JOHNSON SOLID

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 (''J''₁); 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 (''J''₂) is an example that actually has a degree-5 vertex.
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.
Of the Johnson solids, the elongated square gyrobicupola (''J''₃₇) 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.

Contents

Names

Enumeration

Modified pyramids and dipyramids

Modified cupolas and rotunda

Modified Platonic solids

Modified Archimedean solids

Miscellaneous

See also

References

External links

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.)

Enumeration

=== Prismatoids and rotundae ===

★ Pyramids

★ Cupolas

★ Rotunda

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

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					''C''3v
8	Elongated square pyramid or (augmented cube) or (augmented square prism)		9	16	9	4	5					''C''4v
9	Elongated pentagonal pyramid		11	20	11	5	5	1				''C''5v
10	Gyroelongated square pyramid		9	20	13	12	1					''C''4v
11	Gyroelongated pentagonal pyramid or (diminished icosahedron)		11	25	16	15		1				''C''5v
12	Triangular dipyramid		5	9	6	6						''D''3h
13	Pentagonal dipyramid		7	15	10	10						''D''5h
14	Elongated triangular dipyramid		8	15	9	6	3					''D''3h
15	Elongated square dipyramid or (biaugmented cube) or (biaugmented square prism)		10	20	12	8	4					''D''4h
16	Elongated pentagonal dipyramid		12	25	15	10	5					''D''5h
17	Gyroelongated square dipyramid		10	24	16	16						''D''4d

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			''C''3v
19	Elongated square cupola ''(diminished rhombicuboctahedron)''		20	36	18	4	13			1		''C''4v
20	Elongated pentagonal cupola		25	45	22	5	15	1			1	''C''5v
21	Elongated pentagonal rotunda		30	55	27	10	10	6			1	''C''5v
22	Gyroelongated triangular cupola		15	33	20	16	3		1			''C''3v
23	Gyroelongated square cupola		20	44	26	20	5			1		''C''4v
24	Gyroelongated pentagonal cupola		25	55	32	25	5	1			1	''C''5v
25	Gyroelongated pentagonal rotunda		30	65	37	30		6			1	''C''5v
26	Gyrobifastigium		8	14	8	4	4					''D''2d
27	Triangular orthobicupola ''(gyrate cuboctahedron)''		12	24	14	8	6					''D''3h
28	Square orthobicupola		16	32	18	8	10					''D''4h
29	Square gyrobicupola		16	32	18	8	10					''D''4d
30	Pentagonal orthobicupola		20	40	22	10	10	2				''D''5h
31	Pentagonal gyrobicupola		20	40	22	10	10	2				''D''5d
32	Pentagonal orthocupolarotunda		25	50	27	15	5	7				''C''5v
33	Pentagonal gyrocupolarotunda		25	50	27	15	5	7				''C''5v
34	Pentagonal orthobirotunda ''(gyrate icosidodecahedron)''		30	60	32	20		12				''D''5h
35	Elongated triangular orthobicupola		18	36	20	8	12					''D''3h
36	Elongated triangular gyrobicupola		18	36	20	8	12					''D''3d
37	Elongated square gyrobicupola ''(gyrate rhombicuboctahedron)''		24	48	26	8	18					''D''4d
38	Elongated pentagonal orthobicupola		30	60	32	10	20	2				''D''5h
39	Elongated pentagonal gyrobicupola		30	60	32	10	20	2				''D''5d
40	Elongated pentagonal orthocupolarotunda		35	70	37	15	15	7				''C''5v
41	Elongated pentagonal gyrocupolarotunda		35	70	37	15	15	7				''C''5v
42	Elongated pentagonal orthobirotunda		40	80	42	20	10	12				''D''5h
43	Elongated pentagonal gyrobirotunda		40	80	42	20	10	12				''D''5d
44	Gyroelongated triangular bicupola (2 chiral forms)		18	42	26	20	6					''D''3
45	Gyroelongated square bicupola (2 chiral forms)		24	56	34	24	10					''D''4
46	Gyroelongated pentagonal bicupola (2 chiral forms)		30	70	42	30	10	2				''D''5
47	Gyroelongated pentagonal cupolarotunda (2 chiral forms)		35	80	47	35	5	7				''C''5
48	Gyroelongated pentagonal birotunda (2 chiral forms)		40	90	52	40		12				''D''5

=== 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					''C''2v
50	Biaugmented triangular prism		8	17	11	10	1					''C''2v
51	Triaugmented triangular prism		9	21	14	14						''D''3h
52	Augmented pentagonal prism		11	19	10	4	4	2				''C''2v
53	Biaugmented pentagonal prism		12	23	13	8	3	2				''C''2v
54	Augmented hexagonal prism		13	22	11	4	5		2			''C''2v
55	Parabiaugmented hexagonal prism		14	26	14	8	4		2			''D''2h
56	Metabiaugmented hexagonal prism		14	26	14	8	4		2			''C''2v
57	Triaugmented hexagonal prism		15	30	17	12	3		2			''D''3h

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				''C''5v
59	Parabiaugmented dodecahedron		22	40	20	10		10				''D''5d
60	Metabiaugmented dodecahedron		22	40	20	10		10				''C''2v
61	Triaugmented dodecahedron		23	45	24	15		9				''C''3v
62	Metabidiminished icosahedron		10	20	12	10		2				''C''2v
63	Tridiminished icosahedron		9	15	8	5		3				''C''3v
64	Augmented tridiminished icosahedron		10	18	10	7		3				''C''3v

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			''C''3v
66	Augmented truncated cube		28	48	22	12	5			5		''C''4v
67	Biaugmented truncated cube		32	60	30	16	10			4		''D''4h
68	Augmented truncated dodecahedron		65	105	42	25	5	1			11	''C''5v
69	Parabiaugmented truncated dodecahedron		70	120	52	30	10	2			10	''D''5d
70	Metabiaugmented truncated dodecahedron		70	120	52	30	10	2			10	''C''2v
71	Triaugmented truncated dodecahedron		75	135	62	35	15	3			9	''C''3v
72	Gyrate rhombicosidodecahedron		60	120	62	20	30	12				''C''5v
73	Parabigyrate rhombicosidodecahedron		60	120	62	20	30	12				''D''5d
74	Metabigyrate rhombicosidodecahedron		60	120	62	20	30	12				''C''2v
75	Trigyrate rhombicosidodecahedron		60	120	62	20	30	12				''C''3v
76	Diminished rhombicosidodecahedron		55	105	52	15	25	11			1	''C''5v
77	Paragyrate diminished rhombicosidodecahedron		55	105	52	15	25	11			1	''C''5v
78	Metagyrate diminished rhombicosidodecahedron		55	105	52	15	25	11			1	''C''s
79	Bigyrate diminished rhombicosidodecahedron		55	105	52	15	25	11			1	''C''s
80	Parabidiminished rhombicosidodecahedron		50	90	42	10	20	10			2	''D''5d
81	Metabidiminished rhombicosidodecahedron		50	90	42	10	20	10			2	''C''2v
82	Gyrate bidiminished rhombicosidodecahedron		50	90	42	10	20	10			2	''C''s
83	Tridiminished rhombicosidodecahedron		45	75	32	5	15	9			3	''C''3v

Miscellaneous

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

See also

★ Near-miss Johnson solid

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.

★ Convex Polyhedra with Regular Faces, Victor A. Zalgaller, , , Consultants Bureau, 1969, No ISBN The first proof that there are only 92 Johnson solids.

External links

★ Sylvain Gagnon, "Convex polyhedra with regular faces", Structural Topology, No. 6, 1982, 83-95.

★ Paper Models of Polyhedra Many links

★ Johnson Solids by George W. Hart.

★ Images of all 92 solids, categorized, on one page

★ MathWorld

★ VRML models

★ Educational toy system for making Johnson solids and other polyhedra Magnetic Blocks

★ VRML models of Johnson Solids by Vladimir Bulatov