The 122 polytope contains 72 vertices, and 54
5-demicubic facets. It has a
birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the
simple Lie group E6.
Alternate names •
Pentacontatetrapeton (Acronym: mo) - 54-facetted polypeton (Jonathan Bowers)
Images Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green. The multiplicities of vertices by color are given in parentheses.
Construction It is created by a
Wythoff construction upon a set of 6
hyperplane mirrors in 6-dimensional space. The facet information can be extracted from its
Coxeter-Dynkin diagram, . Removing the node on either of 2-length branches leaves the
5-demicube, 121, . The
vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the
birectified 5-simplex, 022, . Seen in a
configuration matrix, the element counts can be derived by mirror removal and ratios of
Coxeter group orders.
Related complex polyhedron The
regular complex polyhedron 3{3}3{4}2, , in \mathbb{C}^2 has a real representation as the
122 polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its
complex reflection group is 3[3]3[4]2, order 1296. It has a half-symmetry quasiregular construction as , as a
rectification of the
Hessian polyhedron, .
Related polytopes and honeycomb Along with the semiregular polytope,
221, it is also one of a family of 39 convex
uniform polytopes in 6-dimensions, made of
uniform polytope facets and
vertex figures, defined by all permutations of rings in this
Coxeter-Dynkin diagram: .
Geometric folding The
122 is related to the
24-cell by a geometric
folding E6 → F4 of
Coxeter-Dynkin diagrams, E6 corresponding to
122 in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the
Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 122.
Tessellations This polytope is the
vertex figure for a
uniform tessellation of 6-dimensional space,
222, . == Rectified 122 polytope ==