 28 18 16 2 C2v
89 Hebesphenomegacorona 14 33 21 18 3 C2v
90 Disphenocingulum 16 38 24 20 4 D2d
91 Bilunabirotunda 14 26 14 8 2 4 D2h
92 Triangular hebesphenorotunda 18 36 20 13 3 3 1 C3v

In geometry, the Snub disphenoid is one of the Johnson solids (J84). ... Image File history File links Snub_disphenoid. ... In geometry, the Snub square antiprism is one of the Johnson solids (J85). ... Image File history File links Snub_square_antiprism. ... In geometry, the Sphenocorona is one of the Johnson solids (J86). ... Image File history File links Sphenocorona. ... In geometry, the Augmented sphenocorona is one of the Johnson solids (J87), and is obtained by adding a square pyramid to one of the square faces of the sphenocorona. ... Image File history File links Augmented_sphenocorona. ... In geometry, the Sphenomegacorona is one of the Johnson solids (J88). ... Image File history File links Sphenomegacorona. ... In geometry, the Hebesphenomegacorona is one of the Johnson solids (J89). ... Image File history File links Hebesphenomegacorona. ... In geometry, the Disphenocingulum is one of the Johnson solids (J90). ... Image File history File links Disphenocingulum. ... In geometry, the Bilunabirotunda is one of the Johnson solids (J91). ... Image File history File links Bilunabirotunda. ... In geometry, the Triangular hebesphenorotunda is one of the Johnson solids (J92). ... Image File history File links Triangular_hebesphenorotunda. ...

See also

  • Near-miss Johnson solid

In geometry, a near-miss Johnson solid is a convex polyhedron, where every face is a regular or nearly regular polygon, and excluding the 5 Platonic solids, the 13 Archimedean solids, the infinite set of prisms, the infinite set of antiprisms, and the 92 Johnson solids. ...

References

  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN.  The first proof that there are only 92 Johnson solids.

Norman W. Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. ... Victor (Viktor) Abramovich Zalgaller (Russian: ) (born on December 25, 1920 in Parfino, Novgorod oblast) is a mathematician in the fields of geometry and optimization. ...

External links


  Results from FactBites:
 
Johnson solid - Wikipedia, the free encyclopedia (744 words)
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism.
Of the Johnson solids, the elongated square gyrobicupola (J
Most of the Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundae), together with the Platonic and Archimedean solids, prisms, and antiprisms.
Brick-solid win - Roanoke.com (735 words)
Johnson's comeback victory fulfilled a quest for him, Knaus and the team, a journey that tormented them, taught them and rewarded them.
Johnson blew a left front tire on lap 39 of the 160-lap event as he ran in the top five.
Johnson led when a caution for debris slowed the race 18 laps from the finish.
  More results at FactBites »


 

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