# Elongated square cupola

Elongated square cupola
Rank3
TypeCRF
Notation
Bowers style acronymEscu
Coxeter diagramoxx4xxx&#xt
Elements
Faces
Edges4+4+4+4+4+8+8
Vertices4+8+8
Vertex figures4+8 isosceles trapezoids, edge lengths 1, 2, 2, 2
8 isosceles triangles, edge lengths 2+2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}}}{2}}\approx 1.39897}$
Volume${\displaystyle {\frac {9+8{\sqrt {2}}}{3}}\approx 6.77124}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
4–4: 135°
4–8: 90°
Central density1
Number of external pieces18
Level of complexity18
Related polytopes
ArmyEscu
RegimentEscu
DualOctakis order-8 truncated semibisected tetragonal trapezohedron
Abstract & topological properties
Flag count144
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB2×I, order 8
Flag orbits18
ConvexYes
NatureTame

The elongated square cupola (OBSA: escu) 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:

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

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