Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences.
He also studied[2] its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 221.
The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope: Its construction is based on the E6 group.
Seen in a configuration matrix, the element counts can be derived from the Coxeter group orders.
[5] Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram: .
Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.
Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .
Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t1(211), .
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple.