Cupola (geometry)

Family of convex cupolae

n	2	3	4	5	6
Name	{2} || t{2}	{3} || t{3}	{4} || t{4}	{5} || t{5}	{6} || t{6}
Cupola	Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)
Related uniform polyhedra	Triangular prism	Cubocta- hedron	Rhombi- cubocta- hedron	Rhomb- icosidodeca- hedron	Rhombi- trihexagonal tiling

Plane "hexagonal cupolae" in the rhombitrihexagonal tiling

The above-mentioned three polyhedra are the only non-trivial convex cupolae with regular faces: The "hexagonal cupola" is a plane figure, and the triangular prism might be considered a "cupola" of degree 2 (the cupola of a line segment and a square). However, cupolae of higher-degree polygons may be constructed with irregular triangular and rectangular faces.

A 40-sided cupola has 40 isosceles triangles (blue), 40 rectangles (yellow), a top regular 40-gon (red) and a bottom regular 80-gon (hidden).

The definition of the cupola does not require the base (or the side opposite the base, which can be called the top) to be a regular polygon, but it is convenient to consider the case where the cupola has its maximal symmetry, C_nv. In that case, the top is a regular n-gon, while the base is either a regular 2n-gon or a 2n-gon which has two different side lengths alternating and the same angles as a regular 2n-gon. It is convenient to fix the coordinate system so that the base lies in the xy-plane, with the top in a plane parallel to the xy-plane. The z-axis is the n-fold axis, and the mirror planes pass through the z-axis and bisect the sides of the base. They also either bisect the sides or the angles of the top polygon, or both. (If n is even, half of the mirror planes bisect the sides of the top polygon and half bisect the angles, while if n is odd, each mirror plane bisects one side and one angle of the top polygon.) The vertices of the base can be designated V₁ through V_2n, while the vertices of the top polygon can be designated V_2n+1 through V_3n. With these conventions, the coordinates of the vertices can be written as:

• V_2j−1: (r_b cos[2π(j − 1) / n + α], r_b sin[2π(j − 1) / n + α], 0)
• V_2j: (r_b cos(2πj / n − α), r_b sin(2πj / n − α), 0)
• V_2n+j: (r_t cos(πj / n), r_t sin(πj / n), h)

where j = 1, 2, ..., n.

Since the polygons V₁V₂V_2n+2V_2n+1, etc. are rectangles, this puts a constraint on the values of r_b, r_t, and α. The distance V₁V₂ is equal to

r_b{[cos(2π / n − α) − cos α]² + [sin(2π / n − α) − sin α]²}^1/2

= r_b{[cos²(2π / n − α) − 2cos(2π / n − α)cos α + cos² α] + [sin²(2π / n − α) − 2sin(2π / n − α)sin α + sin² α]}^1/2

= r_b{2[1 − cos(2π / n − α)cos α − sin(2π / n − α)sin α]}^1/2

= r_b{2[1 − cos(2π / n − 2α)]}^1/2

while the distance V_2n+1V_2n+2 is equal to

r_t{[cos(π / n) − 1]² + sin²(π / n)}^1/2

= r_t{[cos²(π / n) − 2cos(π / n) + 1] + sin²(π / n)}^1/2

= r_t{2[1 − cos(π / n)]}^1/2.

These are to be equal, and if this common edge is denoted by s,

r_b = s / {2[1 − cos(2π / n − 2α)]}^1/2

r_t = s / {2[1 − cos(π / n)]}^1/2

These values are to be inserted into the expressions for the coordinates of the vertices given earlier.

Family of star-cupolae

n / d	4	5	7	8
3	{4/3}	{5/3}	{7/3}	{8/3}
5	—	—	{7/5}	{8/5}

Family of star-cuploids

​n⁄d	3	5	7
2	Crossed triangular cuploid	Pentagrammic cuploid	Heptagrammic cuploid
4	—	Crossed pentagonal cuploid	Crossed heptagrammic cuploid

Star cupolae exist for all bases {n/d} where ⁶/₅ < ⁿ/_d < 6 and d is odd. At the limits the cupolae collapse into plane figures: beyond the limits the triangles and squares can no longer span the distance between the two polygons. When d is even, the bottom base {2n/d} becomes degenerate: we can form a cuploid or semicupola by withdrawing this degenerate face and instead letting the triangles and squares connect to each other here. In particular, the tetrahemihexahedron may be seen as a {3/2}-cuploid. The cupolae are all orientable, while the cuploids are all nonorientable. When n/d > 2 in a cuploid, the triangles and squares do not cover the entire base, and a small membrane is left in the base that simply covers empty space. Hence the {5/2} and {7/2} cuploids pictured above have membranes (not filled in), while the {5/4} and {7/4} cuploids pictured above do not.

The height h of an {n/d}-cupola or cuploid is given by the formula ${\displaystyle h={\sqrt {1-{\frac {1}{4\sin ^{2}({\frac {\pi d}{n}})}}}}}$. In particular, h = 0 at the limits of n/d = 6 and n/d = 6/5, and h is maximized at n/d = 2 (the triangular prism, where the triangles are upright).[1][2]

In the images above, the star cupolae have been given a consistent colour scheme to aid identifying their faces: the base n/d-gon is red, the base 2n/d-gon is yellow, the squares are blue, and the triangles are green. The cuploids have the base n/d-gon red, the squares yellow, and the triangles blue, as the other base has been withdrawn.

Set of anticupolas

Pentagonal example
Schläfli symbol	| t{n}
Faces	3n triangles 1 n-gon, 1 2n-gon
Edges	6n
Vertices	3n
Symmetry group	Cnv, [1,n], (*nn), order 2n
Rotation group	Cn, [1,n]+, (nn), order n
Dual	?
Properties	convex

An n-gonal anticupola is constructed from a regular 2n-gonal base, 3n triangles as two types, and a regular n-gonal top. For n = 2, the top digon face is reduced to a single edge. The vertices of the top polygon are aligned with vertices in the lower polygon. The symmetry is C_nv, order 2n.

An anticupola can't be constructed with all regular faces, although some can be made regular. If the top n-gon and triangles are regular, the base 2n-gon can not be planar and regular. In such a case, n=6 generates a regular hexagon and surrounding equilateral triangles of a snub hexagonal tiling, which can be closed into a zero volume polygon with the base a symmetric 12-gon shaped like a larger hexagon, having adjacent pairs of colinear edges.

Two anticupola can be augmented together on their base as a bianticupola.

Family of convex anticupolae

n	2	3	4	5	6...
Name	s{2} || t{2}	s{3} || t{3}	s{4} || t{4}	s{5} || t{5}	s{6} || t{6}
Image	Digonal	Triangular	Square	Pentagonal	Hexagonal
Transparent
Net

The hypercupolae or polyhedral cupolae are a family of convex nonuniform polychora (here four-dimensional figures), analogous to the cupolas. Each one's bases are a Platonic solid and its expansion.[3]

Name	Tetrahedral cupola		Cubic cupola		Octahedral cupola		Dodecahedral cupola		Hexagonal tiling cupola
Schläfli symbol	{3,3} || rr{3,3}		{4,3} || rr{4,3}		{3,4} || rr{3,4}		{5,3} || rr{5,3}		{6,3} || rr{6,3}
Segmentochora index[3]	K4.23		K4.71		K4.107		K4.152
circumradius	1		sqrt((3+sqrt(2))/2) = 1.485634		sqrt(2+sqrt(2)) = 1.847759		3+sqrt(5) = 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}	194	80 {3} + 90 {4} + 24 {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	∞	1 hexagonal tiling ∞ hexagonal prisms ∞ triangular prisms ∞ triangular pyramids 1 rhombitrihexagonal tiling
Related uniform polychora	runcinated 5-cell		runcinated tesseract		runcinated 24-cell		runcinated 120-cell		runcinated hexagonal tiling honeycomb

• Orthobicupola
• Gyrobicupola
• Rotunda

• Johnson, N.W. Convex Polyhedra with Regular Faces. Can. J. Math. 18, 169–200, 1966.

• Weisstein, Eric W. "Cupola". MathWorld.
• Segmentotopes