Johnson solid

The naming of Johnson solids follows a flexible and precise descriptive formula, such that many solids can be named in different ways without compromising their accuracy as a description. Most Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundas), together with the Platonic and Archimedean solids, prisms, and antiprisms; the centre of a particular solid's name will reflect these ingredients. From there, a series of prefixes are attached to the word to indicate additions, rotations, and transformations:

• Bi-[<>] indicates that two copies of the solid in question are joined base-to-base. For cupolae and rotundas, the solids can be joined so that either like faces (ortho-) or unlike faces (gyro-[*]) meet. Using this nomenclature, an octahedron can be described as a square bipyramid[4<>], a cuboctahedron as a triangular gyrobicupola[3cc*], and an icosidodecahedron as a pentagonal gyrobirotunda[5rr*].
• Elongated[=] indicates a prism is joined to the base of the solid in question, or between the bases in the case of Bi- solids. A rhombicuboctahedron can thus be described as an elongated square orthobicupola.
• Gyroelongated[z] indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids. An icosahedron can thus be described as a gyroelongated pentagonal bipyramid.
• Augmented[+] indicates another polyhedron, namely a pyramid or cupola, is joined 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, as in the difference between ortho- and gyrobicupolae.

The last 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 cupolae, and a tridiminished solid has three removed pyramids or cupolae.

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 2 oblique faces gyrated.

The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson[2] with the following nomenclature:

• A lune is a complex of two triangles attached to opposite sides of a square.
• Spheno- indicates a wedgelike complex formed by two adjacent lunes. Dispheno- indicates two such complexes.
• Hebespheno- indicates a blunt complex of two lunes separated by a third lune.
• Corona is a crownlike complex of eight triangles.
• Megacorona is a larger crownlike complex of 12 triangles.
• The suffix -cingulum indicates a belt of 12 triangles.

25 of the Johnson solids have vertices that exist on the surface of a sphere: 1–6,11,19,27,34,37,62,63,72–83. All of them can be seen to be related to a regular or uniform polyhedra by gyration, diminishment, or dissection.[3]

Octahedron	Cuboctahedron		Rhombicuboctahedron
J1	J3	J27	J4	J19	J37

Icosahedron				Icosidodecahedron
J2	J11	J62	J63	J6	J34

Rhombicosidodecahedron
J5	J72	J73	J74	J75	J76	J77
	J78	J79	J80	J81	J82	J83

• Near-miss Johnson solid
• Catalan solid
• Toroidal polyhedron

• Johnson, Norman W. (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. ISSN 0008-414X. Zbl 0132.14603. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
• Zalgaller, Victor A. (1967). "Convex Polyhedra with Regular Faces". Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (in Russian). 2: 1–221. ISSN 0373-2703. Zbl 0165.56302. The first proof that there are only 92 Johnson solids. English translation: Zalgaller, Victor A. (1969). "Convex Polyhedra with Regular Faces". Seminars in Mathematics, V. A. Steklov Math. Inst., Leningrad. Consultants Bureau. 2. ISSN 0080-8873. Zbl 0177.24802.
• Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 3 Further Convex polyhedra
• Timofeenko, A.V. (2009). "The Non-Platonic and Non-Archimedean Noncomposite Polyhedra". J. Math. Sci. 162 (5): 710–729. doi:10.1007/s10958-009-9655-0. S2CID 120114341.

olyhedra." J. Math. Sci. 162, 710-729, 2009.

• Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières" [Convex polyhedra with regular faces] (PDF). Structural Topology (6): 83–95.
• Paper Models of Polyhedra Archived 2013-02-26 at the Wayback Machine Many links
• Johnson Solids by George W. Hart.
• Images of all 92 solids, categorized, on one page
• Weisstein, Eric W. "Johnson Solid". MathWorld.
• VRML models of Johnson Solids by Jim McNeill
• VRML models of Johnson Solids by Vladimir Bulatov
• 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.