Square cupola

Square cupola
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

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

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

Other convex cupolae

Family of convex cupolae
n23456
Name{2} || t{2}{3} || t{3}{4} || t{4}{5} || t{5}{6} || t{6}
Cupola
Digonal cupola
Triangular cupola
Square cupola
Pentagonal cupola
Hexagonal cupola
(Flat)
Related
uniform
polyhedra		 Triangular prism
 Cubocta-
hedron
 Rhombi-
cubocta-
hedron
 Rhomb-
icosidodeca-
hedron
 Rhombi-
trihexagonal
tiling

Dual polyhedron

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

Dual square cupola Net of dual

Crossed square cupola

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

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

References

  1. ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603 .
  2. ↑ Stephen Wolfram, "Square cupola" from Wolfram Alpha. Retrieved July 20, 2010.
  3. ↑ http://woodenpolyhedra.web.fc2.com/J4.html


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