In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose facesBy definition, each face is the intersection of the convex polyhedron with a different bounding plane, so no two faces are coplanar — any two adjacent faces form an angle less than 180 degrees. If instead a convex polyhedron is presented by giving a collection of polygons that a priori may be coplanar (e.g., by subdividing a face), one could write "strictly convex polyhedron" here to indicate the condition that no two of the polygons are coplanar, that no two meet in a 180-degree angle. This notion of "strictly convex" for polyhedra is not the same as the standard notion used for general convex sets: no convex polyhedra are strictly convex in the latter sense; see p. 263 of A. G. Khovanskii, Geometry of generalized virtual polyhedra, J. Math. Sciences 269 (2023), 256–269. are regular polygons and that is not a uniform polyhedron. There are 92 such solids: 48 composed of the primitive pyramids, cupolas, and rotundas assembled in various ways together with prisms and antiprisms; 35 formed by modifying uniform polyhedra, by augmenting, diminishing, or gyrating with primitives; and 9 which are not derived from "cut-and-paste" manipulations of uniform solids.
Definition and background
A convex polyhedron is the convex hull of a finite set of points in 3-dimensional space, not all in a plane. Its boundary is a finite union of polygons, no two in the same plane; those polygons are called the faces. A Johnson solid is a convex polyhedron whose faces are all regular polygons, but not a uniform polyhedron; the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms.
The solids are named after Norman Johnson and Victor Zalgaller. published a list of 92 such solids and assigned them their names and numbers. proved Johnson's conjecture that there were none beyond these 92.
A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a near-miss Johnson solid.
Naming and construction of solids
- invalid, - Platonic, - Archimedean, - Gyrated sections.
The naming of Johnson solids follows a flexible and precise descriptive formula that allows many solids to be named in multiple different ways without compromising the accuracy of each name as a description. The names of the Johnson solids are described in the following sections.
Pyramids, cupolas, rotundas The first 48 Johnson solids are constructed from pyramids, cupolas, or rotundas, combined with prisms or antiprisms. The following prefixes are attached to the word to indicate specific combinations of shapes:
Bi- indicates that two copies of the solid are joined base-to-base. For cupolas and rotundas, ortho- indicates that like faces meet. For cupolas and rotundas, gyro- indicates that unlike faces meet. Elongated indicates a prism is joined to the base of the solid, or between the bases. Gyroelongated indicates an antiprism is joined to the base of the solid, or between the bases.
Using this nomenclature, a pentagonal bipyramid is a solid constructed by attaching two bases of pentagonal pyramids. Triangular orthobicupola is constructed by two triangular cupolas along their bases.
Excluded solids: - coplanar, - Platonic, - Archimedean.
Pyramids Cupolas Cupola-Rotunda Rotundas Tetrahedron "triangular pyramid" 1 Square pyramid 2 Pentagonal pyramid 3 Triangular cupola 4 Square cupola 5 Pentagonal cupola 6 Pentagonal rotunda Elongated 7 Elongated triangular pyramid 8 Elongated square pyramid 9 Elongated pentagonal pyramid 18 Elongated triangular cupola 19 Elongated square cupola 20 Elongated pentagonal cupola 21 Elongated pentagonal rotunda Gyroelongated Augmented octahedron "Gyroelongated triangular pyramid" 10 Gyroelongated square pyramid 11 Gyroelongated pentagonal pyramid 22 Gyroelongated triangular cupola 23 Gyroelongated square cupola 24 Gyroelongated pentagonal cupola 25 Gyroelongated pentagonal rotunda orthobi- 12 Triangular bipyramid Octahedron "Square bipyramid" 13 Pentagonal bipyramid 27 Triangular orthobicupola 28 Square orthobicupola 30 Pentagonal orthobicupola 32 Pentagonal orthocupolarotunda 34 Pentagonal orthobirotunda gyrobi- Cuboctahedron "Triangular gyrobicupola" 29 Square gyrobicupola 31 Pentagonal gyrobicupola 33 Pentagonal gyrocupolarotunda Icosidodecahedron "pentagonal gyrobirotunda" Elongated orthobi- 14 Elongated triangular bipyramid 15 Elongated square bipyramid 16 Elongated pentagonal bipyramid 35 Elongated triangular orthobicupola Rhombicuboctahedron "Elongated square orthobicupola" 38 Elongated pentagonal orthobicupola 40 Elongated pentagonal orthocupolarotunda 42 Elongated pentagonal orthobirotunda Elongated gyrobi- 36 Elongated triangular gyrobicupola 37 Elongated square gyrobicupola 39 Elongated pentagonal gyrobicupola 41 Elongated pentagonal gyrocupolarotunda 43 Elongated pentagonal gyrobirotunda Gyroelongated bi- Trigonal trapezohedron "Gyroelongated triangular bipyramid" 17 Gyroelongated square bipyramid Icosahedron "Gyroelongated pentagonal bipyramid" 44 Gyroelongated triangular bicupola 45 Gyroelongated square bicupola 46 Gyroelongated pentagonal bicupola 47 Gyroelongated pentagonal cupolarotunda 48 Gyroelongated pentagonal birotunda
Fastigium gyrobi- 26 Gyrobifastigium
Modified uniform polyhedra
The next 35 Johnson solids are constructed by modifying uniform polyhedra such as prisms, Platonic, or Archimedean solids by adding, subtracting, or rotating pyramids or cupolas. The following prefixes are attached to the word to indicate additions, subtractions, or rotations:
Augmented indicates a pyramid or cupola is added to one or more faces of the solid in question. Diminished indicates a pyramid or cupola is removed from one or more faces of the solid in question. Gyrate indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up.
The three operations—augmentation, diminution, and gyration—can be performed multiple times for certain large solids. Bi- & Tri- indicate a double and triple operation respectively. For example, a bigyrate solid has two rotated cupolas, and a tridiminished solid has three removed pyramids or cupolas. In certain large solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. Para- indicates the former, that the solid in question has altered parallel faces, and meta- the latter, altered oblique faces. For example, a parabiaugmented solid has had two parallel faces augmented, and a metabigyrate solid has had two oblique cupolas gyrated.
Augmented from Prisms augmented by pyramids Triangular prism 49 Augmented triangular prism 50 Biaugmented triangular prism 51 Triaugmented triangular prism Pentagonal prism 52 Augmented pentagonal prism 53 Biaugmented pentagonal prism Hexagonal prism 54 Augmented hexagonal prism 55 Parabiaugmented hexagonal prism 56 Metabiaugmented hexagonal prism 57 Triaugmented hexagonal prism Modified from Platonics modified by pyramids Regular dodecahedron 58 Augmented dodecahedron 59 Parabiaugmented dodecahedron 60 Metabiaugmented dodecahedron 61 Triaugmented dodecahedron Regular icosahedron 62 Metabidiminished icosahedron 63 Tridiminished icosahedron 64 Augmented tridiminished icosahedron Modified from Archimedeans modified by cupolas Truncated tetrahedron 65 Augmented truncated tetrahedron Truncated cube 66 Augmented truncated cube 67 Biaugmented truncated cube Truncated dodecahedron 68 Augmented truncated dodecahedron 69 Parabiaugmented truncated dodecahedron 70 Metabiaugmented truncated dodecahedron 71 Triaugmented truncated dodecahedron Rhombicosidodecahedron 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
Non cut-and-paste
The last 9 Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson with the following nomenclature: A lune is a figure of two triangles attached to opposite sides of a square. Prefixes indicating a complex of lunes are: Spheno- is a wedgelike complex of two adjacent lunes. Dispheno- indicates two such complexes. Hebespheno- is a blunt complex of three adjacent lunes. Suffixes indicating a complex of triangles are: -corona is a crownlike complex of eight triangles. -megacorona is a larger crownlike complex of twelve triangles. -cingulum is a belt of twelve triangles. Suffix -rotunda indicates a complex of two or three pentagons with triangles between them, bearing a structural resemblance to the pentagonal rotunda.
Snub polyhedra 84 Snub disphenoid 85 Snub square antiprism Formed from lunes and triangles 86 Sphenocorona 87 Augmented sphenocorona 88 Sphenomegacorona 89 Hebesphenomegacorona 90 Disphenocingulum Rotundoids 91 Bilunabirotunda 92 Triangular hebesphenorotunda
Notable subsets Deltahedra
Five Johnson solids are deltahedra, with only triangle faces. J12 Triangular bipyramid J13 Pentagonal bipyramid J17 Gyroelongated square bipyramid J51 Triaugmented triangular prism J84 Snub disphenoid
Elementary solids Seventeen 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 first six Johnson solids satisfy this criterion: J1 square pyramid J2 pentagonal pyramid J3 triangular cupola J4 square cupola J5 pentagonal cupola J6 pentagonal rotunda
The criterion is also satisfied by eleven other Johnson solids: J63 tridiminished icosahedron J80 parabidiminished rhombicosidodecahedron J83 tridiminished rhombicosidodecahedron J84 snub disphenoid J85 snub square antiprism J86 sphenocorona J88 sphenomegacorona J89 hebesphenomegacorona J90 disphenocingulum J91 bilunabirotunda J92 triangular hebesphenorotunda
Chiral solids
The five gyroelongated bicupolas or birotundas are chiral and have distinct left-handed and right-handed forms. J44 Gyroelongated triangular bicupola J45 Gyroelongated square bicupola J46 Gyroelongated pentagonal bicupola J47 Gyroelongated pentagonal cupolarotunda J48 Gyroelongated pentagonal birotunda
Circumscribable solids
Twenty five of the Johnson solids have vertices that exist on the surface of a sphere. All of them can be seen to be related to a Platonic or Archimedean solid by gyration, diminishment, or dissection. OctahedronIcosahedron J1 J2 J11 J62 J63
CuboctahedronRhombicuboctahedronIcosidodecahedron J3 J27 J4 J19 J37 J6 J34Rhombicosidodecahedron J5 J72 J73 J74 J75 J76 J77 J78 J79 J80 J81 J82 J83
Characteristics of solids