Square cupola

Square cupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Squacu
Coxeter diagram	ox4xx&#x
Stewart notation	Q4
Elements
Faces	4 triangles, 1+4 squares, 1 octagon
Edges	4+4+4+8
Vertices	4+8
Vertex figures	4 isosceles trapezoids, edge lengths 1, √2, √2, √2
	8 scalene triangles, edge lengths 1, √2, √2+√2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {5+2{\sqrt {2}}}}{2}}\approx 1.39897$
Volume	${\frac {3+2{\sqrt {2}}}{3}}\approx 1.94281$
Dihedral angles	3–4: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	4–4: 135°
	3–8: $\arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }$
	4–8: 45°
Height	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Number of external pieces	10
Level of complexity	10
Related polytopes
Army	Squacu
Regiment	Squacu
Dual	Semibisected tetragonal trapezohedron
Conjugate	Retrograde square cupola
Abstract & topological properties
Flag count	80
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B2×I, order 8
Convex	Yes
Nature	Tame

The square cupola is one of the 92 Johnson solids (J₄). It consists of 4 triangles, 1+4 squares, and 1 octagon. It is a cupola based on the square, and is one of three Johnson solid cupolas, the other two being the triangular cupola and the pentagonal cupola.

It can be obtained as a segment of the small rhombicuboctahedron, when considered as an elongated square orthobicupola.

Vertex coordinates

A square cupola of edge length 1 has vertices given by the following coordinates:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {2}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,0\right).$

These can be obtained from placing a square and octagon in parallel planes.

Representations

A square cupola has the following Coxeter diagrams:

• ox4xx&#x
• so8ox&#x
• oqxw qowx&#xr (bases in digonal symmetry)

Related polyhedra

Two square cupolas can be attached at their octagonal bases in the same orientation to form a square orthobicupola. If the second cupola is rotated by 45º the result is the square gyrobicupola.

An octagonal prism can be attached to the square cupola's octagonal base to form the elongated square cupola. If an octagonal antiprism is attached instead, the result is the gyroelongated square cupola.

External links

• Klitzing, Richard. "squacu".

• Quickfur. "The Square Cupola".

• Weisstein, Eric W. "Square Cupola" ("Johnson solid") at MathWorld.

• Wikipedia contributors. "Square cupola".
• McCooey, David. "Square Cupola"

• Hi.gher.Space Wiki Contributors. "Square cupola".