Gyroelongated cupola

Set of gyroelongated cupolae
Example pentagonal form
Faces	3n triangles n squares 1 n-gon 1 2n-gon
Edges	9n
Vertices	5n
Symmetry group	Cnv, [n], (*nn)
Rotational group	Cn, [n]+, (nn)
Dual polyhedron
Properties	convex

In geometry, the gyroelongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to an 2n-gonal antiprism.

There are three gyroelongated cupolae that are Johnson solids made from regular triangles and square, 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	2+8 triangles, 2+1 square
	gyroelongated triangular cupola (J22)	9+1 triangles, 3 squares, 1 hexagon
	gyroelongated square cupola (J23)	12 triangles, 4+1 squares, 1 octagon
	gyroelongated pentagonal cupola (J24)	15 triangles, 5 squares, 1 pentagon, 1 decagon
	gyroelongated hexagonal cupola	18 triangles, 6 squares, 1 hexagon, 1 dodecagon

See also

• Elongated cupola
• Elongated bicupola

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.
• Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN.  The first proof that there are only 92 Johnson solids.