Английская Википедия:Cupola (geometry)
Шаблон:Short description Шаблон:Other uses Шаблон:Infobox polyhedron
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 Шаблон:Math representing a regular polygon Шаблон:Math joined by a parallel of its truncation, Шаблон:Math or Шаблон:Math
Cupolae are a subclass of the prismatoids.
Its dual contains a shape that is sort of a weld between half of an Шаблон:Mvar-sided trapezohedron and a Шаблон:Math-sided pyramid.
Examples
The above-mentioned three polyhedra 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 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, Шаблон:Math. In that case, the top is a regular Шаблон:Mvar-gon, while the base is either a regular Шаблон:Math-gon or a Шаблон:Math-gon which has two different side lengths alternating and the same angles as a regular Шаблон:Math-gon. It is convenient to fix the coordinate system so that the base lies in the Шаблон:Mvar-plane, with the top in a plane parallel to the Шаблон:Mvar-plane. The Шаблон:Mvar-axis is the Шаблон:Mvar-fold axis, and the mirror planes pass through the Шаблон: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 Шаблон:Mvar is even, half of the mirror planes bisect the sides of the top polygon and half bisect the angles, while if Шаблон:Mvar is odd, each mirror plane bisects one side and one angle of the top polygon.) The vertices of the base can be designated Шаблон:Tmath through Шаблон:Tmath while the vertices of the top polygon can be designated Шаблон:Tmath through Шаблон: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>
where Шаблон:Math.
Since the polygons Шаблон:Tmath etc. are rectangles, this puts a constraint on the values of Шаблон: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} \\
=\ & 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] } \\
=\ & 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]} \\
=\ & 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} } \\
=\ & r_t \sqrt{ \left[ \cos^2\tfrac{\pi}{n} - 2\cos\tfrac{\pi}{n} + 1 \right] + \sin^2\tfrac{\pi}{n} } \\
=\ & 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 Шаблон: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-cupolae
Шаблон:Star-cupolae Шаблон:Star-cupoloids Star cupolae exist for all bases Шаблон:Math where Шаблон:Math and Шаблон:Mvar is odd. At the limits the cupolae collapse into plane figures: beyond the limits the triangles and squares can no longer span the distance between the two polygons (it can still be made if the triangles or squares are irregular.). When Шаблон:Mvar is even, the bottom base Шаблон:Math becomes degenerate: we can form a cupoloid or semicupola by withdrawing this degenerate face and instead letting the triangles and squares connect to each other here. In particular, the tetrahemihexahedron may be seen as a {3/2}-cupoloid. The cupolae are all orientable, while the cupoloids are all nonorientable. When Шаблон:Math in a cupoloid, the triangles and squares do not cover the entire base, and a small membrane is left in the base that simply covers empty space. Hence the {5/2} and {7/2} cupoloids pictured above have membranes (not filled in), while the {5/4} and {7/4} cupoloids pictured above do not.
The height Шаблон:Mvar of an Шаблон:Math-cupola or cupoloid is given by the formula <math display=block>h = \sqrt{1-\frac{1}{4 \sin^{2} \frac{\pi d}{n}}}</math> In particular, Шаблон:Math at the limits of Шаблон:Math and Шаблон:Math, and Шаблон:Mvar is maximized at Шаблон:Math (the triangular prism, where the triangles are upright).[1][2]
In the images above, the star cupolae have been given a consistent colour scheme to aid identifying their faces: the base Шаблон:Math-gon is red, the base Шаблон:Math-gon is yellow, the squares are blue, and the triangles are green. The cupoloids have the base Шаблон:Math-gon red, the squares yellow, and the triangles blue, as the other base has been withdrawn.
Anticupola
An Шаблон:Mvar-gonal anticupola is constructed from a regular Шаблон:Math-gonal base, Шаблон:Math triangles as two types, and a regular Шаблон:Mvar-gonal top. For Шаблон:Math, the top digon face is reduced to a single edge. The vertices of the top polygon are aligned with vertices in the lower polygon. The symmetry is Шаблон:Math, order Шаблон:Math.
An anticupola can't be constructed with all regular faces, Шаблон:Fact although some can be made regular. If the top Шаблон:Mvar-gon and triangles are regular, the base Шаблон:Math-gon can not be planar and regular. In such a case, Шаблон:Math generates a regular hexagon and surrounding equilateral triangles of a snub hexagonal tiling, which can be closed into a zero volume polygon with the base a symmetric 12-gon shaped like a larger hexagon, having adjacent pairs of colinear edges.
Two anticupola can be augmented together on their base as a bianticupola.
Hypercupolae
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.[3]
See also
References
- Johnson, N.W. Convex Polyhedra with Regular Faces. Can. J. Math. 18, 169–200, 1966.
External links
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ 3,0 3,1 Convex Segmentochora Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000
