geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex

square cupola In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in t ...

. It can be obtained as a slice of the

nonconvex great rhombicuboctahedron In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin dia ...

or quasirhombicuboctahedron. As in all cupolae, the base

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

has twice as many

edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

and vertices as the top; in this case the base polygon is an octagram. It may be seen as a cupola with a retrograde square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola, hence intersecting each other.

Related polyhedra

The crossed square cupola may be seen as a part of some uniform polyhedra. For example, the

great cubicuboctahedron In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It has 20 faces (8 triangles, 6 squares and 6 octagrams), 48 edges, and 24 vertices. Its square faces and its octagrammic faces are parallel to those of a ...

may be seen as six crossed square cupolae connected at their triangular faces, while the nonconvex great rhombicuboctahedron may be seen as a blend of six cupolae. Additionally, the nonconvex great rhombicuboctahedron may be seen as an octagrammic prism with the octagrams excavated with crossed square cupolae, similarly to how the rhombicuboctahedron may be seen as an

octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by rectangular sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron. Symmetry Images The octagonal prism can also b ...

with the octagons augmented with square cupolae. Rotating one of the cupolae in this construction results in the

pseudo-great rhombicuboctahedron In geometry, the pseudo great rhombicuboctahedron is one of the two pseudo uniform polyhedra, the other being the convex elongated square gyrobicupola or pseudo rhombicuboctahedron. It has the same vertex figure as the nonconvex great rhombicuboc ...

. To this may be added the

great rhombihexahedron In geometry, the great rhombihexahedron (or great rhombicube) is a nonconvex uniform polyhedron, indexed as U21. It has 18 faces (12 squares and 6 octagrams), 48 edges, and 24 vertices. Its dual is the great rhombihexacron. Its vertex figure is ...

, as the exclusive or of all three of these octagrammic prisms which may be used to construct the nonconvex great rhombicuboctahedron. The pictures below show the excavation of the octagrammic prism with crossed square cupolae taking place one step at a time. The crossed square cupolae are always red, while the square sides of the octagrammic prism are in the other colours. All images are oriented approximately the same way for clarity. This series of excavations may be easily compared to the corresponding series of augmentations of the octagonal prism: {, class="wikitable" ,

The octagonal prism (coloured with ''D''_8h symmetry)... , Elongated square cupola

...with one of the octagons augmented with a square cupola. , Small rhombicuboctahedron

There are two choices on the orientation of the other crossed square cupola. One aligns corresponding faces (triangles with triangles, squares with squares) and produces the rhombicuboctahedron. This construction has ''D''_4h symmetry, although the rhombicuboctahedron has full octahedral symmetry. ,

The other choice aligns noncorresponding faces (triangles with squares) and produces the pseudorhombicuboctahedron. This construction has ''D''_4d symmetry.

Dual polyhedron

The dual of the crossed square cupola has 8 triangular and 4 kite faces: Dual crossed square cupola

Due to faces of the crossed square cupola passing close to its centre, this dual is very ''spiky'' in appearance. This also occurs for the dual uniform polyhedra known as the great pentakis dodecahedron (DU₅₈) and medial inverted pentagonal hexecontahedron (DU₆₀).

References

* Jim McNeill
Cupola OR Semicupola
* Jim McNeill

External links

Paper model of this polyhedron by Robert Webb
Prismatoid polyhedra