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List of Johnson solids

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In geometry, polyhedra are three-dimensional objects where points are connected by lines to form polygons. The points, lines, and polygons of a polyhedron are referred to as its vertices, edges, and faces, respectively.[1] A polyhedron is considered to be convex if:[2]

  • The shortest path between any two of its vertices lies either within its interior or on its boundary.
  • None of its faces are coplanar—they do not share the same plane and do not "lie flat".
  • None of its edges are colinear—they are not segments of the same line.

A convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid. Some authors exclude uniform polyhedra from the definition. A uniform polyhedron is a polyhedron in which the faces are regular and they are isogonal; examples include Platonic and Archimedean solids as well as prisms and antiprisms.[3] The Johnson solids are named after American mathematician Norman Johnson (1930–2017), who published a list of 92 such polyhedra in 1966. His conjecture that the list was complete and no other examples existed was proven by Russian-Israeli mathematician Victor Zalgaller (1920–2020) in 1969.[4]

Some of the Johnson solids may be categorized as elementary polyhedra, meaning they cannot be separated by a plane to create two small convex polyhedra with regular faces. The Johnson solids satisfying this criteria are the first six—equilateral square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda. The criteria is also satisfied by eleven other Johnson solids, specifically the tridiminished icosahedron, parabidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron, snub disphenoid, snub square antiprism, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum, bilunabirotunda, and triangular hebesphenorotunda.[5] The rest of the Johnson solids are not elementary, and they are constructed using the first six Johnson solids together with Platonic and Archimedean solids in various processes. Augmentation involves attaching the Johnson solids onto one or more faces of polyhedra, while elongation or gyroelongation involve joining them onto the bases of a prism or antiprism, respectively. Some others are constructed by diminishment, the removal of one of the first six solids from one or more of a polyhedron's faces.[6]

The following table contains the 92 Johnson solids, with edge length . The table includes the solid's enumeration (denoted as ).[7] It also includes the number of vertices, edges, and faces of each solid, as well as its symmetry group, surface area , and volume . Every polyhedron has its own characteristics, including symmetry and measurement. An object is said to have symmetry if there is a transformation that maps it to itself. All of those transformations may be composed in a group, alongside the group's number of elements, known as the order. In two-dimensional space, these transformations include rotating around the center of a polygon and reflecting an object around the perpendicular bisector of a polygon. A polygon that is rotated symmetrically by is denoted by , a cyclic group of order ; combining this with the reflection symmetry results in the symmetry of dihedral group of order .[8] In three-dimensional symmetry point groups, the transformations preserving a polyhedron's symmetry include the rotation around the line passing through the base center, known as the axis of symmetry, and the reflection relative to perpendicular planes passing through the bisector of a base, which is known as the pyramidal symmetry of order . The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry of order . The antiprismatic symmetry of order preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.[9] The symmetry group of order preserves the symmetry by rotation around the axis of symmetry and reflection on the horizontal plane; the specific case preserving the symmetry by one full rotation is of order 2, often denoted as .[10] The mensuration of polyhedra includes the surface area and volume. An area is a two-dimensional measurement calculated by the product of length and width; for a polyhedron, the surface area is the sum of the areas of all of its faces.[11] A volume is a measurement of a region in three-dimensional space.[12] The volume of a polyhedron may be ascertained in different ways: either through its base and height (like for pyramids and prisms), by slicing it off into pieces and summing their individual volumes, or by finding the root of a polynomial representing the polyhedron.[13]

The 92 Johnson solids
Solid name Image Vertices Edges Faces Symmetry group and its order[14] Surface area and volume[15]
1 Equilateral square pyramid 5 8 5 of order 8
2 Pentagonal pyramid 6 10 6 of order 10
3 Triangular cupola 9 15 8 of order 6
4 Square cupola 12 20 10 of order 8
5 Pentagonal cupola 15 25 12 of order 10
6 Pentagonal rotunda 20 35 17 of order 10
7 Elongated triangular pyramid 7 12 7 of order 6
8 Elongated square pyramid 9 16 9 of order 8
9 Elongated pentagonal pyramid 11 20 11 of order 10
10 Gyroelongated square pyramid 9 20 13 of order 8
11 Gyroelongated pentagonal pyramid 11 25 16 of order 10
12 Triangular bipyramid 5 9 6 of order 12
13 Pentagonal bipyramid 7 15 10 of order 20
14 Elongated triangular bipyramid 8 15 9 of order 12
15 Elongated square bipyramid 10 20 12 of order 16
16 Elongated pentagonal bipyramid 12 25 15 of order 20
17 Gyroelongated square bipyramid 10 24 16 of order 16
18 Elongated triangular cupola 15 27 14 of order 6
19 Elongated square cupola 20 36 18 of order 8
20 Elongated pentagonal cupola 25 45 22 of order 10
21 Elongated pentagonal rotunda 30 55 27 of order 10
22 Gyroelongated triangular cupola 15 33 20 of order 6
23 Gyroelongated square cupola 20 44 26 of order 8
24 Gyroelongated pentagonal cupola 25 55 32 of order 10
25 Gyroelongated pentagonal rotunda 30 65 37 of order 10
26 Gyrobifastigium 8 14 8 of order 8
27 Triangular orthobicupola 12 24 14 of order 12
28 Square orthobicupola 16 32 18 of order 16
29 Square gyrobicupola 16 32 18 of order 16
30 Pentagonal orthobicupola 20 40 22 of order 20
31 Pentagonal gyrobicupola 20 40 22 of order 20
32 Pentagonal orthocupolarotunda 25 50 27 of order 10
33 Pentagonal gyrocupolarotunda 25 50 27 of order 10
34 Pentagonal orthobirotunda 30 60 32 of order 20
35 Elongated triangular orthobicupola 18 36 20 of order 12
36 Elongated triangular gyrobicupola 18 36 20 of order 12
37 Elongated square gyrobicupola 24 48 26 of order 16
38 Elongated pentagonal orthobicupola 30 60 32 of order 20
39 Elongated pentagonal gyrobicupola 30 60 32 of order 20
40 Elongated pentagonal orthocupolarotunda 35 70 37 of order 10
41 Elongated pentagonal gyrocupolarotunda 35 70 37 of order 10
42 Elongated pentagonal orthobirotunda 40 80 42 of order 20
43 Elongated pentagonal gyrobirotunda 40 80 42 of order 20
44 Gyroelongated triangular bicupola 18 42 26 of order 6
45 Gyroelongated square bicupola 24 56 34 of order 8
46 Gyroelongated pentagonal bicupola 30 70 42 of order 10
47 Gyroelongated pentagonal cupolarotunda 35 80 47 of order 5
48 Gyroelongated pentagonal birotunda 40 90 52 of order 10
49 Augmented triangular prism 7 13 8 of order 4
50 Biaugmented triangular prism 8 17 11 of order 4
51 Triaugmented triangular prism 9 21 14 of order 12
52 Augmented pentagonal prism 11 19 10 of order 4
53 Biaugmented pentagonal prism 12 23 13 of order 4
54 Augmented hexagonal prism 13 22 11 of order 4
55 Parabiaugmented hexagonal prism 14 26 14 of order 8
56 Metabiaugmented hexagonal prism 14 26 14 of order 4
57 Triaugmented hexagonal prism 15 30 17 of order 12
58 Augmented dodecahedron 21 35 16 of order 10
59 Parabiaugmented dodecahedron 22 40 20 of order 20
60 Metabiaugmented dodecahedron 22 40 20 of order 4
61 Triaugmented dodecahedron 23 45 24 of order 6
62 Metabidiminished icosahedron 10 20 12 of order 4
63 Tridiminished icosahedron 9 15 8 of order 6
64 Augmented tridiminished icosahedron 10 18 10 of order 6
65 Augmented truncated tetrahedron 15 27 14 of order 6
66 Augmented truncated cube 28 48 22 of order 8
67 Biaugmented truncated cube 32 60 30 of order 16
68 Augmented truncated dodecahedron 65 105 42 of order 10
69 Parabiaugmented truncated dodecahedron 70 120 52 of order 20
70 Metabiaugmented truncated dodecahedron 70 120 52 of order 4
71 Triaugmented truncated dodecahedron 75 135 62 of order 6
72 Gyrate rhombicosidodecahedron 60 120 62 of order 10
73 Parabigyrate rhombicosidodecahedron 60 120 62 of order 20
74 Metabigyrate rhombicosidodecahedron 60 120 62 of order 4
75 Trigyrate rhombicosidodecahedron 60 120 62 of order 6
76 Diminished rhombicosidodecahedron 55 105 52 of order 10
77 Paragyrate diminished rhombicosidodecahedron 55 105 52 of order 10
78 Metagyrate diminished rhombicosidodecahedron 55 105 52 of order 2
79 Bigyrate diminished rhombicosidodecahedron 55 105 52 of order 2
80 Parabidiminished rhombicosidodecahedron 50 90 42 of order 20
81 Metabidiminished rhombicosidodecahedron 50 90 42 of order 4
82 Gyrate bidiminished rhombicosidodecahedron 50 90 42 of order 2
83 Tridiminished rhombicosidodecahedron 45 75 32 of order 6
84 Snub disphenoid 8 18 12 of order 8
85 Snub square antiprism 16 40 26 of order 16
86 Sphenocorona 10 22 14 of order 4
87 Augmented sphenocorona 11 26 17 of order 2
88 Sphenomegacorona 12 28 18 of order 4
89 Hebesphenomegacorona 14 33 21 of order 4
90 Disphenocingulum 16 38 24 of order 8
91 Bilunabirotunda 14 26 14 of order 8
92 Triangular hebesphenorotunda 18 36 20 of order 6

References

  1. ^ Meyer (2006), p. 418.
  2. ^
  3. ^
  4. ^
  5. ^
  6. ^
  7. ^ Uehara (2020), p. 62.
  8. ^
  9. ^ Flusser, Suk & Zitofa (2017), p. 126.
  10. ^
  11. ^ Walsh (2014), p. 284.
  12. ^ Parker (1997), p. 264.
  13. ^
  14. ^ Johnson (1966).
  15. ^ Berman (1971).

Bibliography