In 6-dimensional geometry, the 1<sub>22</sub> polytope is a uniform polytope, constructed from the E<sub>6</sub> group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V<sub>72</sub> (for its 72 vertices).
Its Coxeter symbol is 1<sub>22</sub>, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1<sub>22</sub>, constructed by positions points on the elements of 1<sub>22</sub>. The rectified 1<sub>22</sub> is constructed by points at the mid-edges of the 1<sub>22</sub>. The birectified 1<sub>22</sub> is constructed by points at the triangle face centers of the 1<sub>22</sub>.
These polytopes are from 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: .
The 1<sub>22</sub> 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 E<sub>6</sub>.
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.
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, 1<sub>21</sub>, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0<sub>22</sub>, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
The regular complex polyhedron <sub>3</sub>{3}<sub>3</sub>{4}<sub>2</sub>, , in has a real representation as the 1<sub>22</sub> polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is <sub>3</sub>[3]<sub>3</sub>[4]<sub>2</sub>, order 1296. It has a half-symmetry quasiregular construction as , as a rectification of the Hessian polyhedron, .
Along with the semiregular polytope, 2<sub>21</sub>, 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: .
The 1<sub>22</sub> is related to the 24-cell by a geometric folding E6 â F4 of Coxeter-Dynkin diagrams, E6 corresponding to 1<sub>22</sub> 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 1<sub>22</sub>.
This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 2<sub>22</sub>, .
The rectified 1<sub>22</sub> polytope (also called 0<sub>221</sub>) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple.
Its construction is based on the E<sub>6</sub> group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .
Removing the ring on the short branch leaves the birectified 5-simplex, .
Removing the ring on either of 2-length branches leaves the birectified 5-orthoplex in its alternated form: t<sub>2</sub>(2<sub>11</sub>), .
The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}ÃÂ{3}ÃÂ{}, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
Its construction is based on the E<sub>6</sub> group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue.
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet.