Set of gyroelongated cupolaeExample of a pentagonal formFaces5n trianglesn squares 1 n-gon1 2n-gonEdges11nVertices5nSymmetry groupCnv, [n], (*nn)Rotational groupCn, [n]+, (nn)Dual polyhedronPropertiesconvex
In geometry, the gyroelongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to a 2n-gonal antiprism.
There are three gyroelongated cupolae that are Johnson solids made from regular triangles, squares, and pentagons. Higher forms can be constructed with isosceles triangles. Adjoining a triangular prism to a square antiprism also generates a polyhedron, but has adjacent parallel faces, so is not a Johnson solid. The hexagonal form can be constructed from regular polygons, but the cupola faces are all in the same plane. Topologically other forms can be constructed without regular faces.
Forms name faces gyroelongated digonal cupola 10 triangles, 2+1 squares gyroelongated triangular cupola (J22) 15+1 triangles, 3 squares, 1 hexagon gyroelongated square cupola (J23) 20 triangles, 4+1 squares, 1 octagon gyroelongated pentagonal cupola (J24) 25 triangles, 5 squares, 1 pentagon, 1 decagon gyroelongated hexagonal cupola 30 triangles, 6 squares, 1 hexagon, 1 dodecagon
See also Bicupola Elongated cupola Elongated bicupola Gyroelongated bicupola Rotunda
References Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. The first proof that there are only 92 Johnson solids.