Cryptext 3.21 series#

Removing the node on the end of the 3-length branch leaves the rectified 2 21 polytope in its alternated form: t 1(2 21). Removing the node on the end of the 2-length branch leaves the birectified 6-orthoplex in its alternated form: t 2(3 11). Removing the node on the short branch leaves the birectified 6-simplex. Birectified hecatonicosihexa-pentacosiheptacontihexa-exon as a birectified 126-576 facetted polyexon (acronym branq) (Jonathan Bowers).īirectified 3 21 polytope Birectified 3 21 polytope The vertex figure is determined by removing the ringed node and ringing the neighboring node. Removing the node on the end of the 3-length branch leaves the 2 21. Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: t 13 11. Coxeter named it as 3 21 by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence. Its construction is based on the E7 group. Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers).Rectified 3 21 polytope Rectified 3 21 polytope

Cryptext 3.21 series#

Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.Į 9 = E ~ 8 =E 7 ++ Each progressive uniform polytope is constructed vertex figure of the previous polytope. The 3 21 is fifth in a dimensional series of semiregular polytopes. Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders. Įvery simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex. Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 3 11. Removing the node on the short branch leaves the 6-simplex. The facet information can be extracted from its Coxeter-Dynkin diagram. Coxeter named it as 3 21 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence. The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite: Hecatonicosihexa-pentacosiheptacontihexa-exon (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers).Coxeter called it 3 21 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch. Elte named it V 56 (for its 56 vertices) in his 1912 listing of semiregular polytopes. He called it an 7-ic semi-regular figure. It was enumerated by Thorold Gosset in his 1900 paper.It is also called the Hess polytope for Edmund Hess who first discovered it.This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 3 31 and Coxeter-Dynkin diagram. The 1- skeleton of the 3 21 polytope is the Gosset graph. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection. Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. It has 56 vertices, and 702 facets: 126 3 11 and 576 6-simplexes.įor visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). In 7-dimensional geometry, the 3 21 is a uniform polytope.