Johnson Solids

Convex polyhedra with regular faces that are neither Platonic nor Archimedean

Pyramids

Square Pyramid 5 faces
Pyramid with square base - fundamental Johnson solid
Pentagonal Pyramid 6 faces
Pyramid with pentagonal base - classic Johnson form

Cupolas and Rotundas

Triangular Cupola 8 faces
Dome-like structure with triangular top and hexagonal base
Square Cupola 10 faces
Dome with square top and octagonal base
Pentagonal Cupola 12 faces
Dome with pentagonal top and decagonal base
Rotunda Half Icosidodecahedron
Half of an icosidodecahedron - complex cupola form

Elongated Forms

Elongated Triangular Pyramid 7 faces
Triangular prism with triangular pyramid on top
Elongated Square Pyramid 9 faces
Square prism with square pyramid on top
Elongated Pentagonal Pyramid 11 faces
Pentagonal prism with pentagonal pyramid on top

Bipyramids

Triangular Bipyramid 6 faces
Two triangular pyramids joined at their bases
Pentagonal Bipyramid 10 faces
Two pentagonal pyramids joined at their bases
Elongated Square Bipyramid 12 faces
Square bipyramid with elongated middle section

Complex Johnson Forms

Gyrobifastigium Twisted
Twisted double wedge - unique Johnson solid
Gyroelongated Square Dipyramid Rotated
Square bipyramid with twisted elongation
Snub Disphenoid Chiral
Twisted tetrahedron variant - highly irregular Johnson solid

About Johnson Solids

Johnson solids are convex polyhedra with regular faces that don't fit into other well-known categories:

  • Architecture - Complex building forms
  • Engineering - Specialized components
  • Crystallography - Unusual crystal forms
  • Mathematics - Polyhedral theory
  • 3D Modeling - Complex shapes
  • Education - Geometric examples
Johnson Solid Properties
Regular Faces
All faces are regular polygons
(triangles, squares, pentagons, etc.)
Convexity
All vertices point outward
No concave surfaces
Non-Uniform
Not vertex-transitive
Different vertex types
Completeness
Exactly 92 Johnson solids
Enumerated by Norman Johnson
Historical Note: Norman Johnson enumerated all 92 solids in 1966, and Victor Zalgaller proved the list complete in 1969.

Categories and Applications

Pyramids & Bipyramids
  • Architecture: Roof structures, spires
  • Engineering: Structural joints
  • Art: Sculptural forms
Cupolas & Rotundas
  • Architecture: Dome structures
  • Engineering: Pressure vessels
  • Design: Decorative elements
Elongated Forms
  • Manufacturing: Container shapes
  • Architecture: Extended structures
  • Packaging: Specialized containers
Complex Forms
  • Research: Mathematical models
  • 3D Printing: Test geometries
  • Education: Geometry examples
Quick Reference
92 Solids
Johnson Solids
Regular
Faces
Convex
Shape
1966
Enumerated
1969
Proven Complete
Historical Context

Norman Johnson (1966): Systematically enumerated all 92 convex polyhedra with regular faces.

Victor Zalgaller (1969): Proved that Johnson's list of 92 solids was complete.

Modern Use: Applications in architecture, 3D modeling, and mathematical research.