Johnson Solid

The Johnson solids are the convex polyhedra having regular faces and equal edge lengths (with the exception of the completely regular Platonic solids, the "semiregular" Archimedean solids, and the two infinite families of prisms and antiprisms). There are 28 simple (i.e., cannot be dissected into two other regular-faced polyhedra by a plane) regular-faced polyhedra in addition to the prisms and antiprisms (Zalgaller 1969), and Johnson (1966) proposed and Zalgaller (1969) proved that there exist exactly 92 Johnson solids in all.

They are implemented in the Wolfram Language as PolyhedronData["Johnson", n].

The sketelons of the Johnson solids may be termed Johnson skeleton graphs.

There is a near-Johnson solid which can be constructed by inscribing regular nonagons inside the eight triangular faces of a regular octahedron, then joining the free edges to the 24 triangles and finally the remaining edges of the triangles to six squares, with one square for each octahedral vertex. It turns out that the triangles are not quite equilateral, making the edges that bound the squares a slightly different length from that of the enneagonal edge. However, because the differences in edge lengths are so small, the flexing of an average model allows the solid to be constructed with all edges equal.

A database of solids and polyhedron vertex nets of these solids is maintained on the Sandia National Laboratories Netlib server (http://netlib.sandia.gov/polyhedra/), but a few errors exist in several entries. Corrected versions are implemented in the Wolfram Language via PolyhedronData. The following list summarizes the names of the Johnson solids and gives their images and nets.

1. Square pyramid

2. Pentagonal pyramid

3. Triangular cupola

4. Square cupola

5. Pentagonal cupola

6. Pentagonal rotunda

7. Elongated triangular pyramid

8. Elongated square pyramid

9. Elongated pentagonal pyramid

10. Gyroelongated square pyramid

11. Gyroelongated pentagonal pyramid

12. Triangular dipyramid

13. Pentagonal dipyramid

14. Elongated triangular dipyramid

15. Elongated square dipyramid

16. Elongated pentagonal dipyramid

17. Gyroelongated square dipyramid

18. Elongated triangular cupola

19. Elongated square cupola

20. Elongated pentagonal cupola

21. Elongated pentagonal rotunda

22. Gyroelongated triangular cupola

23. Gyroelongated square cupola

24. Gyroelongated pentagonal cupola

25. Gyroelongated pentagonal rotunda

26. Gyrobifastigium

27. Triangular orthobicupola

28. Square orthobicupola

29. Square gyrobicupola

30. Pentagonal orthobicupola

31. Pentagonal gyrobicupola

32. Pentagonal orthocupolarotunda

33. Pentagonal gyrocupolarotunda

34. Pentagonal orthobirotunda

35. Elongated triangular orthobicupola

36. Elongated triangular gyrobicupola

37. Elongated square gyrobicupola

38. Elongated pentagonal orthobicupola

39. Elongated pentagonal gyrobicupola

40. Elongated pentagonal orthocupolarotunda

41. Elongated pentagonal gyrocupolarotunda

42. Elongated pentagonal orthobirotunda

43. Elongated pentagonal gyrobirotunda

44. Gyroelongated triangular bicupola

45. Gyroelongated square bicupola

46. Gyroelongated pentagonal bicupola

47. Gyroelongated pentagonal cupolarotunda

48. Gyroelongated pentagonal birotunda

49. Augmented triangular prism

50. Biaugmented triangular prism

51. Triaugmented triangular prism

52. Augmented pentagonal prism

53. Biaugmented pentagonal prism

54. Augmented hexagonal prism

55. Parabiaugmented hexagonal prism

56. Metabiaugmented hexagonal prism

57. Triaugmented hexagonal prism

58. Augmented dodecahedron

59. Parabiaugmented dodecahedron

60. Metabiaugmented dodecahedron

61. Triaugmented dodecahedron

62. Metabidiminished icosahedron

63. Tridiminished icosahedron

64. Augmented tridiminished icosahedron

65. Augmented truncated tetrahedron

66. Augmented truncated cube

67. Biaugmented truncated cube

68. Augmented truncated dodecahedron

69. Parabiaugmented truncated dodecahedron

70. Metabiaugmented truncated dodecahedron

71. Triaugmented truncated dodecahedron

72. Gyrate rhombicosidodecahedron

73. Parabigyrate rhombicosidodecahedron

74. Metabigyrate rhombicosidodecahedron

75. Trigyrate rhombicosidodecahedron

76. Diminished rhombicosidodecahedron

77. Paragyrate diminished rhombicosidodecahedron

78. Metagyrate diminished rhombicosidodecahedron

79. Bigyrate diminished rhombicosidodecahedron

80. Parabidiminished rhombicosidodecahedron

81. Metabidiminished rhombicosidodecahedron

82. Gyrate bidiminished rhombicosidodecahedron

83. Tridiminished rhombicosidodecahedron

84. Snub disphenoid

85. Snub square antiprism

86. Sphenocorona

87. Augmented sphenocorona

88. Sphenomegacorona

89. Hebesphenomegacorona

90. Disphenocingulum

91. Bilunabirotunda

92. Triangular hebesphenorotunda

The number of constituent -gons () for each Johnson solid are given in the following table.


1	4	1					47	35	5	7
2	5		1				48	40		12
3	4	3		1			49	6	2
4	4	5			1		50	10	1
5	5	5	1			1	51	14
6	10		6			1	52	4	4	2
7	4	3					53	8	3	2
8	4	5					54	4	5		2
9	5	5	1				55	8	4		2
10	12	1					56	8	4		2
11	15		1				57	12	3		2
12	6						58	5		11
13	10						59	10		10
14	6	3					60	10		10
15	8	4					61	15		9
16	10	5					62	10		2
17	16						63	5		3
18	4	9		1			64	7		3
19	4	13			1		65	8	3		3
20	5	15	1			1	66	12	5			5
21	10	10	6			1	67	16	10			4
22	16	3		1			68	25	5	1			11
23	20	5			1		69	30	10	2			10
24	25	5	1			1	70	30	10	2			10
25	30		6			1	71	35	15	3			9
26	4	4					72	20	30	12
27	8	6					73	20	30	12
28	8	10					74	20	30	12
29	8	10					75	20	30	12
30	10	10	2				76	15	25	11			1
31	10	10	2				77	15	25	11			1
32	15	5	7				78	15	25	11			1
33	15	5	7				79	15	25	11			1
34	20		12				80	10	20	10			2
35	8	12					81	10	20	10			2
36	8	12					82	10	20	10			2
37	8	18					83	5	15	9			3
38	10	20	2				84	12
39	10	20	2				85	24	2
40	15	15	7				86	12	2
41	15	15	7				87	16	1
42	20	10	12				88	16	2
43	20	10	12				89	18	3
44	20	6					90	20	4
45	24	10					91	8	2	4
46	30	10	2				92	13	3	3	1

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References

Bulatov, V. "V. Bulatov's Polyhedra Collection: Johnson Solids." http://bulatov.org/polyhedra/johnson/.Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 86-92, 1997.Hart, G. "NetLib Polyhedra DataBase." http://www.georgehart.com/virtual-polyhedra/netlib-info.html.Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.Hume, A. Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals. Computer Science Technical Report #130. Murray Hill, NJ: AT&T Bell Laboratories, 1986.Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169-200, 1966.Pedagoguery Software. Poly. http://www.peda.com/poly/.Pugh, A. "Further Convex Polyhedra with Regular Faces." Ch. 3 in Polyhedra: A Visual Approach. Berkeley, CA: University of California Press, pp. 28-35, 1976.Sandia National Laboratories. "Polyhedron Database." http://netlib.sandia.gov/polyhedra/.Webb, R. "Miscellaneous Polyhedra: Johnson Solids and Their Duals." http://www.software3d.com/Misc.html#Johnson.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 70-71, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. New York: Penguin Books, pp. 88-89, 1986.Zalgaller, V. Convex Polyhedra with Regular Faces. New York: Consultants Bureau, 1969.

Referenced on Wolfram|Alpha

Johnson Solid

Cite this as:

Weisstein, Eric W. "Johnson Solid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JohnsonSolid.html

Johnson Solid

See also

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References

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Subject classifications


1	4	1					47	35	5	7
2	5		1				48	40		12
3	4	3		1			49	6	2
4	4	5			1		50	10	1
5	5	5	1			1	51	14
6	10		6			1	52	4	4	2
7	4	3					53	8	3	2
8	4	5					54	4	5		2
9	5	5	1				55	8	4		2
10	12	1					56	8	4		2
11	15		1				57	12	3		2
12	6						58	5		11
13	10						59	10		10
14	6	3					60	10		10
15	8	4					61	15		9
16	10	5					62	10		2
17	16						63	5		3
18	4	9		1			64	7		3
19	4	13			1		65	8	3		3
20	5	15	1			1	66	12	5			5
21	10	10	6			1	67	16	10			4
22	16	3		1			68	25	5	1			11
23	20	5			1		69	30	10	2			10
24	25	5	1			1	70	30	10	2			10
25	30		6			1	71	35	15	3			9
26	4	4					72	20	30	12
27	8	6					73	20	30	12
28	8	10					74	20	30	12
29	8	10					75	20	30	12
30	10	10	2				76	15	25	11			1
31	10	10	2				77	15	25	11			1
32	15	5	7				78	15	25	11			1
33	15	5	7				79	15	25	11			1
34	20		12				80	10	20	10			2
35	8	12					81	10	20	10			2
36	8	12					82	10	20	10			2
37	8	18					83	5	15	9			3
38	10	20	2				84	12
39	10	20	2				85	24	2
40	15	15	7				86	12	2
41	15	15	7				87	16	1
42	20	10	12				88	16	2
43	20	10	12				89	18	3
44	20	6					90	20	4
45	24	10					91	8	2	4
46	30	10	2				92	13	3	3	1


1	4	1					47	35	5	7
2	5		1				48	40		12
3	4	3		1			49	6	2
4	4	5			1		50	10	1
5	5	5	1			1	51	14
6	10		6			1	52	4	4	2
7	4	3					53	8	3	2
8	4	5					54	4	5		2
9	5	5	1				55	8	4		2
10	12	1					56	8	4		2
11	15		1				57	12	3		2
12	6						58	5		11
13	10						59	10		10
14	6	3					60	10		10
15	8	4					61	15		9
16	10	5					62	10		2
17	16						63	5		3
18	4	9		1			64	7		3
19	4	13			1		65	8	3		3
20	5	15	1			1	66	12	5			5
21	10	10	6			1	67	16	10			4
22	16	3		1			68	25	5	1			11
23	20	5			1		69	30	10	2			10
24	25	5	1			1	70	30	10	2			10
25	30		6			1	71	35	15	3			9
26	4	4					72	20	30	12
27	8	6					73	20	30	12
28	8	10					74	20	30	12
29	8	10					75	20	30	12
30	10	10	2				76	15	25	11			1
31	10	10	2				77	15	25	11			1
32	15	5	7				78	15	25	11			1
33	15	5	7				79	15	25	11			1
34	20		12				80	10	20	10			2
35	8	12					81	10	20	10			2
36	8	12					82	10	20	10			2
37	8	18					83	5	15	9			3
38	10	20	2				84	12
39	10	20	2				85	24	2
40	15	15	7				86	12	2
41	15	15	7				87	16	1
42	20	10	12				88	16	2
43	20	10	12				89	18	3
44	20	6					90	20	4
45	24	10					91	8	2	4
46	30	10	2				92	13	3	3	1


1	4	1					47	35	5	7
2	5		1				48	40		12
3	4	3		1			49	6	2
4	4	5			1		50	10	1
5	5	5	1			1	51	14
6	10		6			1	52	4	4	2
7	4	3					53	8	3	2
8	4	5					54	4	5		2
9	5	5	1				55	8	4		2
10	12	1					56	8	4		2
11	15		1				57	12	3		2
12	6						58	5		11
13	10						59	10		10
14	6	3					60	10		10
15	8	4					61	15		9
16	10	5					62	10		2
17	16						63	5		3
18	4	9		1			64	7		3
19	4	13			1		65	8	3		3
20	5	15	1			1	66	12	5			5
21	10	10	6			1	67	16	10			4
22	16	3		1			68	25	5	1			11
23	20	5			1		69	30	10	2			10
24	25	5	1			1	70	30	10	2			10
25	30		6			1	71	35	15	3			9
26	4	4					72	20	30	12
27	8	6					73	20	30	12
28	8	10					74	20	30	12
29	8	10					75	20	30	12
30	10	10	2				76	15	25	11			1
31	10	10	2				77	15	25	11			1
32	15	5	7				78	15	25	11			1
33	15	5	7				79	15	25	11			1
34	20		12				80	10	20	10			2
35	8	12					81	10	20	10			2
36	8	12					82	10	20	10			2
37	8	18					83	5	15	9			3
38	10	20	2				84	12
39	10	20	2				85	24	2
40	15	15	7				86	12	2
41	15	15	7				87	16	1
42	20	10	12				88	16	2
43	20	10	12				89	18	3
44	20	6					90	20	4
45	24	10					91	8	2	4
46	30	10	2				92	13	3	3	1