# Johnson solid

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A Johnson solid is a strictly convex regular-faced polyhedron that is not uniform. They are named after Norman W. Johnson, who in 1966 first listed all 92 such polyhedra.[1] Before him, Duncan Sommerville discovered the subset of them that are circumscribable.[2] In 1969, Victor Zalgaller proved that the list was complete.[3]

Even though they are not allowed to be uniform, Johnson solids can have just one type of polygon for their faces, as the triangular bipyramid does, or have only one vertex figure, as the elongated square gyrobicupola does.

Johnson solids only have triangles, squares, pentagons, hexagons, octagons, or decagons as faces. All have at least some degree of symmetry.

## Generalizations

Relaxing the requirement of convexity (and allowing uniforms) results in the acrohedra, which also encompass the Stewart toroids. The entire set of acrohedra is not generally studied, but it is interesting to ask if an acrohedron exists under certain constraints, such as having a certain configuration of faces around at least one vertex.

The higher-dimensional generalizations of the Johnson solids include the non-uniform CRF polytopes, where faces must be regular, and the non-uniform Blind polytopes, where facets (e.g. cells in a 4-polytope) are regular. The Blind polytopes have been completely enumerated, while the CRF polytopes are a much broader class.

While all Johnson solids are symmetrical, the Blind polytopes contain asymmetrical polytopes, all of which are special cuts, which are 4D analogues of the diminished icosahedra in the Johnson solids.

Roger Kaufman investigated the convex triamond polyhedra, which are convex polyhedra with all regular faces except for at least one "triamond," defined as a trapezoid with edge lengths 1:1:1:2.

A large amount of near-miss Johnson solids may also be constructed. These polyhedra are convex and all of their faces are either regular or almost regular. They may also use polygons unavailable to the proper Johnson solids, such as the heptagon, enneagon, hendecagon, or dodecagon.

## References

1. 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.
2. Sommerville, D. M. Y. (1905), "Semi-regular networks of the plane in absolute geometry", Transactions of the Royal Society of Edinburgh, 41: 725–747, doi:10.1017/s0080456800035560.
3. Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. Zbl 0177.24802. No ISBN. The first proof that there are only 92 Johnson solids: see also 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.