Johnson solid

From Free net encyclopedia

In geometry, a Johnson solid is a 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. An example of a Johnson solid is the square-based pyramid with equilateral sides (J₁); it has one square face and four triangular faces.

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.

Although there is no obvious restriction on any regular polygon's being 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.

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.

1 Names
2 Names and Johnson numbers
3 See also
4 References
5 External links

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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.

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.)

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.)

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Names and Johnson numbers

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J1 - J12

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	F6	F8	F10	Symmetry
1	Square pyramid	Image:Square pyramid.png	pyramid	5	8	5	4	1	0	0	0	0	C_4v
2	Pentagonal pyramid	Image:Pentagonal pyramid.png	pyramid	6	10	6	5	0	1	0	0	0	C_5v
3	Triangular cupola	Image:Triangular cupola.png	cupola	9	15	8	4	3	0	1	0	0	C_3v
4	Square cupola	Image:Square cupola.png	cupola	12	20	10	4	5	0	0	1	0	C_4v
5	Pentagonal cupola	Image:Pentagonal cupola.png	cupola	15	25	12	5	5	1	0	0	1	C_5v
6	Pentagonal rotunda	Image:Pentagonal rotunda.png	rotunda	20	35	17	10	0	6	0	0	1	C_5v
7	Elongated triangular pyramid	Image:Elongated triangular pyramid.png	elongated pyramid	7	12	7	4	3	0	0	0	0	C_3v
8	Elongated square pyramid	Image:Elongated square pyramid.png	elongated pyramid	9	16	9	4	5	0	0	0	0	C_4v
9	Elongated pentagonal pyramid	Image:Elongated pentagonal pyramid.png	elongated pyramid	11	20	11	5	5	1	0	0	0	C_5v
10	Gyroelongated square pyramid	Image:Gyroelongated square pyramid.png	gyroelongated pyramid	9	20	13	12	1	0	0	0	0	C_4v
11	Gyroelongated pentagonal pyramid	Image:Gyroelongated pentagonal pyramid.png	gyroelongated pyramid	11	25	16	15	0	1	0	0	0	C_5v
12	Triangular dipyramid	Image:Triangular dipyramid.png	bipyramid	5	9	6	6	0	0	0	0	0	D_3h

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J13 - J24

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	F6	F8	F10	Symmetry
13	Pentagonal dipyramid	Image:Pentagonal dipyramid.png	bipyramid	7	15	10	10	0	0	0	0	0	D_5h
14	Elongated triangular dipyramid	Image:Elongated triangular dipyramid.png	Elongated dipyramid	8	15	9	6	3	0	0	0	0	D_3h
15	Elongated square dipyramid	Image:Elongated square dipyramid.png	Elongated dipyramid	10	20	12	8	4	0	0	0	0	D_4h
16	Elongated pentagonal dipyramid	Image:Elongated pentagonal dipyramid.png	Elongated dipyramid	12	25	15	10	5	0	0	0	0	D_5h
17	Gyroelongated square dipyramid	Image:Gyroelongated square dipyramid.png	Gyroelongated dipyramid	10	24	16	16	0	0	0	0	0	D_4d
18	Elongated triangular cupola	Image:Elongated triangular cupola.png	elongated cupola	15	27	14	4	9	0	1	0	0	C_3v
19	Elongated square cupola	Image:Elongated square cupola.png	elongated cupola	20	36	18	4	13	0	0	1	0	C_4v
20	Elongated pentagonal cupola	Image:Elongated pentagonal cupola.png	elongated cupola	25	45	22	5	15	1	0	0	1	C_5v
21	Elongated pentagonal rotunda	Image:Elongated pentagonal rotunda.png	elongated rotunda	30	55	27	10	10	6	0	0	1	C_5v
22	Gyroelongated triangular cupola	Image:Gyroelongated triangular cupola.png	gyroelongated cupola	15	33	20	16	3	0	1	0	0	C_3v
23	Gyroelongated square cupola	Image:Gyroelongated square cupola.png	gyroelongated cupola	20	44	26	20	5	0	0	1	0	C_4v
24	Gyroelongated pentagonal cupola	Image:Gyroelongated pentagonal cupola.png	gyroelongated cupola	25	55	32	25	5	1	0	0	1	C_5v

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J25 - J36

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	F10	Symmetry
25	Gyroelongated pentagonal rotunda	Image:Gyroelongated pentagonal rotunda.png	gyroelongated rotunda	30	65	37	30	0	6	1	C_5v
26	Gyrobifastigium	Image:Gyrobifastigium.png	bicupola	8	14	8	4	4	0	0	D_2d
27	Triangular orthobicupola	Image:Triangular orthobicupola.png	bicupola	12	24	14	8	6	0	0	D_3h
28	Square orthobicupola	Image:Square orthobicupola.png	bicupola	16	32	18	8	10	0	0	D_4h
29	Square gyrobicupola	Image:Square gyrobicupola.png	bicupola	16	32	18	8	10	0	0	D_4d
30	Pentagonal orthobicupola	Image:Pentagonal orthobicupola.png	bicupola	20	40	22	10	10	2	0	D_5h
31	Pentagonal gyrobicupola	Image:Pentagonal gyrobicupola.png	bicupola	20	40	22	10	10	2	0	D_5d
32	Pentagonal orthocupolarotunda	Image:Pentagonal orthocupolarotunda.png	cupolarotunda	25	50	27	15	5	7	0	C_5v
33	Pentagonal gyrocupolarotunda	Image:Pentagonal gyrocupolarotunda.png	cupolarotunda	25	50	27	15	5	7	0	C_5v
34	Pentagonal orthobirotunda	Image:Pentagonal orthobirotunda.png	birotunda	30	60	32	20	0	12	0	C_5v
35	Elongated triangular orthobicupola	Image:Elongated triangular orthobicupola.png	elongated bicupola	18	36	20	8	12	0	0	D_3h
36	Elongated triangular gyrobicupola	Image:Elongated triangular gyrobicupola.png	elongated bicupola	18	36	20	8	12	0	0	D_3d

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J37 - J48

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	Symmetry
37	Elongated square gyrobicupola	Image:Elongated square gyrobicupola.png	elongated bicupola	24	48	26	8	18	0	D_4h
38	Elongated pentagonal orthobicupola	Image:Elongated pentagonal orthobicupola.png	elongated bicupola	30	60	32	10	20	2	D_5h
39	Elongated pentagonal gyrobicupola	Image:Elongated pentagonal gyrobicupola.png	elongated bicupola	30	60	32	10	20	2	D_5v
40	Elongated pentagonal orthocupolarotunda	Image:Elongated pentagonal orthocupolarotunda.png	elongated cupolarotunda	35	70	37	15	15	7	C_5v
41	Elongated pentagonal gyrocupolarotunda	Image:Elongated pentagonal gyrocupolarotunda.png	elongated cupolarotunda	35	70	37	15	15	7	C_5v
42	Elongated pentagonal orthobirotunda	Image:Elongated pentagonal orthobirotunda.png	elongated birotunda	40	80	42	20	10	12	D_5h
43	Elongated pentagonal gyrobirotunda	Image:Elongated pentagonal gyrobirotunda.png	elongated birotunda	40	80	42	20	10	12	D_5v
44	Gyroelongated triangular bicupola	Image:Gyroelongated triangular bicupola.png	gyroelongated bicupola	18	42	26	20	6	0	D_3v
45	Gyroelongated square bicupola	Image:Gyroelongated square bicupola.png	gyroelongated bicupola	24	56	34	24	10	0	D_4v
46	Gyroelongated pentagonal bicupola	Image:Gyroelongated pentagonal bicupola.png	gyroelongated bicupola	30	70	42	30	10	2	D_5v
47	Gyroelongated pentagonal cupolarotunda	Image:Gyroelongated pentagonal cupolarotunda.png	gyroelongated cupolarotunda	35	80	47	35	5	7	C_5v
48	Gyroelongated pentagonal birotunda	Image:Gyroelongated pentagonal birotunda.png	gyroelongated birotunda	40	90	52	40	0	12	C_5v

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J49 - J59

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	F6	Symmetry
49	Augmented triangular prism	Image:Augmented triangular prism.png	augmented prism	7	13	8	6	2	0	0	-
50	Biaugmented triangular prism	Image:Biaugmented triangular prism.png	augmented prism	8	17	11	10	1	0	0	-
51	Triaugmented triangular prism	Image:Triaugmented triangular prism.png	augmented prism	9	21	14	14	0	0	0	-
52	Augmented pentagonal prism	Image:Augmented pentagonal prism.png	augmented prism	11	19	10	4	4	2	0	-
53	Biaugmented pentagonal prism	Image:Biaugmented pentagonal prism.png	augmented prism	12	23	13	8	3	2	0	-
54	Augmented hexagonal prism	Image:Augmented hexagonal prism.png	augmented prism	13	22	11	4	5	0	2	-
55	Parabiaugmented hexagonal prism	Image:Parabiaugmented hexagonal prism.png	augmented prism	14	26	14	8	4	0	2	-
56	Metabiaugmented hexagonal prism	Image:Metabiaugmented hexagonal prism.png	augmented prism	14	26	14	8	4	0	2	-
57	Triaugmented hexagonal prism	Image:Triaugmented hexagonal prism.png	augmented prism	15	30	17	12	3	0	2	-
58	Augmented dodecahedron	Image:Augmented dodecahedron.png	augmented dodecahedron	21	35	16	5	0	11	0	-
59	Parabiaugmented dodecahedron	Image:Parabiaugmented dodecahedron.png	augmented dodecahedron	22	40	20	10	0	10	0	-
60	Metabiaugmented dodecahedron	Image:Metabiaugmented dodecahedron.png	augmented dodecahedron	22	40	20	10	0	10	0	-

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J61 - J72

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	F6	F8	F10	Symmetry
61	Triaugmented dodecahedron	Image:Triaugmented dodecahedron.png	augmented dodecahedron	23	45	24	15	0	9	0	0	0	-
62	Metabidiminished icosahedron	Image:Metabidiminished icosahedron.png	diminished icosahedron	10	20	12	10	0	2	0	0	0	-
63	Tridiminished icosahedron	Image:Tridiminished icosahedron.png	diminished icosahedron	9	15	8	5	0	3	0	0	0	-
64	Augmented tridiminished icosahedron	Image:Augmented tridiminished icosahedron.png	-	10	18	10	7	0	3	0	0	0	-
65	Augmented truncated tetrahedron	Image:Augmented truncated tetrahedron.png	-	15	27	14	8	3	0	3	0	0	-
66	Augmented truncated cube	Image:Augmented truncated cube.png	augmented truncated cube	28	48	22	12	5	0	0	5	0	-
67	Biaugmented truncated cube	Image:Biaugmented truncated cube.png	augmented truncated cube	32	60	30	16	10	0	0	4	0	-
68	Augmented truncated dodecahedron	Image:Augmented truncated dodecahedron.png	augmented truncated dodecahedron	65	105	42	25	5	1	0	0	11	-
69	Parabiaugmented truncated dodecahedron	Image:Parabiaugmented truncated dodecahedron.png	augmented truncated dodecahedron	70	120	52	30	10	2	0	0	10	-
70	Metabiaugmented truncated dodecahedron	Image:Metabiaugmented truncated dodecahedron.png	augmented truncated dodecahedron	70	120	52	30	10	2	0	0	10	-
71	Triaugmented truncated dodecahedron	Image:Triaugmented truncated dodecahedron.png	augmented truncated dodecahedron	75	35	62	35	15	3	0	0	9	-
72	Gyrate rhombicosidodecahedron	Image:Gyrate rhombicosidodecahedron.png	gyrate rhombicosadodecahedron	60	120	62	20	30	12	0	0	0	-

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J73 - J84

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	F10	Symmetry
73	Parabigyrate rhombicosidodecahedron	Image:Parabigyrate rhombicosidodecahedron.png	gyrate rhombicosadodecahedron	60	120	62	20	30	12	0	-
74	Metabigyrate rhombicosidodecahedron	Image:Metabigyrate rhombicosidodecahedron.png	gyrate rhombicosadodecahedron	60	120	62	20	30	12	0	-
75	Trigyrate rhombicosidodecahedron	Image:Trigyrate rhombicosidodecahedron.png	gyrate rhombicosadodecahedron	60	120	62	20	30	12	0	-
76	Diminished rhombicosidodecahedron	Image:Diminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	55	105	52	15	25	11	1	-
77	Paragyrate diminished rhombicosidodecahedron	Image:Paragyrate diminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	55	105	52	15	25	11	1	-
78	Metagyrate diminished rhombicosidodecahedron	Image:Metagyrate diminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	55	105	52	15	25	11	1	-
79	Bigyrate diminished rhombicosidodecahedron	Image:Bigyrate diminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	55	105	52	15	25	11	1	-
80	Parabidiminished rhombicosidodecahedron	Image:Parabidiminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	50	90	42	10	20	10	2	-
81	Metabidiminished rhombicosidodecahedron	Image:Metabidiminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	50	90	42	10	20	10	2	-
82	Gyrate bidiminished rhombicosidodecahedron	Image:Gyrate bidiminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	50	90	42	10	20	10	2	-
83	Tridiminished rhombicosidodecahedron	Image:Tridiminished rhombicosidodecahedron.png	diminished rhombicosadodecahedron	45	75	32	5	15	9	3	-
84	Snub disphenoid	Image:Snub disphenoid.png	-	8	18	12	12	0	0	0	-

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J85 - J92

Jn	Solid name	Image	Type	V	E	F	F3	F4	F5	F6	Symmetry
85	Snub square antiprism	Image:Snub square antiprism.png	-	16	40	26	24	2	0	0	-
86	Sphenocorona	Image:Sphenocorona.png	-	10	22	14	12	2	0	0	-
87	Augmented sphenocorona	Image:Augmented sphenocorona.png	-	11	26	17	16	1	0	0	-
88	Sphenomegacorona	Image:Sphenomegacorona.png	-	12	28	18	16	2	0	0	-
89	Hebesphenomegacorona	Image:Hebesphenomegacorona.png	-	14	33	21	18	3	0	0	-
90	Disphenocingulum	Image:Disphenocingulum.png	-	16	38	24	20	4	0	0	-
91	Bilunabirotunda	Image:Bilunabirotunda.png	-	14	26	14	8	2	0	0	-
92	Triangular hebesphenorotunda	Image:Triangular hebesphenorotunda.png	-	18	36	20	13	3	3	1	-

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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.
Eric W. Weisstein. Johnson Solid at MathWorld.
Template:Cite book The first proof that there are only 92 Johnson solids.

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