Semiregular

Semiregular polyhedra are a class of convex polyhedra that combine elements of both regular and uniform polyhedra. They are characterized by their vertices, edges, and faces being arranged in a way that is neither fully regular (where all faces are identical regular polygons and all vertices are identical) nor fully uniform (where faces are congruent but vertices may vary). Instead, semiregular polyhedra exhibit a consistent pattern of vertex figures, meaning that the arrangement of faces around each vertex is identical throughout the polyhedron.

The most well-known semiregular polyhedra include the Catalan solids, which are duals of Archimedean solids. These include the truncated tetrahedron, truncated cube, truncated octahedron, rhombicuboctahedron, and others. Another notable group is the Johnson solids, which are convex polyhedra with regular polygonal faces and identical vertex figures, though they are not necessarily uniform. The Johnson solids include shapes like the rhombic dodecahedron, snub dodecahedron, and truncated square pyramid.

Semiregular polyhedra are often studied in geometry for their aesthetic appeal and mathematical complexity. They serve as a bridge between simpler regular polyhedra and more complex uniform polyhedra, offering a rich variety of shapes that can be constructed using regular polygons. Their properties are often explored in fields such as crystallography, where they appear in certain crystal structures, and in architectural design, where their symmetry can inspire innovative forms.