Muoctahedron

Muoctahedron
Rank	3
Space	Euclidean
Notation
Bowers style acronym	Muo
Schläfli symbol	{6,4∣4}, $\left\{6,{\frac {4}{1,2}}\mid 4\right\}$
Elements
Faces	4N  hexagons
Edges	12N
Vertices	6N
Vertex figure	Skew square
Petrie polygons	Triangular helices
Holes	3N squares
Measures (edge length 1)
Dihedral angle	$\arccos \left(-{\dfrac {1}{3}}\right)\approx 109.47122^{\circ }$
Related polytopes
Army	Batch
Regiment	Batch
Dual	Mucube
Petrie dual	Petrial muoctahedron
Skewing	Skewed muoctahedron
κ ?	Petrial muoctahedron
Convex hull	Bitruncated cubic honeycomb
Abstract & topological properties
Schläfli type	{6,4}
Orientable	Yes
Genus	∞
Properties
Symmetry	R4×2
Convex	No
Dimension vector	(2,1,2)
History
Discovered by	John Finders Petrie
First discovered	1926

The muoctahedron or muo, short for multiple octahedron, is a regular skew apeirohedron in Euclidean 3-space. Its faces are hexagons, with 4 meeting at each vertex.

Note that on visual inspection, the muoctahedron may at first appear to have two types of edges, "convex" and "concave". However, its "interior" and "exterior" are congruent and not distinguished by the polyhedron's symmetry group, so the two apparent types of edges are not actually distinct, and the muoctahedron is in fact isotoxal like all other regular polyhedra.

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the bitruncated cubic honeycomb.

Constructions

As with other regular polyhedra with Schläfli types of the form {p,4}, the muoctahedron can also be constructed as the rectification of a polyhedron with the type {p,p}, in this case the halved mucube.

The muoctahedron can be constructed from the bitruncated cubic honeycomb. Its faces are the hexagonal faces of the bitruncated cubic honeycomb.

Abstractly it can also be formed as a quotient of the order-4 hexagonal tiling by identifying every 4th vertex on each hole. It can also be constructed as the first-order kappa κ 1  of the octahedron.

Gallery

• 

  A view of a complete muoctahedron.

Related polytopes

{6,4∣3} is one of several skew polyhedra of the form {6,4∣n}.

• {6,4∣3}, 4D spherical
• {6,4∣4} (muoctahedron), 3D parabolic
• {6,4∣5}, 3D compact hyperbolic
• {6,4∣6}, 3D paracompact hyperbolic

There is a direct correspondence between these and the regular polyhedra with triangular faces.

• {3,3} (tetrahedron)
• {3,4} (octahedron)
• {3,5} (icosahedron)
• {3,6} (triangular tiling)

External links

• Wikipedia contributors. "Regular skew apeirohedron".
• Klitzing, Richard. "shexat".
• Klitzing, Richard. "Skew polytopes".
• jan Misali (2020). "there are 48 regular polyhedra".

Bibliography

• Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. pp. 333–335.
• Coxeter, H.S.M. (1999). The Beauty of Geometry: Twelve essays. Dover Publications, Inc. pp. 157 ff.
• Garner, Cyril (1967), "Regular skew polyhedra in hyperbolic three-space" (PDF), Canadian Journal of Mathematics, 19, doi:10.4153/CJM-1967-106-9
• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space". Discrete Computational Geometry. 17: 449–478.