The
221 has 27 vertices, and 99 facets: 27
5-orthoplexes and 72
5-simplices. Its
vertex figure is a
5-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a
Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection. The
Schläfli graph is the
1-skeleton of this polytope.
Alternate names •
E. L. Elte named it V27 (for its 27 vertices) in his 1912 listing of semiregular polytopes. •
Icosihepta-heptacontadi-peton - 27-72 facetted polypeton (Acronym: jak) (Jonathan Bowers)
Coordinates The 27 vertices can be expressed in 8-space as an edge-figure of the
421 polytope: :(−2, 0, 0, 0,−2, 0, 0, 0), :( 0,−2, 0, 0,−2, 0, 0, 0), :( 0, 0,−2, 0,−2, 0, 0, 0), :( 0, 0, 0,−2,−2, 0, 0, 0), :( 0, 0, 0, 0,−2, 0, 0,−2), :( 0, 0, 0, 0, 0,−2,−2, 0) :( 2, 0, 0, 0,−2, 0, 0, 0), :( 0, 2, 0, 0,−2, 0, 0, 0), :( 0, 0, 2, 0,−2, 0, 0, 0), :( 0, 0, 0, 2,−2, 0, 0, 0), :( 0, 0, 0, 0,−2, 0, 0, 2) :(−1,−1,−1,−1,−1,−1,−1,−1), :(−1,−1,−1, 1,−1,−1,−1, 1), :(−1,−1, 1,−1,−1,−1,−1, 1), :(−1,−1, 1, 1,−1,−1,−1,−1), :(−1, 1,−1,−1,−1,−1,−1, 1), :(−1, 1,−1, 1,−1,−1,−1,−1), :(−1, 1, 1,−1,−1,−1,−1,−1), :( 1,−1,−1,−1,−1,−1,−1, 1), :( 1,−1, 1,−1,−1,−1,−1,−1), :( 1,−1,−1, 1,−1,−1,−1,−1), :( 1, 1,−1,−1,−1,−1,−1,−1), :(−1, 1, 1, 1,−1,−1,−1, 1), :( 1,−1, 1, 1,−1,−1,−1, 1), :( 1, 1,−1, 1,−1,−1,−1, 1), :( 1, 1, 1,−1,−1,−1,−1, 1), :( 1, 1, 1, 1,−1,−1,−1,−1)
Construction Its construction is based on the
E6 group. The facet information can be extracted from its
Coxeter-Dynkin diagram, . Removing the node on the short branch leaves the
5-simplex, . Removing the node on the end of the 2-length branch leaves the
5-orthoplex in its alternated form: (
211), . Every simplex facet touches a 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex. The
vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes
5-demicube (121 polytope), . The edge-figure is the vertex figure of the vertex figure, a
rectified 5-cell, (021 polytope), . Seen in a
configuration matrix, the element counts can be derived from the
Coxeter group orders.
Images Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The multiplicities of vertices by color are given in parentheses.
Geometric folding The
221 is related to the
24-cell by a geometric
folding of the E6/F4
Coxeter-Dynkin diagrams. 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 221. This polytope can tessellate Euclidean 6-space, forming the
222 honeycomb with this Coxeter-Dynkin diagram: .
Related complex polyhedra The
regular complex polygon 3{3}3{3}3, , in \mathbb{C}^2 has a real representation as the
221 polytope, , in 4-dimensional space. It is called a
Hessian polyhedron after
Edmund Hess. It has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its
complex reflection group is 3[3]3[3]3, order 648.
Related polytopes The 221 is fourth in a dimensional series of
semiregular polytopes. Each progressive
uniform polytope is constructed
vertex figure of the previous polytope.
Thorold Gosset identified this series in 1900 as containing all
regular polytope facets, containing all
simplexes and
orthoplexes. The 221 polytope is fourth in dimensional series 2
k1. The 221 polytope is second in dimensional series 22
k. == Rectified 221 polytope ==