Cupola (geometry)

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For other uses, see Cupola (disambiguation).
Pentagonal cupola (example)
Pentagonal cupola
Type Set of cupolas
Faces n triangles,
n squares1 n-gon,
1 2n-gon
Edges 5n
Vertices 3n
Symmetry group Cnv, [1,n], (*nn), order 2n
Rotation group Cn, [1,n]+, (nn), order n
Dual polyhedron ?
Properties convex

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.

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

Cupolae are a subclass of the prismatoids.

Examples[edit]

Family of convex cupolae
2 3 4 5 6
Triangular prism wedge.png
Digonal cupola
Triangular cupola.png
Triangular cupola
Square cupola.png
Square cupola
Pentagonal cupola.png
Pentagonal cupola
Hexagonal cupola flat.png
Hexagonal cupola
(Flat)
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.


Coordinates of the vertices[edit]

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[edit]

Family of star-cupolae and star-cuploids
n / d 3 4 5 6 7 8
2 Tetrahemihexahedron.png
Crossed triangular cuploid
Pentagrammic cuploid.png
Pentagrammic cuploid
Heptagrammic cuploid.png
Heptagrammic cuploid
3 Crossed square cupola.png
Crossed square cupola
Crossed pentagrammic cupola.png
Crossed pentagrammic cupola
Heptagrammic cupola.png
Heptagrammic cupola
Octagrammic cupola.png
Octagrammic cupola
4 Crossed pentagonal cuploid.png
Crossed pentagonal cuploid
Crossed heptagrammic cuploid.png
Crossed heptagrammic cuploid
5 Crossed hexagonal cupola
(Flat)
Crossed heptagrammic cupola.png
Crossed heptagrammic cupola
Crossed octagrammic cupola.png
Crossed octagrammic cupola

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 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 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.

Hypercupolae[edit]

The hypercupolae 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]

Tetrahedral cupola Cubic cupola Octahedral cupola Dodecahedral cupola Icosahedral cupola
Image 4D Tetrahedral Cupola-perspective-cuboctahedron-first.png 4D Cubic Cupola-perspective-cube-first.png 4D octahedral cupola-perspective-octahedron-first.png
Vertices 16 32 30 80 72
Edges 42 84 84 210 210
Faces 42 24 triangles
18 squares
80 32 triangles
48 squares
82 40 triangles
42 squares
194 80 triangles
90 squares
24 pentagons
202 100 triangles
90 squares
12 pentagons
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
64  1 icosahedron
20 triangular prisms
30 triangular prisms
12 pentagonal pyramids
 1 rhombicosidodecahedron

References[edit]

  1. ^ http://www.orchidpalms.com/polyhedra/cupolas/cupola1.html
  2. ^ http://www.orchidpalms.com/polyhedra/cupolas/cupola2.html
  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. Canad. J. Math. 18, 169–200, 1966.

External links[edit]