Cupola

(Redirected from Cuploid)

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 .

Convex cupolae

Convex polygonal cupolae
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

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

Star polygonal cupolae
Polygon Cupola Cap of
4/3

Quasirhombicuboctahedron
5/3
Pentagrammic cupola

Quasirhombicosidodecahedron
7/3
Great heptagrammic cupola
7/5
Heptagrammic cupola
8/3
Octagrammic cupola
8/5

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.

Polygonal cuploids
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 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

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.

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

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 ${\displaystyle 1}$ ${\displaystyle {\sqrt {\frac {3+{\sqrt {2}}}{2}}}}$
${\displaystyle \approx 1.485634}$
${\displaystyle {\sqrt {2+{\sqrt {2}}}}}$
${\displaystyle \approx 1.847759}$
${\displaystyle 3+{\sqrt {5}}}$
${\displaystyle \approx 5.236068}$
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.