 Bicupola (geometry)

Set of bicupolae Faces 2n triangles,
2n squares
2 nagonEdges 8n Vertices 4n Symmetry group Ortho: D_{nh}
Gyro: D_{nd}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 in 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 orthoform, 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.
Contents
Forms
Set of orthobicupolae
Symmetry Picture Description D_{3h}
[2,3]
*223Triangular orthobicupola (J27): 8 triangles, 6 squares; its dual is the trapezorhombic dodecahedron D_{4h}
[2,4]
*224Square orthobicupola (J28): 8 triangles, 10 squares D_{5h}
[2,5]
*225Pentagonal orthobicupola (J30): 10 triangles, 10 squares, 2 pentagons D_{nh}
[2,n]
*22nngonal orthobicupola: 2n triangles, 2n squares, 2 ngons Set of gyrobicupolae
Symmetry Picture Description D_{2d}
[2+,4]
2*2Gyrobifastigium (J26): 4 triangles, 4 squares D_{3d}
[2+,6]
2*3Triangular gyrobicupola or cuboctahedron: 8 triangles, 6 squares; its dual is the rhombic dodecahedron D_{4d}
[2+,8]
2*4Square gyrobicupola (J29): 8 triangles, 10 squares D_{5d}
[2+,10]
2*5Pentagonal gyrobicupola (J31): 10 triangles, 10 squares, 2 pentagons D_{nd}
[2+,2n]
2*nngonal gyrobicupola: 2n triangles, 2n squares, 2 ngons 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.
Polyhedron navigator Platonic solids (regular) Archimedean solids
(Semiregular/Uniform)Catalan solids
(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular Dihedral uniform Duals of dihedral uniform Dihedral others Degenerate polyhedra are in italics.Categories:
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