Elongated square cupola

Elongated square cupola
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Escu
Coxeter diagram	oxx4xxx&#xt
Elements
Faces	4 triangles, 1+4+4+4 squares, 1 octagon
Edges	4+4+4+4+4+8+8
Vertices	4+8+8
Vertex figures	4+8 isosceles trapezoids, edge lengths 1, √2, √2, √2
	8 isosceles triangles, edge lengths √2+√2, √2, √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{5+2\sqrt2}}{2} ≈ 1.39897$
Volume	$\frac{9+8\sqrt2}{3} ≈ 6.77124$
Dihedral angles	3–4: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	4–4: 135°
	4–8: 90°
Central density	1
Related polytopes
Army	Escu
Regiment	Escu
Dual	Octakis order-8 truncated semibisected tetragonal trapezohedron
Conjugate	Elongated retrograde square cupola
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B2×I, order 8
Convex	Yes
Nature	Tame

The elongated square cupola is one of the 92 Johnson solids (J₁₉). It consists of 4 triangles, 1+4+4+4 squares, and 1 octagon. It can be constructed by attaching an octagonal prism to the octagonal base of the square cupola.

It can also be seen as a diminished small rhombicuboctahedron, formed by cutting off one of its square cupola segments. Conversely, attaching a second square cupola to the other octagonal base of the prism in the same orientation leads to the small rhombicuboctahedron, seen as an elongated square orthobicupola. If the second cupola is rotated by 45º instead, the result is an elongated square gyrobicupola.

Vertex coordinates

An elongated square cupola of edge length 1 has the following vertices:

• $\left(±\frac12,\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$
• $\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right),$
• $\left(±\frac12,\,±\frac12\,\frac{1+\sqrt2}{2}\right).$

These are simply the coordinates of the small rhombicuboctahedron with the vertices of one square face removed.

External links

• Klitzing, Richard. "escu".

• Quickfur. "The Elongated Square Cupola".

• Wikipedia Contributors. "Elongated square cupola".
• McCooey, David. "Elongated Square Cupola"