---
tag: ZSA992C21V
kind: object
name: Johnson solid
msc: [51M20, 52B10]
instance_of: [300K6CB1KH]
uses: [01KJJ9A400, 017E5PRS1Q, BZXHGNGQAX]
sources:
  - url: https://en.wikipedia.org/wiki/Johnson_solid
same_as:
  wikidata: Q1059659
  wikipedia: Johnson solid
---

## Description

A *Johnson solid* is a strictly convex polyhedron whose faces are all [[regular polygons|01KJJ9A400]].
It must not be a [[Platonic solid|017E5PRS1Q]], 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|BZXHGNGQAX]], 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.

## Explanations

### L1 human

Take regular polygons, such as equilateral triangles, squares and regular pentagons, and glue them into a convex solid.
If every corner looks the same, the result is a Platonic solid, an Archimedean solid, a prism or an antiprism.
The Johnson solids are all the other ones, where some corners differ, and there are exactly $92$ of them.
A pyramid with a square base and four equilateral sides is the simplest example.