Cupola (geometry)
In geometry, a cupola is a solid formed by joining two polygons, one 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. Cupolae are a subclass of the prismatoids.
A cupola can also 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 representing a regular polygon joined by a parallel of its truncation, or
The dual of a cupola contains a shape that is sort of a weld between half of an -sided trapezohedron and a -sided pyramid.
Examples
Image:Tile 3464.svg|thumb|Plane "hexagonal cupolae" in the rhombitrihexagonal tilingThe triangular, square, and pentagonal cupolae 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. 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 to be a regular polygon, but it is convenient to consider the case where the cupola has its maximal symmetry,. In that case, the top is a regular -gon, while the base is either a regular -gon or a -gon which has two different side lengths alternating and the same angles as a regular -gon. It is convenient to fix the coordinate system so that the base lies in the -plane, with the top in a plane parallel to the -plane. The -axis is the -fold axis, and the mirror planes pass through the -axis and bisect the sides of the base. They also either bisect the sides or the angles of the top polygon, or both. The vertices of the base can be designated through while the vertices of the top polygon can be designated through With these conventions, the coordinates of the vertices can be written as:for.
Since the polygons etc. are rectangles, this puts a constraint on the values of The distance is equal to
while the distance is equal to
These are to be equal, and if this common edge is denoted by,
These values are to be inserted into the expressions for the coordinates of the vertices given earlier.
Star-cupolae
Star cupolae exist for any top base where and is odd. At these limits, the cupolae collapse into plane figures. Beyond these limits, the triangles and squares can no longer span the distance between the two base polygons. If is even, the bottom base becomes degenerate; then we can form a cupoloid or semicupola by withdrawing this degenerate face and letting the triangles and squares connect to each other here rather than to the late bottom base. In particular, the tetrahemihexahedron may be seen as a -cupoloid.The cupolae are all orientable, while the cupoloids are all non-orientable. For a cupoloid, if, then the triangles and squares do not cover the entire base, and a small membrane is placed in this base -gon that simply covers empty space. Hence the - and -cupoloids pictured above have membranes, while the - and -cupoloids pictured above do not.
The height of an -cupola or cupoloid is given by the formula:
In particular, at the limits and, and is maximized at .
In the images above, the star cupolae have been given a consistent colour scheme to aid identifying their faces: the base -gon is red, the base -gon is yellow, the squares are blue, and the triangles are green. The cupoloids have the base -gon red, the squares yellow, and the triangles blue, as the base -gon has been withdrawn.