# 3 21 polytope

| 3$_{21}$ | 3$_{21}$ | 2$_{31}$ | 2$_{31}$ | 1$_{32}$ | 1$_{32}$ |
| Rectified 3$_{21}$ | Rectified 3$_{21}$ | Rectified 3$_{21}$ | birectified 3$_{21}$ | birectified 3$_{21}$ | birectified 3$_{21}$ |
| Rectified 2$_{31}$ | Rectified 2$_{31}$ | Rectified 2$_{31}$ | Rectified 1$_{32}$ | Rectified 1$_{32}$ | Rectified 1$_{32}$ |
| Orthogonal projections in E$_7$ Coxeter plane ||||||
In 7-dimensional geometry, the 3$_{21}$ polytope is a uniform 7-polytope, constructed within the symmetry of the E$_7$ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.
Its Coxeter symbol is 3$_{21}$, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.
The rectified 3$_{21}$ is constructed by points at the mid-edges of the 3$_{21}$. The birectified 3$_{21}$ is constructed by points at the triangle face centers of the 3$_{21}$. The trirectified 3$_{21}$ is constructed by points at the tetrahedral centers of the 3$_{21}$, and is the same as the rectified 1$_{32}$.
These polytopes are part of a family of 127 (2$^7$-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## 3$_{21}$ polytope

3$_{21}$ polytope :
| Type | Uniform 7-polytope |
| Family | k$_{21}$ polytope |
| Schläfli symbol | {3,3,3,3$^{2,1}$} |
| Coxeter symbol | 3$_{21}$ |
| Coxeter diagram | |
| 6-faces | 702 total: 126 3$_{11}$ 576 {3$^5$} |
| 5-faces | 6048: 4032 {3$^4$} 2016 {3$^4$} |
| 4-faces | 12096 {3$^3$} |
| Cells | 10080 {3,3} |
| Faces | 4032 {3} |
| Edges | 756 |
| Vertices | 56 |
| Vertex figure | 2$_{21}$ polytope |
| Petrie polygon | octadecagon |
| Coxeter group | E$_7$, [3$^{3,2,1}$], order 2903040 |
| Properties | convex |
In 7-dimensional geometry, the 3$_{21}$ polytope is a uniform polytope. It has 56 vertices, and 702 facets: 126 3$_{11}$ and 576 6-simplexes.
For 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). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
The 1-skeleton of the 3$_{21}$ polytope is the Gosset graph.
This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 3$_{31}$ and Coxeter-Dynkin diagram: .

### Alternate names

* It is also called the Hess polytope for Edmund Hess who first discovered it.
* It was enumerated by Thorold Gosset in his 1900 paper. He called it an 7-ic semi-regular figure.
* E. L. Elte named it V$_{56}$ (for its 56 vertices) in his 1912 listing of semiregular polytopes.
* H.S.M. 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.
* Hecatonicosihexa-pentacosiheptacontihexa-exon (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)

### Coordinates

The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:
    ± (-3, -3, 1, 1, 1, 1, 1, 1)

### Construction

Its construction is based on the E7 group. 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 facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 6-simplex, .
Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 3$_{11}$, .
Every simplex facet touches a 6-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 2$_{21}$ polytope, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
| E$_7$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_3$ | f$_4$ | f$_5$ | f$_5$ | f$_6$ | f$_6$ | k-figures | notes |
| E$_6$ | ( ) | f$_0$ | 56 | 27 | 216 | 720 | 1080 | 432 | 216 | 72 | 27 | 2$_{21}$ | E$_7$/E$_6$ = 72x8!/72x6! = 56 |
| D$_5$A$_1$ | { } | f$_1$ | 2 | 756 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 5-demicube | E$_7$/D$_5$A$_1$ = 72x8!/16/5!/2 = 756 |
| A$_4$A$_2$ | {3} | f$_2$ | 3 | 3 | 4032 | 10 | 30 | 20 | 10 | 5 | 5 | rectified 5-cell | E$_7$/A$_4$A$_2$ = 72x8!/5!/2 = 4032 |
| A$_3$A$_2$A$_1$ | {3,3} | f$_3$ | 4 | 6 | 4 | 10080 | 6 | 6 | 3 | 2 | 3 | triangular prism | E$_7$/A$_3$A$_2$A$_1$ = 72x8!/4!/3!/2 = 10080 |
| A$_4$A$_1$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 5 | 12096 | 2 | 1 | 1 | 2 | isosceles triangle | E$_7$/A$_4$A$_1$ = 72x8!/5!/2 = 12096 |
| A$_5$A$_1$ | {3,3,3,3} | f$_5$ | 6 | 15 | 20 | 15 | 6 | 4032 | * | 1 | 1 | { } | E$_7$/A$_5$A$_1$ = 72x8!/6!/2 = 4032 |
| A$_5$ | {3,3,3,3} | f$_5$ | 6 | 15 | 20 | 15 | 6 | * | 2016 | 0 | 2 | { } | E$_7$/A$_5$ = 72x8!/6! = 2016 |
| A$_6$ | {3,3,3,3,3} | f$_6$ | 7 | 21 | 35 | 35 | 21 | 10 | 0 | 576 | * | ( ) | E$_7$/A$_6$ = 72x8!/7! = 576 |
| D$_6$ | {3,3,3,3,4} | f$_6$ | 12 | 60 | 160 | 240 | 192 | 32 | 32 | * | 126 | ( ) | E$_7$/D$_6$ = 72x8!/32/6! = 126 |

### Images

Coxeter plane projections :
| E7 | E6 / F4 | B7 / A6 |
| [18] | [12] | [7x2] |
| A5 | D7 / B6 | D6 / B5 |
| [6] | [12/2] | [10] |
| D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |
| [8] | [6] | [4] |

### Related polytopes

The 3$_{21}$ is fifth 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.
k$_{21}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| E$_n$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Coxeter group | E$_3$=A$_2$A$_1$ | E$_4$=A$_4$ | E$_5$=D$_5$ | E$_6$ | E$_7$ | E$_8$ | E$_9$ = ${\tilde {E}}_{8}$ = E$_8$$^+$ | E$_{10}$ = ${\bar {T}}_{8}$ = E$_8$$^{++}$ |
| Coxeter diagram | | | | | | | | |
| Symmetry | [3$^{−1,2,1}$] | [3$^{0,2,1}$] | [3$^{1,2,1}$] | [3$^{2,2,1}$] | [3$^{3,2,1}$] | [3$^{4,2,1}$] | [3$^{5,2,1}$] | [3$^{6,2,1}$] |
| Order | 12 | 120 | 1,920 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ∞ |
| Graph | | | | | | | - | - |
| Name | −1$_{21}$ | 0$_{21}$ | 1$_{21}$ | 2$_{21}$ | 3$_{21}$ | 4$_{21}$ | 5$_{21}$ | 6$_{21}$ |
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3$_{k1}$ series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)
3$_{k1}$ dimensional figures :
| Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$=E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| Coxeter diagram | | | | | | |
| Symmetry | [3$^{−1,3,1}$] | [3$^{0,3,1}$] | [[3$^{1,3,1}$]] = [4,3,3,3,3] | [3$^{2,3,1}$] | [3$^{3,3,1}$] | [3$^{4,3,1}$] |
| Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | ∞ |
| Graph | | | | | - | - |
| Name | 3$_{1,-1}$ | 3$_{10}$ | 3$_{11}$ | 3$_{21}$ | 3$_{31}$ | 3$_{41}$ |

## Rectified 3$_{21}$ polytope

Rectified 3$_{21}$ polytope :
| Type | Uniform 7-polytope |
| Schläfli symbol | t$_1${3,3,3,3$^{2,1}$} |
| Coxeter symbol | t$_1$(3$_{21}$) |
| Coxeter diagram | |
| 6-faces | 758 |
| 5-faces | 44352 |
| 4-faces | 70560 |
| Cells | 48384 |
| Faces | 11592 |
| Edges | 12096 |
| Vertices | 756 |
| Vertex figure | 5-demicube prism |
| Petrie polygon | octadecagon |
| Coxeter group | E$_7$, [3$^{3,2,1}$], order 2903040 |
| Properties | convex |

### Alternate names

* Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers)

### Construction

Its construction is based on the E7 group. Coxeter named it as 3$_{21}$ by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 6-simplex, .
Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: t$_1$3$_{11}$, .
Removing the node on the end of the 3-length branch leaves the 2$_{21}$, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-demicube prism, .

### Images

Coxeter plane projections :
| E7 | E6 / F4 | B7 / A6 |
| [18] | [12] | [7x2] |
| A5 | D7 / B6 | D6 / B5 |
| [6] | [12/2] | [10] |
| D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |
| [8] | [6] | [4] |

## Birectified 3$_{21}$ polytope

Birectified 3$_{21}$ polytope :
| Type | Uniform 7-polytope |
| Schläfli symbol | t$_2${3,3,3,3$^{2,1}$} |
| Coxeter symbol | t$_2$(3$_{21}$) |
| Coxeter diagram | |
| 6-faces | 758 |
| 5-faces | 12348 |
| 4-faces | 68040 |
| Cells | 161280 |
| Faces | 161280 |
| Edges | 60480 |
| Vertices | 4032 |
| Vertex figure | 5-cell-triangle duoprism |
| Petrie polygon | octadecagon |
| Coxeter group | E$_7$, [3$^{3,2,1}$], order 2903040 |
| Properties | convex |

### Alternate names

* Birectified hecatonicosihexa-pentacosiheptacontihexa-exon as a birectified 126-576 facetted polyexon (acronym branq) (Jonathan Bowers)

### Construction

Its construction is based on the E7 group. Coxeter named it as 3$_{21}$ by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the birectified 6-simplex, .
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 end of the 3-length branch leaves the rectified 2$_{21}$ polytope in its alternated form: t$_1$(2$_{21}$), .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-cell-triangle duoprism, .

### Images

Coxeter plane projections :
| E7 | E6 / F4 | B7 / A6 |
| [18] | [12] | [7x2] |
| A5 | D7 / B6 | D6 / B5 |
| [6] | [12/2] | [10] |
| D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |
| [8] | [6] | [4] |