Johnson Solid — Definition, Formula & Examples
A Johnson solid is one of 92 convex polyhedra whose faces are all regular polygons but which are not Platonic solids, Archimedean solids, prisms, or antiprisms. They are named after Norman Johnson, who catalogued the complete list in 1966.
A Johnson solid is a strictly convex polyhedron each of whose faces is a regular polygon (equilateral triangle, square, regular pentagon, etc.) and which does not belong to any of the other families of regular-faced convex polyhedra: the 5 Platonic solids, the 13 Archimedean solids, the infinite families of prisms, or the infinite families of antiprisms. Victor Zalgaller proved in 1969 that exactly 92 such solids exist.
How It Works
Each Johnson solid is labeled through . The simplest is the square pyramid (): a square base topped by four equilateral triangles. Unlike Archimedean solids, Johnson solids do not require the same arrangement of faces around every vertex. You can identify a Johnson solid by checking that all faces are regular polygons, the shape is convex, and it does not fit into any of the other named families.
Worked Example
Problem: Verify that the square pyramid (J₁) satisfies Euler's formula for polyhedra, given it has 5 faces, 8 edges, and 5 vertices.
Recall Euler's formula: For any convex polyhedron, the number of vertices minus the number of edges plus the number of faces equals 2.
Substitute the values: The square pyramid has , , and (1 square base and 4 triangular faces).
Answer: The result is 2, confirming the square pyramid satisfies Euler's formula.
Why It Matters
Johnson solids appear in advanced geometry courses and in fields like crystallography, architecture, and 3D modeling, where understanding the full catalogue of regular-faced convex shapes matters. Recognizing them deepens your grasp of how restrictions on face regularity, convexity, and vertex uniformity classify polyhedra.
Common Mistakes
Mistake: Assuming any convex solid built from regular polygons must be a Platonic or Archimedean solid.
Correction: Platonic solids require identical regular faces, and Archimedean solids require identical vertex arrangements. Johnson solids have regular polygon faces but lack these symmetry constraints, giving 92 additional shapes.