Set of bicupolae
Faces	2n triangles, 2n square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... s 2 n-agon Polygon In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
Edges	8n
Vertices	4n
Symmetry group	Ortho: Dnh Gyro: Dnd
Properties	convex

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a bicupola is a solid formed by connecting two cupolae

Cupola (geometry)

In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of triangles and rectangles...

 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 pyramid

Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....

s, bipyramid

Bipyramid

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...

s, prisms

Prism (geometry)

In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

, and trapezohedra

Trapezohedron

The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...

.

Six bicupolae have regular polygon

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 faces: triangular

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

, square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

 and pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

al ortho- and gyrobicupolae. The triangular gyrobicupola is an Archimedean solid

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

, the cuboctahedron

Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

; the other five are Johnson solid

Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

s.

Bicupolae of higher order can be constructed if the flank faces are allowed to stretch into rectangle

Rectangle

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

s and isosceles triangles.

Bicupolae are special in having four faces on every vertex. This means that their dual polyhedra will have all quadrilateral

Quadrilateral

In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

 faces. The best known example is the rhombic dodecahedron

Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

 composed of 12 rhombic faces. The dual of the ortho-form, triangular orthobicupola

Triangular orthobicupola

In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases...

, is also a dodecahedron, similar to rhombic dodecahedron, but it has 6 trapezoid faces which alternate long and short edges around the circumference.

Set of orthobicupolae

Symmetry	Picture	Description
D3h [2,3] *223		Triangular orthobicupola Triangular orthobicupola In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases...  (J27): 8 triangles, 6 squares; its dual is the trapezo-rhombic dodecahedron Trapezo-rhombic dodecahedron The trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces.This shape could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom...
D4h [2,4] *224		Square orthobicupola Square orthobicupola In geometry, the square orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces...  (J28): 8 triangles, 10 squares
D5h [2,5] *225		Pentagonal orthobicupola Pentagonal orthobicupola In geometry, the pentagonal orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by joining two pentagonal cupolae along their decagonal bases, matching like faces...  (J30): 10 triangles, 10 squares, 2 pentagons
Dnh [2,n] *22n		n-gonal orthobicupola: 2n triangles, 2n squares, 2 n-gons

Set of gyrobicupolae

Symmetry	Picture	Description
D2d [2+,4] 2*2		Gyrobifastigium Gyrobifastigium In geometry, the gyrobifastigium is the 26th Johnson solid . It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a half-turn to one prism....  (J26): 4 triangles, 4 squares
D3d [2+,6] 2*3		Triangular gyrobicupola or cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,... : 8 triangles, 6 squares; its dual is the rhombic dodecahedron Rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...
D4d [2+,8] 2*4		Square gyrobicupola Square gyrobicupola In geometry, the square gyrobicupola is one of the Johnson solids . Like the square orthobicupola , it can be obtained by joining two square cupolae along their bases...  (J29): 8 triangles, 10 squares
D5d [2+,10] 2*5		Pentagonal gyrobicupola Pentagonal gyrobicupola In geometry, the pentagonal gyrobicupola is one of the Johnson solids . Like the pentagonal orthobicupola , it can be obtained by joining two pentagonal cupolae along their bases... (J31): 10 triangles, 10 squares, 2 pentagons
Dnd [2+,2n] 2*n		n-gonal gyrobicupola: 2n triangles, 2n squares, 2 n-gons