Johnson solid

Builds on Regular polygon Platonic solid Uniform polytope

Description

A Johnson solid is a strictly convex polyhedron whose faces are all regular polygons. It must not be a Platonic solid, an Archimedean solid, a prism or an antiprism. Strictly convex means that no two adjacent faces lie in a common plane. There are exactly $92$ Johnson solids.

Example: the square pyramid with a unit square base and four equilateral triangles is a Johnson solid. The apex lies above the centre of the base at height $h$ with $h^2 + (\sqrt2/2)^2 = 1$ , so $h = \sqrt2/2$ , and all eight edges have length $1$ . Its faces are regular, but its apex lies on $4$ edges while each base vertex lies on $3$ . So it is not vertex-transitive, hence not Platonic or Archimedean, and it is not a prism or an antiprism either.

Remarks

The excluded families are the convex uniform polyhedra, whose symmetries act transitively on vertices. So the Johnson solids are exactly the strictly convex polyhedra with regular faces that are not vertex-transitive. Norman Johnson published the list of $92$ in 1966, and Victor Zalgaller proved that the list is complete. Other examples are the pentagonal pyramid, the triangular cupola and the elongated square bipyramid.

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Tag	`ZSA992C21V`
URI	`m2://ZSA992C21V`
Kind	object
Writing	W0 stub
MSC	51M20, 52B10
Instance of	Convex polytope
Uses	Regular polygon, Platonic solid, Uniform polytope
Same as	Wikidata Q1059659 Wikipedia Johnson solid

Johnson solid

Description

Remarks

Reading path

Further reading

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