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To visually understand cupolas, it helps to start with 3D CRF ones.

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/isosceles trapezoids in non-regular faced variants). Three cupolas are CRF: the triangular cupola, square cupola and pentagonal cupola.

A p-gonal cupola can be seen as the cap of the uniform polyhedron CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node 1.png.

Convex cupolae[edit | edit source]

Convex polygonal cupolae
Polygon Cupola Cap of
2 Triangular prism wedge.png
Triangular prism
Triangular prism wedge.png
Triangular prism
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
3 Triangular cupola 2.png
Triangular cupola
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4 Square cupola 2.png
Square cupola
Small rhombicuboctahedron.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
5 Pentagonal cupola 2.png
Pentagonal cupola
Small rhombicosidodecahedron.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
6 Hexagonal cupola flat.png
Hexagonal cupola
(CRF version is flat)
1-uniform n6.svg
Rhombitrihexagonal tiling
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
7 Heptagonal cupola
(CRF version is hyperbolic)
Rhombitriheptagonal tiling.svg
Rhombitriheptagonal tiling
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png

Star cupolae[edit | edit source]

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

Star polygonal cupolae
Polygon Cupola Cap of
4/3 Crossed square cupola.png
Retrograde square cupola
Uniform great rhombicuboctahedron.png
CDel node 1.pngCDel 4.pngCDel rat.pngCDel 3x.pngCDel node.pngCDel 3.pngCDel node 1.png
5/3 Crossed pentagrammic cupola.png
Pentagrammic cupola
Uniform great rhombicosidodecahedron.png
CDel node 1.pngCDel 5.pngCDel rat.pngCDel 3x.pngCDel node.pngCDel 3.pngCDel node 1.png
7/3 Heptagrammic cupola.png
Great heptagrammic cupola
7/5 Crossed heptagrammic cupola.png
Heptagrammic cupola
8/3 Octagrammic cupola.png
Octagrammic cupola
8/5 Crossed octagrammic cupola.png
Retrograde octagrammic cupola

Cuploids[edit | edit source]

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.

Polygonal cuploids
Polygon Cuploid
3/2 Tetrahemihexahedron.png
5/2 Pentagrammic cuploid.png
Pentagrammic cuploid
5/4 Crossed pentagonal cuploid.png
Pentagonal cuploid
7/2 Heptagrammic cuploid.png
Heptagrammic cuploid
7/4 Crossed heptagrammic cuploid.png
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 non-orientable.

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 7-7-3 acrohedron.

Cupolaic blends[edit | edit source]

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[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
K4.23 K4.71 K4.107 K4.152

Image 4D Tetrahedral Cupola-perspective-cuboctahedron-first.png 4D Cubic Cupola-perspective-cube-first.png 4D octahedral cupola-perspective-octahedron-first.png Dodecahedral cupola.png
Cap cells Uniform polyhedron-33-t0.pngUniform polyhedron-33-t02.png Uniform polyhedron-43-t0.pngUniform polyhedron-43-t02.png Uniform polyhedron-43-t2.pngUniform polyhedron-43-t02.png Uniform polyhedron-53-t0.pngUniform polyhedron-53-t02.png
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
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png

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 Attribution-ShareAlike 3.0 Unported License (view authors). Wikipedia logo