Cupola (geometry)

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For other uses, see cupola (disambiguation).

Pentagonal cupola
Type	Set of cupolas
Faces	n triangles, n squares 1 n-agon, 1 2n-agon
Edges	5n
Vertices	3n
Symmetry group	Cnv
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 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.

[edit] Examples

The digonal cupola (wedge)	The triangular cupola with regular faces (J3)
The square cupola with regular faces (J4)	The pentagonal cupola with regular faces (J5)

The above-mentioned three polyhedra are the only non-trivial 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.

[edit] 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, 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_bcos [2π(j-1)/n + α], r_bsin [2π(j-1)/n + α], 0) (where j=1, 2, …, n);
V_2j: (r_bcos (2πj/n - α), r_bsin (2πj/n - α), 0) (where j=1, 2, …, n);

V_2n+j: (r_tcos (πj/n), r_tsin (πj/n), h) (where j=1, 2, …, n).

Since the polygons V₁V₂V_2n+1V_2n+2, 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{[cos ² (2π/n - α) – 2cos (2π/n - α)cos α + cos² α] + [sin ² (2π/n - α) – 2 sin (2π/n - α)sin α + sin ²α]}^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.