Pentagonal cupola (example) | |
---|---|
Type | Set of cupolas |
Faces | n triangles, n squares 1 n-agon, 1 2n-agon |
Edges | 5n |
Vertices | 3n |
Symmetry group | C_{nv} |
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
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 V_{1} 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 V_{1}V_{2}V_{2n+2}V_{2n+1}, etc. are rectangles, this puts a constraint on the values of r_{b}, r_{t}, and α. The distance V_{1}V_{2 }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 + cos^{2} α] + [sin ^{2} (2π/n - α) – 2 sin (2π/n - α)sin α + sin ^{2}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+1}V_{2n+2} is equal to
r_{t}{[cos (π/n) – 1]^{2} + sin^{2}(π/n)}^{1/2}
= r_{t}{[cos^{2} (π/n) – 2cos (π/n) + 1] + sin^{2}(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.
External links
- Weisstein, Eric W., "Cupola" from MathWorld.
References
- Johnson, N. W. Convex Polyhedra with Regular Faces. Canad. J. Math. 18, 169-200, 1966.eo:Kupolo (geometrio)it:Cupola (geometria)