Cupola (geometry)

Pentagonal cupola (example)
Type	Set of cupolas
Faces	n triangles, n squares<rb>1 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 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

Family of cupolae

2	3	4	5	6
Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)

Plane "hexagonal cupolae" in the rhombitrihexagonal tiling

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.

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_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.

Hypercupolas

The hypercupolas 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.^[1]

	Tetrahedral cupola		Cubic cupola		Octahedral cupola		Dodecahedral cupola		Icosahedral cupola
Image
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

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

External links

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