Grünbaum polyhedron

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Grünbaum polyhedron
Grünbaum polyhedron.png
Faces8+8 equilateral triangles, 24+24 scalene triangles
Measures ​
Central density0
Related polytopes
ArmySnub cube
Convex hullnon-uniform snub cube
Abstract & topological properties
Flag count384
Euler characteristic–8
Schläfli type{3,8}
SymmetryB3+, order 24

The Grünbaum polyhedron is a realization of the Fricke-Klein map in 3-dimensional Euclidean space with the maximum possible symmetry. The Fricke-Klein map is regular, and thus the Grünbaum polyhedron is regular under its automorphism group, however it is not regular under its symmetry group.

Construction[edit | edit source]

The Grünbaum polyhedron can be constructed by blending two non-identical snub cubes with coincident vertices. The inner snub cube is the same as the outer snub cube with each of its square faces rotated about its own center a quarter turn. If a uniform snub cube is used the inner snub cube will self-intersect. Typically a non-uniform version is used to avoid this.

Gallery[edit | edit source]

A view with the outermost faces hidden, in order to better see the inner faces.

External links[edit | edit source]

Bibliography[edit | edit source]

  • Gévay, Gábor; Schulte, Egon; Wills, Jörg (2016). "The Regular Grünbaum Polyhedron of Genus 5". Advances in Geometry. arXiv:1212.6588.