- Semiregular 4-polytope
In
geometry , a semiregular 4-polytope (orpolychoron ) is a 4-dimensionalpolytope which isvertex-transitive (i.e. thesymmetry group of the polytope acts transitively on the vertices) and whose cells are regular polyhedra. These represent a subset of theuniform polychora which are composed of both regular and uniform polyhedra cells.A further constraint can require edge-transitivity. Polychora that fail this contraint are listed and noted as such. The regular and semiregular honeycombs, and regular polychora are also listed here for completeness.
Summary
Polychora
* 6 regular polychora
* 2 Vertex-transitive AND edge-transitive:rectified 5-cell ,Rectified 600-cell
* 1 Vertex-transitive:snub 24-cell Honeycombs
* 1 Regular honeycomb:cubic honeycomb
* 1 Vertex-transitive AND edge-transitive:tetrahedral-octahedral honeycomb
* 1 Vertex-transitive:gyrated tetrahedral-octahedral honeycomb Regular polytopes
The 6
convex regular 4-polytope s are:There are two semiregular
honeycomb s and they contain the same edge cells, but the second is not edge-transitive as the order changes.
Vertex figuresDual honeycombs
The "regular"
cubic honeycomb is self-dual.The "semiregular"
tetrahedral-octahedral honeycomb dual is called arhombic dodecahedral honeycomb .The "semiregular"
gyrated tetrahedral-octahedral honeycomb dual is called arhombo-hexagonal dodecahedron honeycomb.Existence enumeration by edge configurations
Semiregular polytopes are constructed by vertex figures which are regular, semiregular or johnson polyhedra.
# If the vertex figure is a regular
Platonic solid polyhedron, the polytope will be regular.
# If the vertex figure is asemiregular polyhedron , the polytope will have one type of edge configuration.
# If the vertex figure is aJohnson solid polyhedron, then the polytope will have more than one edge configuration.Edge configurations are limited by the sum of the dihedral angles of the cells along the edge. The sum of dihedral angles must be 360 degrees or less. If it is equal to 360, the vertex figure will stay within 3D space and can be a part of an infinite tessellation.
The
dihedral angle of each Platonic solid is:
where φ = (1 + √5)/2 is the golden mean.Name exact dihedral angle (in radians) approximate dihedral angle (in degrees) {3,3} Tetrahedron arccos(1/3) 70.53° {3,4} Octahedron π − arccos(1/3) 109.47° {4,3} Hexahedron or Cubeπ/2 90° {3,5} Icosahedron 2·arctan(φ + 1) 138.19° {5,3} Dodecahedron 2·arctan(φ) 116.56° There are 17 possible edge configurations formed by the 5
platonic solids that have angle defects of zero or greater.
* Three cells/edge:
# {3,3}3
# {3,3}2.{3.4}
# {3,3}2.{3.5}
# {3,3}.{3,4}2
# {3,3}.{3.4}.{3.5}
# {3,3}.{3.5}2
# {3,4}3
# {3,4}2.{3,5}
# {4,3}3
# {5,3}3
* Four cells/edge:
# {3,3}4
# {3,3}2.{3.4}2 [Angle defect zero]
# [{3,3}.{3.4}] 2 [Angle defect zero]
# {3,3}3.{3,4}
# {3,3}3.{3,5}
# {4,3}4 [Angle defect zero]
* Five cells/edge:
# {3,3}5As listed above, from these 17 edge configurations and a single vertex figure, there are 6 regular polytopes, and 3 semiregular polytopes, 1 regular honeycomb, and 2 semiregular honeycombs.
ee also
*
Convex regular 4-polytope
*Uniform polychora
*Semiregular polyhedron
*List of regular polytopes External links
* Vertex/Edge/Face/Cell data
** [http://members.aol.com/Polycell/section1.html Dispentachoron] [2]
** [http://members.aol.com/Polycell/section4.html Icosahedral hexacosihecatonicosachoron] [34]
** [http://members.aol.com/Polycell/section3.html Snub icositetrachoron] [31]
* Exploded/Unfolded cell images
** [http://members.aol.com/Polycell/nets.html Snub icositetrachoron]
** [http://members.aol.com/Polycell/nets.html Icosahedral hexacosihecatonicosachoron]
* Data and Images (www.polytope.de)
** [http://www.polytope.de/nr01.html 5-cell]
** [http://www.polytope.de/nr06.html 8-cell]
** [http://www.polytope.de/nr02.html 16-cell]
** [http://www.polytope.de/nr07.html 24-cell]
** [http://www.polytope.de/nr49.html 120-cell]
** [http://www.polytope.de/nr24.html 600-cell]
** [http://www.polytope.de/nr03.html Rectified 5-cell]
** [http://www.polytope.de/nr45.html Rectified 600-cell]
** [http://www.polytope.de/nr22.html Snub icositetrachoron]
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