# Gyrobifastigium

Gyrobifastigium Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymGybef
Coxeter diagramxxo oxx&#xt
Elements
Faces4 triangles, 4 squares
Edges2+4+8
Vertices4+4
Vertex figures4 isosceles triangles, edge lengths 1, 2, 2
4 kites, edge lengths 1 and 2
Measures (edge length 1)
Volume$\frac{\sqrt3}2 \approx 0.86603$ Dihedral angles4–3 join: 150°
4–3 prismatic: 90°
4–4: 60°
Central density1
Number of external pieces8
Level of complexity7
Related polytopes
ArmyGybef
RegimentGybef
DualElongated tetragonal disphenoid
ConjugateNone
Abstract & topological properties
Flag count56
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(B2×A1)/2, order 8
ConvexYes
NatureTame

The gyrobifastigium is one of the 92 Johnson solids (J26). It consists of 4 triangles and 4 squares. It can be constructed by attaching two triangular prisms, seen as digonal cupolas, at one of their square faces so that their opposite edges are perpendicular. As such, it could be considered to be a digonal gyrobicupola.

## Vertex coordinates

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

• $\left(\pm\frac12,\,0,\,\frac{\sqrt3}2\right)$ ,
• $\left(\pm\frac12,\,\pm\frac12,\,0\right)$ ,
• $\left(0,\,\pm\frac12,\,-\frac{\sqrt3}2\right)$ .

## In vertex figures

The gyrobifastigium appears as the vertex figure of the nonuniform triangular duoantiprism. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson gyrobifastigium has no circumscribed sphere.

Variants made by changing the two edges perpendicular to the symmetry axis also appear as the vertex figure of the nonuniform duoantiprisms made out of two regular polygons. The symmetry of the gyrofastigium is (B2×A1)/2, order 8 if the two polygons are identical, otherwise the symmetry is K2×I, order 4.