Johnson SolidEdit
Johnson solids are a well-defined class of convex polyhedra distinguished by having all faces as regular polygons, while not belonging to the uniform family of solids. In total, there are 92 such shapes, a finite collection that sits between the Platonic and Archimedean solids on one side and the wide universe of general convex polyhedra on the other. The name and cataloging of these solids reflect a collaboration between mathematical description and geometric construction, and they are studied for the way regular faces can be combined in diverse, non-uniform ways to yield stable, convex forms. polyhedron convex polyhedron regular polygon Platonic solid Archimedean solid
Definition and scope
A Johnson solid is any convex polyhedron whose faces are regular polygons, provided that the solid is not one of the Platonic solids, not an Archimedean solid, and not formed by the standard infinite families of prisms or antiprisms. In other words, a Johnson solid is a convex polyhedron built from regular polygons that does not have the high degree of symmetry that characterizes the uniform solids. The emphasis on regular faces distinguishes Johnson solids from the broader set of convex polyhedra, while the exclusion of prisms and antiprisms carves out a specific, finite family. polyhedron convex polyhedron regular polygon Platonic solid Archimedean solid prism antiprism
The collection is finite and countable: exactly 92 shapes meet the criteria. These solids are not uniform, meaning their vertex configurations are not identical at every vertex, although each individual solid has regular faces. This non-uniformity is what gives Johnson solids their variety and their particular place in the taxonomy of three-dimensional solids. Not all constructions that produce convex polyhedra with regular faces qualify; the shape must fail to be uniform to be included in the Johnson list. uniform polyhedron vertex figure
Construction and examples
A hallmark of Johnson solids is how they are often built: by augmenting, elongating, or otherwise modifying regular polyhedra in ways that preserve regular faces while breaking uniform symmetry. Common construction themes include: - augmenting a face of a regular polyhedron with a pyramid so that the added faces are regular polygons (typically triangles) while the base remains a regular polygon. This is a standard route to many Johnson solids. pyramid (polyhedron) - attaching a cupola (a polyhedral cap) to a face of a regular polyhedron, yielding new shapes with regular faces. Cupolae are themselves built from regular polygons and often serve as modular building blocks in Johnson solids. cupola - combining several such augmentations, sometimes separated by elongation or other operations, to produce more complex convex shapes that nonetheless preserve regular faces. polyhedron
These methods yield shapes that are robust in the sense of convexity and face regularity, yet diverse in vertex arrangement. While many Johnson solids can be traced back to augmentations of a Platonic solid, the resulting object is not uniform, which is precisely what distinguishes it from a uniform solid. The explicit catalog of 92 solids reflects the full range of such permissible augmentations and combinations. Platonic solid Johnson solids
Notable features across the Johnson family include a variety of vertex configurations, such as mixes of triangle and square faces, and the occasional use of pentagonal or hexagonal faces in carefully balanced arrangements. The study of these solids often focuses on which regular faces can meet at a vertex in a way that preserves convexity, a question tied to angle sums and dihedral angles. dihedral angle vertex figure
Historical background
The concept and enumeration of Johnson solids trace to the work of Norman W. Johnson, who described the set of convex polyhedra with regular faces in 1966. The solids themselves are named in common parlance by later mathematicians, most notably Jonathan F. Bowers, who popularized the term “Johnson solids” and contributed to the broader cataloging and discussion of these shapes. The 92-solid list has since become a standard reference in geometry, often appearing alongside discussions of regular and semi-regular solids and their relationships to more general polyhedral families. Norman W. Johnson Jonathan F. Bowers List of Johnson solids
The Johnson solids occupy a distinct niche in the geometric landscape: they are regular-faced, convex, non-uniform, and finite in number, bridging the gap between highly symmetric solids and the wider class of all convex polyhedra. This layering mirrors broader patterns in geometry where strict regularity meets creative construction, yielding a catalog that is both rigorous and richly imaginative. regular polygon polyhedron