Gyroelongated bicupola

In geometry, the gyroelongated bicupolae are an infinite sets of polyhedra, constructed by adjoining two n-gonal cupolas to an n-gonal Antiprism. The triangular, square, and pentagonal gyroelongated bicupola are three of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form.

Set of gyroelongated bicupolae

Example pentagonal form
Faces	6n triangles 2n squares 2 n-gon
Edges	16n
Vertices	6n
Symmetry group	Dn, [n,2]+, (n22)
Rotation group	Dn, [n,2]+, (n22)
Properties	convex, chiral

Adjoining two triangular prisms to a square antiprism also generates a polyhedron, but not a Johnson solid, as it is not convex. The hexagonal form is also a polygon, but has coplanar faces. Higher forms can be constructed without regular faces.

Image cw	Image ccw	Name	Faces
		Gyroelongated digonal bicupola	4 triangles, 4 squares
		Gyroelongated triangular bicupola (J44)	6+2 triangles, 6 squares
		Gyroelongated square bicupola (J45)	8 triangles, 8+2 squares
		Gyroelongated pentagonal bicupola (J46)	30 triangles, 10 squares, 2 pentagon
		Gyroelongated hexagonal bicupola	12 triangles, 24 squares, 2 hexagon

See also

• Elongated cupola

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.