Cupola

A cupola (plural cupolas or cupolae) is a segmentohedron joining an n-sided upper base to a 2n-sided lower base with a ring of triangles and squares (isosceles triangles and rectangles in non-regular faced variants). Three cupolas are CRF: the triangular cupola, square cupola and pentagonal cupola.

Convex cupolas
A n-cupola can be seen as the cap of the uniform polyhedron x3onx - the triangular prism, cuboctahedron, small rhombicuboctahedron, small rhombicosidodecahedron, small rhombitrihexagonal tiling, small rhombitriheptagonal tiling, small rhombitrioctagonal tiling, etc.
 * Triangular prism (fastigium) (2-cupola) (digons degenerate to edges)
 * Triangular cupola (3-cupola)
 * Square cupola (4-cupola)
 * Pentagonal cupola (5-cupola)
 * Hexagonal cupola (6-cupola) (CRF version is flat)
 * Heptagonal cupola (7-cupola) (CRF version is hyperbolic)
 * Octagonal cupola (8-cupola) (CRF version is hyperbolic)

Star cupolas
These connect a m/n-gon to a (2m)/n-gon.


 * Retrograde square cupola (4/3-cupola) - cap of the quasirhombicuboctahedron
 * Retrograde pentagrammic cupola (5/3-cupola) - cap of the quasirhombicosidodecahedron
 * Great heptagrammic cupola (7/3-cupola)
 * Retrograde heptagrammic cupola (7/5-cupola)
 * Octagrammic cupola (8/3-cupola)
 * Retrograde octagrammic cupola (8/5-cupola)

Cuploids
If n is even, a m/n-cupola has a double-covered lower base, making it degenerate. Removing the degenerate base results in a legitimate polyhedron, called a cuploid (or semicupola). Instead of a lower base, the ring of triangles and rectangles wraps twice around pseudoface with an odd number of sides, connecting to itself.

The edges of the pentagrammic and retrograde pentagonal cuploids are contained within the small ditrigonary icosidodecahedron.
 * Tetrahemihexahedron (3/2-cuploid)
 * Pentagrammic cuploid (5/2-cuploid)
 * Retrograde pentagonal cuploid (5/4-cuploid)
 * Heptagrammic cuploid (7/2-cuploid)
 * Retrograde great heptagrammic cuploid (7/4-cuploid)

Cuploids have a Euler characteristic of 1, making them topologically real projective planes.

Cupolaic blends
Two m/n-cupolas can be built atop a (2m)/n-gonal base in two different orientations. If these are blended, the resulting polyhedron is called a cupolaic blend.

Generalizations to higher dimensions
The most common generalization of a cupola to higher dimensions is to have a polytope atop its expanded version. This produces valid segmentochora for 4 of the 5 Platonic solids:


 * Tetrahedron atop cuboctahedron (tetrahedral cupola)
 * Cube atop small rhombicuboctahedron (cubic cupola)
 * Octahedron atop small rhombicuboctahedron (octahedral cupola)
 * Dodecahedron atop small rhombicosidodecahedron (dodecahedral cupola)

By this definition, the CRF icosahedral cupola (icosahedron atop small rhombicosidodecahedron) is hyperbolic. This definition also generalizes the idea of being a cap of an expanded polytope, with the 4 CRF cupolas above being caps of, respectively, the small prismatodecachoron, small disprismatotesseractihexadecachoron, small prismatotetracontoctachoron, and small disprismatohexacosihecatonicosachoron.

In higher dimensions, the simplex and hypercube always have valid CRF cupolas by this definition. The hexadecachoric and icositetrachoric cupolas are 0 height; all others, including the hecatonicosachoric and hexacosichoric cupolas and higher cupolas of cross polytopes, are only CRF in hyperbolic space.

Another definition sometimes used by Richard Klitzing is to have the base polytope atop the common intersection of the compound of the base and its dual (which gives a rectification for 3D cases). This does give CRF versions for all the Platonic solids, but generalizes less well in higher dimensions.

Even more generally, sometimes the term cupola is used to refer to any segmentotope that is a lace prism that is neither a prism, pyramid, or polytope atop dual polytope antiprism.