Bicupola (geometry)

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Set of bicupola
Faces	2n triangles, 2n squares 2 n-agon
Edges	8n
Vertices	4n
Symmetry group	Ortho: Dnh
Symmetry group	Gyro: Dnd
Dual polyhedron	?
Properties	convex

In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.

There are two classes of bicupola because each cupola half is bordered by alternating triangles and squares. If similar faces are attached together the result is an orthobicupola; if squares are attached to triangles it is a gyrobicupola.

Cupolae and bicupolae categorically exist as infinite sets of polyhedra, just like the pyramids, bipyramids, prisms, and trapezohedra.

Six bicupolae have regular polygon faces: triangular, square and pentagonal ortho- and gyrobicupolae. The triangular gyrobicupola is an Archimedean solid, the cuboctahedron; the other five are Johnson solids.

Bicupolae of higher order can be constructed if the flank faces are allowed to stretch into rectangles and isosceles triangles.

Bicupolae are special for having four faces on every vertex. This means that their dual polyhedra will have all quadrilateral faces. The best known example is the rhombic dodecahedron composed of 12 rhombic faces. The dual of the ortho-form, triangular orthobicupola, is also a dodecahedron, similar to rhombic dodecahedron, but it has 6 trapezoid faces which alternate long and short edges around the circumference.

[edit] Forms

Set of orthobicupolae:

	Triangular orthobicupola (J27): 8 triangles, 6 squares; its dual is the trapezo-rhombic dodecahedron
	Square orthobicupola (J28): 8 triangles, 10 squares
	Pentagonal orthobicupola (J30): 10 triangles, 10 squares, 2 pentagons
	n-gonal orthobicupola: n triangles, n squares, 2 n-gons

Set of gyrobicupolae:

	Triangular gyrobicupola or Cuboctahedron: 8 triangles, 6 squares; its dual is the rhombic dodecahedron
	Square gyrobicupola (J29): 8 triangles, 10 squares
	Pentagonal gyrobicupola (J31): 10 triangles, 10 squares, 2 pentagons
	n-gonal gyrobicupola: 2n triangles, 2n squares, 2 n-gons