Bicupola (geometry)

In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. There are two classes of bicupola because each cupola (bicupola 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 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. A n-gonal gyrobicupola has the same topology as a n-gonal rectified antiprism, Conway polyhedron notation, aAn.

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Related concepts (11)

Square orthobicupola

In geometry, the square orthobicupola is one of the Johnson solids (J_28). As the name suggests, it can be constructed by joining two square cupolae (J_4) along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola (J_29). The square orthobicupola is the second in an infinite set of orthobicupolae.

Square gyrobicupola

In geometry, the square gyrobicupola is one of the Johnson solids (J_29). Like the square orthobicupola (J_28), it can be obtained by joining two square cupolae (J_4) along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another. The square gyrobicupola is the second in an infinite set of gyrobicupolae. Related to the square gyrobicupola is the elongated square gyrobicupola. This polyhedron is created when an octagonal prism is inserted between the two halves of the square gyrobicupola.

Gyrobifastigium

In geometry, the gyrobifastigium is the 26th Johnson solid (J_26). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space. It is also the vertex figure of the nonuniform p-q duoantiprism (if p and q are greater than 2). Despite the fact that p, q = 3 would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices, except for the case p = 5, q = 5/3, which represents a uniform great duoantiprism.