Cupola (geometry)

In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.

Set of cupolae

Pentagonal cupola (example)
Schläfli symbol{n} || t{n}
Facesn triangles,
n squares,
1 n-gon,
1 2n-gon
Edges5n
Vertices3n
Symmetry groupCnv, [1,n], (*nn), order 2n
Rotation groupCn, [1,n]+, (nn), order n
DualTrapyramids
Propertiesconvex

A cupola can be seen as a prism where one of the polygons has been collapsed in half by merging alternate vertices.

A cupola can be given an extended Schläfli symbol {n} || t{n}, representing a regular polygon {n} joined by a parallel of its truncation, t{n} or {2n}.

Cupolae are a subclass of the prismatoids.

Its dual contains a shape that is sort of a weld between half of an n-sided trapezohedron and a 2n-sided pyramid.

Examples

Family of convex cupolae
n23456
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

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.

Coordinates of the vertices

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, Cnv. 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 V2n, while the vertices of the top polygon can be designated V2n+1 through V3n. With these conventions, the coordinates of the vertices can be written as:

  • V2j−1: (rb cos[2π(j − 1) / n + α], rb sin[2π(j − 1) / n + α], 0)
  • V2j: (rb cos(2πj / n − α), rb sin(2πj / n − α), 0)
  • V2n+j: (rt cos(πj / n), rt sin(πj / n), h)

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

Since the polygons V1V2V2n+2V2n+1, etc. are rectangles, this puts a constraint on the values of rb, rt, and α. The distance V1V2 is equal to

rb{[cos(2π / n − α) − cos α]2 + [sin(2π / n − α) − sin α]2}12
= rb{[cos2(2π / n − α) − 2cos(2π / n − α)cos α + cos2 α] + [sin2(2π / n − α) − 2sin(2π / n − α)sin α + sin2 α]}12
= rb{2[1 − cos(2π / n − α)cos α − sin(2π / n − α)sin α]}12
= rb{2[1 − cos(2π / n − 2α)]}12

while the distance V2n+1V2n+2 is equal to

rt{[cos(π / n) − 1]2 + sin2(π / n)}12
= rt{[cos2(π / n) − 2cos(π / n) + 1] + sin2(π / n)}12
= rt{2[1 − cos(π / n)]}12.

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

rb = s / {2[1 − cos(2π / n − 2α)]}12
rt = s / {2[1 − cos(π / n)]}12

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

Star-cupolae

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 . 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.

Anticupola

Set of anticupolas

Pentagonal example
Schläfli symbols{n} || t{n}
Faces3n triangles
1 n-gon,
1 2n-gon
Edges6n
Vertices3n
Symmetry groupCnv, [1,n], (*nn), order 2n
Rotation groupCn, [1,n]+, (nn), order n
Dual?
Propertiesconvex

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 Cnv, 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
n23456...
Names{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

Hypercupolae

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

See also

References

  1. "cupolas". www.orchidpalms.com. Retrieved 21 April 2018.
  2. "semicupolas". www.orchidpalms.com. Retrieved 21 April 2018.
  3. Convex Segmentochora Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000
  • Johnson, N.W. Convex Polyhedra with Regular Faces. Can. J. Math. 18, 169–200, 1966.
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