Muoctahedron
Muoctahedron  

Rank  3 
Space  Euclidean 
Notation  
Bowers style acronym  Muo 
Schläfli symbol  {6,4∣4}, 
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  
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  R_{4}×2 
Convex  No 
Dimension vector  (2,1,2) 
History  
Discovered by  John Finders Petrie^{[2]} 
First discovered  1926^{[1]} 
The muoctahedron or muo, short for multiple octahedron, is a regular skew apeirohedron in Euclidean 3space. 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[edit  edit source]
Its vertices are the same as those of its regiment colonel, the bitruncated cubic honeycomb.
Constructions[edit  edit source]
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 order4 hexagonal tiling by identifying every 4th vertex on each hole. It can also be constructed as the firstorder kappa κ 1 of the octahedron.
Gallery[edit  edit source]

A view of a complete muoctahedron.
Related polytopes[edit  edit source]
{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.
External links[edit  edit source]
 Wikipedia contributors. "Regular skew apeirohedron".
 Klitzing, Richard. "shexat".
 Klitzing, Richard. "Skew polytopes".
 jan Misali (2020). "there are 48 regular polyhedra".
References[edit  edit source]
Bibliography[edit  edit source]
 Conway, John H.; Burgiel, Heidi; GoodmanStrauss, 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 threespace" (PDF), Canadian Journal of Mathematics, 19, doi:10.4153/CJM19671069
 McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space". Discrete Computational Geometry. 17: 449–478.