Cupola
A cupola (plural cupolas or cupolae) is a segmentohedron joining an nsided upper base to a 2nsided lower base with a ring of triangles and squares (isosceles triangles and rectangles/isosceles trapezoids in nonregular faced variants). Three cupolas are CRF: the triangular cupola, square cupola and pentagonal cupola.
A pgonal cupola can be seen as the cap of the uniform polyhedron .
Convex cupolae[edit  edit source]
Polygon  Cupola  Cap of 

2  Triangular prism 
Triangular prism 
3  Triangular cupola 
Cuboctahedron 
4  Square cupola 
Rhombicuboctahedron 
5  Pentagonal cupola 
Rhombicosidodecahedron 
6  Hexagonal cupola (CRF version is flat) 
Rhombitrihexagonal tiling 
7  Heptagonal cupola (CRF version is hyperbolic) 
Rhombitriheptagonal tiling 
Star cupolae[edit  edit source]
These connect a m/ngon to a (2m)/ngon.
Polygon  Cupola  Cap of 

4/3  Retrograde square cupola 
Quasirhombicuboctahedron 
5/3  Pentagrammic cupola 
Quasirhombicosidodecahedron 
7/3  Great heptagrammic cupola 

7/5  Heptagrammic cupola 

8/3  Octagrammic cupola 

8/5  Retrograde octagrammic cupola 
Cuploids[edit  edit source]
If n is even, a m/ncupola has a doublecovered 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.
Polygon  Cuploid 

3/2  Tetrahemihexahedron 
5/2  Pentagrammic cuploid 
5/4  Pentagonal cuploid 
7/2  Heptagrammic cuploid 
7/4  Great heptagrammic cuploid 
The edges of the pentagrammic and pentagonal cuploids are contained within the small ditrigonary icosidodecahedron.
Cuploids have a Euler characteristic of 1, making them topologically real projective planes; they are nonorientable.
The small supersemicupola is a heptagrammic cuploid that has been "twice inflated" vertically, with square faces augmented into pentagons (with triangles added to buffer the heptagram) and then stretched into heptagons. It was found as a 773 acrohedron.
Cupolaic blends[edit  edit source]
Two m/ncupolas can be built atop a (2m)/ngonal base in two different orientations. If these are blended, the resulting polyhedron is called a cupolaic blend.
Cupolaic blends can be thought of as cuploids based on the compound polygon (2m)/(2n), but unlike cuploids, they are orientable and topologically spherical.
Generalizations to higher dimensions[edit  edit source]
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:
Name  Tetrahedral cupola  Cubic cupola  Octahedral cupola  Dodecahedral cupola  

Segmentochoron index 
K4.23  K4.71  K4.107  K4.152  
Circumradius  
Image  
Cap cells  
Vertices  16  32  30  80  
Edges  42  84  84  210  
Faces  42  24 {3} + 18 {4}  80  32 {3} + 48 {4}  82  40 {3} + 42 {4}  202  100 {3} + 90 {4} + 12 {5} 
Cells  16  1 tetrahedron 4 triangular prisms 6 triangular prisms 4 triangular pyramids 1 cuboctahedron 
28  1 cube 6 square prisms 12 triangular prisms 8 triangular pyramids 1 rhombicuboctahedron 
28  1 octahedron 8 triangular prisms 12 triangular prisms 6 square pyramids 1 rhombicuboctahedron 
64  1 dodecahedron 12 pentagonal prisms 30 triangular prisms 20 triangular pyramids 1 rhombicosidodecahedron 
Cap of  spid 
sidpith 
spic 
sidpixhi 
There are also a retrograde cubic cupola and a retrograde octahedral cupola, their bases being quasirhombicuboctahedra.
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 icosahedral, 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.
External links[edit  edit source]
Wikipedia contributors. "Cupola (geometry)".
This article uses material from the Wikipedia article Cupola (geometry), which is released under the Creative Commons AttributionShareAlike 3.0 Unported License (view authors). 