In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces 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. Johnson (1966) published a list of 92 such solids and assigned them their names and numbers. Zalgaller (1969) 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.
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.
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.
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.
Characteristics of solids 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. 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. A volume is a measurement of a region in three-dimensional space. 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. A polygon that is rotated symmetrically by 360 ∘ n {\textstyle {\frac {360^{\circ }}{n}}} is denoted by C n {\displaystyle C_{n}} , a cyclic group of order n {\displaystyle n} ; combining this with the reflection symmetry results in the symmetry of dihedral group D n {\displaystyle D_{n}} of order 2 n {\displaystyle 2n} . 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 C n v {\displaystyle C_{n\mathrm {v} }} of order 2 n {\displaystyle 2n} . The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry D n h {\displaystyle D_{n\mathrm {h} }} of order 4 n {\displaystyle 4n} . The antiprismatic symmetry D n d {\displaystyle D_{n\mathrm {d} }} of order 4 n {\displaystyle 4n} preserves the symmetry by rotating its half bottom and reflection across the horizontal plane. The symmetry group C n h {\displaystyle C_{n\mathrm {h} }} of order 2 n {\displaystyle 2n} 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 C 1 h {\displaystyle C_{1\mathrm {h} }} of order 2, often denoted as C s {\displaystyle C_{s}} .
The table below lists the properties of the 92 (non-uniform) Johnson solids. The table includes each solid's enumeration (denoted as J n {\displaystyle J_{n}} ). It also includes each solid's symmetry group and number of vertices, edges, and faces, as well as its surface area and volume when constructed with edge length 1. For simplicity, the table uses the quantity α = 5 + 2 5 {\displaystyle \alpha =5+2{\sqrt {5}}} .
See also Near-miss Johnson solid Blind polytope
References
Bibliography
External links Paper Models of Polyhedra Archived 2013-02-26 at the Wayback Machine Many links Hart, George W. "Johnson Solids". Visual Polyhedra, with 3D models and data for all 92 solids, by David I. McCooey. Images of all 92 solids, categorized, on one page Weisstein, Eric W. "Johnson Solid". MathWorld. VRML models of Johnson Solids by Jim McNeill Bulatov, Vladimir. "Johnson solids". – VRML models of Johnson solids CRF polychora discovery project attempts to discover CRF polychora Archived 2020-10-31 at the Wayback Machine (Convex 4-dimensional polytopes with Regular polygons as 2-dimensional Faces), a generalization of the Johnson solids to 4-dimensional space https://levskaya.github.io/polyhedronisme/ a generator of polyhedrons and Conway operations applied to them, including Johnson solids.