Gyrobifastigium

Gyrobifastigium
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Gybef
Coxeter diagram	xxo oxx&#xt
Elements
Faces	4 triangles, 4 squares
Edges	2+4+8
Vertices	4+4
Vertex figures	4 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 angles	4–3 join: 150°
	4–3 prismatic: 90°
	4–4: 60°
Central density	1
Number of external pieces	8
Level of complexity	7
Related polytopes
Army	Gybef
Regiment	Gybef
Dual	Elongated tetragonal disphenoid
Conjugate	None
Abstract & topological properties
Flag count	56
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(B2×A1)/2, order 8
Convex	Yes
Nature	Tame

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.

External links

• Klitzing, Richard. "gybef".

• Quickfur. "The gyrobifastigium".

• Wikipedia Contributors. "Gyrobifastigium".
• McCooey, David. "Gyrobifastigium"