Square cupola

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Square cupola
Square cupola.png
Type Johnson
J3 - J4 - J5
Faces 4 triangles
1+4 squares
1 octagon
Edges 20
Vertices 12
Vertex configuration 8(3.4.8)
4(3.43)
Symmetry group C4v, [4], (*44)
Rotation group C4, [4]+, (44)
Dual polyhedron -
Properties convex
Net
Johnson solid 4 net.png

In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (J4). 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.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

Formulae[edit]

The following formulae for volume, surface area, and circumradius can be used if all faces are regular, with edge length a:[2]

Related polyhedra and Honeycombs[edit]

Other convex cupolae[edit]

Family of convex cupolae
n 2 3 4 5 6
Name {2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6}
Cupola Triangular prism wedge.png
Digonal cupola
Triangular cupola.png
Triangular cupola
Square cupola.png
Square cupola
Pentagonal cupola.png
Pentagonal cupola
Hexagonal cupola flat.png
Hexagonal cupola
(Flat)
Related
uniform
polyhedra
Triangular prism
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
Cubocta-
hedron

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhombi-
cubocta-
hedron

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhomb-
icosidodeca-
hedron

CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhombi-
trihexagonal
tiling

CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png

Dual polyhedron[edit]

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

Dual square cupola Net of dual
Dual square cupola.png Dual square cupola net.png

Crossed square cupola[edit]

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.

Honeycombs[edit]

The square cupola is a component of several nonuniform space-filling lattices:

References[edit]

External links[edit]