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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.
A cupola can be given an extended Schläfli symbol Template:Math representing a regular polygon Template:Math joined by a parallel of its truncation, Template:Math or Template:Math
Cupolae are a subclass of the prismatoids.
Its dual contains a shape that is sort of a weld between half of an Template:Mvar-sided trapezohedron and a Template:Math-sided pyramid.
ExamplesEdit
The 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 (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 verticesEdit
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, Template:Math. In that case, the top is a regular Template:Mvar-gon, while the base is either a regular Template:Math-gon or a Template:Math-gon which has two different side lengths alternating and the same angles as a regular Template:Math-gon. It is convenient to fix the coordinate system so that the base lies in the Template:Mvar-plane, with the top in a plane parallel to the Template:Mvar-plane. The Template:Mvar-axis is the Template:Mvar-fold axis, and the mirror planes pass through the Template:Mvar-axis and bisect the sides of the base. They also either bisect the sides or the angles of the top polygon, or both. (If Template:Mvar is even, half of the mirror planes bisect the sides of the top polygon and half bisect the angles, while if Template:Mvar is odd, each mirror plane bisects one side and one angle of the top polygon.) The vertices of the base can be designated Template:Tmath through Template:Tmath while the vertices of the top polygon can be designated Template:Tmath through Template:Tmath With these conventions, the coordinates of the vertices can be written as: <math display=block>\begin{array}{rllcc}
V_{2j-1} :& \biggl( r_b \cos\left(\frac{2\pi(j-1)}{n} + \alpha\right), & r_b \sin\left(\frac{2\pi(j-1)}{n} + \alpha\right), & 0 \biggr) \\[2pt]
V_{2j} :& \biggl( r_b \cos\left(\frac{2\pi j}{n} - \alpha\right), & r_b \sin\left(\frac{2\pi j}{n} - \alpha\right), & 0 \biggr) \\[2pt]
V_{2n+j} :& \biggl( r_t \cos\frac{\pi j}{n}, & r_t \sin\frac{\pi j}{n}, & h \biggr)
\end{array}</math>
for Template:Math.
Since the polygons Template:Tmath etc. are rectangles, this puts a constraint on the values of Template:Tmath The distance <math>\bigl|V_1 V_2 \bigr|</math> is equal to <math display=block>\begin{align}
& r_b \sqrt{ \left[\cos\left(\tfrac{2\pi}{n} - \alpha\right) - \cos \alpha\right]^2 + \left[\sin\left(\tfrac{2\pi}{n} - \alpha\right) - \sin\alpha\right]^2} \\[5pt]
=\ & r_b \sqrt{ \left[\cos^2 \left(\tfrac{2\pi}{n} - \alpha\right) - 2\cos\left(\tfrac{2pi}{n} - \alpha\right)\cos\alpha + \cos^2 \alpha \right] + \left[\sin^2 \left(\tfrac{2\pi}{n} - \alpha\right) - 2\sin\left(\tfrac{2\pi}{n} - \alpha\right) \sin\alpha + \sin^2 \alpha \right] } \\[5pt]
=\ & r_b \sqrt{ 2\left[1 - \cos\left(\tfrac{2\pi}{n} - \alpha\right) \cos\alpha - \sin\left(\tfrac{2\pi}{n} - \alpha\right)\sin\alpha \right]} \\[5pt]
=\ & r_b \sqrt{ 2\left[1 - \cos\left(\tfrac{2\pi}{n} - 2\alpha\right)\right]}
\end{align}</math>
while the distance <math>\bigl| V_{2n+1}V_{2n+2} \bigr|</math> is equal to <math display=block>\begin{align}
& r_t \sqrt{ \left[ \cos\tfrac{\pi}{n} - 1 \right]^2 + \sin^2 \tfrac{\pi}{n} } \\[5pt]
=\ & r_t \sqrt{ \left[ \cos^2\tfrac{\pi}{n} - 2\cos\tfrac{\pi}{n} + 1 \right] + \sin^2\tfrac{\pi}{n} } \\[5pt]
=\ & r_t \sqrt{2 \left[1 - \cos\tfrac{\pi}{n} \right]}
\end{align}</math>
These are to be equal, and if this common edge is denoted by Template:Mvar, <math display=block>\begin{align}
r_b &= \frac{s}{ \sqrt{2\left[1 - \cos\left(\tfrac{2\pi}{n} - 2\alpha \right) \right] }} \\[4pt]
r_t &= \frac{s}{ \sqrt{2\left[1 - \cos\tfrac{\pi}{n} \right] }}
\end{align}</math>
These values are to be inserted into the expressions for the coordinates of the vertices given earlier.
Star-cupolaeEdit
Template:Star-cupolae Template:Star-cupoloids Star cupolae exist for any top base Template:Math where Template:Math and Template:Mvar 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 (it can still be made with non-equilateral isosceles triangles and non-square rectangles). If Template:Mvar is even, the bottom base Template:Math 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 (through single edges) rather than to the late bottom base (through its double edges). In particular, the tetrahemihexahedron may be seen as a Template:Math-cupoloid.
The cupolae are all orientable, while the cupoloids are all non-orientable. For a cupoloid, if Template:Math, then the triangles and squares do not cover the entire (single) base, and a small membrane is placed in this base Template:Math-gon that simply covers empty space. Hence the Template:Math- and Template:Math-cupoloids pictured above have membranes (not filled in), while the Template:Math- and Template:Math-cupoloids pictured above do not.
The height Template:Mvar of an Template:Math-cupola or cupoloid is given by the formula: <math display=block>h = \sqrt{1 - \frac{1}{4 \sin^{2} \left( \frac{\pi d}{n} \right)}}.</math> In particular, Template:Math at the limits Template:Math and Template:Math, and Template:Mvar is maximized at Template:Math (in the digonal cupola: the triangular prism, where the triangles are upright).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
In the images above, the star cupolae have been given a consistent colour scheme to aid identifying their faces: the base Template:Math-gon is red, the base Template:Math-gon is yellow, the squares are blue, and the triangles are green. The cupoloids have the base Template:Math-gon red, the squares yellow, and the triangles blue, as the base Template:Math-gon has been withdrawn.
HypercupolaeEdit
The hypercupolae or polyhedral cupolae 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.<ref name="Segmentochora">Convex Segmentochora Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000</ref>
See alsoEdit
ReferencesEdit
- Johnson, N.W. Convex Polyhedra with Regular Faces. Can. J. Math. 18, 169–200, 1966.
External linksEdit
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Cupola%7CCupola.html}} |title = Cupola |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}