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Concept
Square cupola
Summary
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (J_4). 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 this case the base polygon is an octagon.
The following formulae for the circumradius, surface area, volume, and height can be used if all faces are regular, with edge length a:
The dual of the square cupola has 8 triangular and 4 kite faces:
The crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron, analogously to how the square cupola may 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 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.
The square cupola is a component of several nonuniform space-filling lattices:
with tetrahedra;
with cubes and cuboctahedra; and
with tetrahedra, square pyramids and various combinations of cubes, elongated square pyramids and elongated square bipyramids.
Official source
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