2 41 polytope

In 8-dimensional geometry, the 2₄₁ is a uniform 8-polytope, constructed within the symmetry of the E₈ group.

Its Coxeter symbol is 2₄₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.

The rectified 2₄₁ is constructed by points at the mid-edges of the 2₄₁. The birectified 2₄₁ is constructed by points at the triangle face centers of the 2₄₁, and is the same as the rectified 1₄₂.

These polytopes are part of a family of 255 (2⁸ − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2₄₁ polytope

{, class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2, 2₄₁ polytope
, -
, bgcolor=#e7dcc3, Type, , Uniform 8-polytope
, -
, bgcolor=#e7dcc3, Family, , 2_k1 polytope
, -
, bgcolor=#e7dcc3, Schläfli symbol, , {3,3,3^4,1}
, -
, bgcolor=#e7dcc3, Coxeter symbol, , 2₄₁
, -
, bgcolor=#e7dcc3, Coxeter diagram, ,
, -
, bgcolor=#e7dcc3, 7-faces, , 17520:
240 2₃₁
17280 {3⁶}
, -
, bgcolor=#e7dcc3, 6-faces, , 144960:
6720 2₂₁
138240 {3⁵}
, -
, bgcolor=#e7dcc3, 5-faces, , 544320:
60480 2₁₁
483840 {3⁴}
, -
, bgcolor=#e7dcc3, 4-faces, , 1209600:
241920 {2₀₁
967680 {3³}
, -
, bgcolor=#e7dcc3, Cells, , 1209600 {3²}
, -
, bgcolor=#e7dcc3, Faces, , 483840 {3}
, -
, bgcolor=#e7dcc3, Edges, , 69120
, -
, bgcolor=#e7dcc3, Vertices, , 2160
, -
, bgcolor=#e7dcc3, Vertex figure, , 1₄₁
, -
, bgcolor=#e7dcc3, Petrie polygon, , 30-gon
, -
, bgcolor=#e7dcc3, Coxeter group, , E₈, ^4,2,1, -
, bgcolor=#e7dcc3, Properties, , convex

The 2₄₁ is composed of 17,520 facets (240 2₃₁ polytopes and 17,280 7-simplices), 144,960 ''6-faces'' (6,720 2₂₁ polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 2₁₁ and 483,840 5-simplices), 1,209,600 ''4-faces'' ( 4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.

This polytope is a facet in the uniform tessellation, 2₅₁ with Coxeter-Dynkin diagram:
:

Alternate names

* E. L. Elte named it V₂₁₆₀ (for its 2160 vertices) in his 1912 listing of semiregular polytopes.
*It is named 2₄₁ by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
* Diacositetracont-myriaheptachiliadiacosioctaconta-zetton (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)

Coordinates

The 2160 vertices can be defined as follows:
: 16 permutations of (±4,0,0,0,0,0,0,0) of (8-orthoplex)
: 1120 permutations of (±2,±2,±2,±2,0,0,0,0) of ( trirectified 8-orthoplex)
: 1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) ''with an odd number of minus-signs''

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram: .

Removing the node on the short branch leaves the 7-simplex: . There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 2₃₁, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4₂₁ polytope.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 1₄₁, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
{, class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="2",
! Configuration matrix
, - valign=top
!E₈, , , , ''k''-face, , f_k , , f₀ , , f₁, , f₂, , f₃, , colspan=2, f₄, , colspan=2, f₅, , colspan=2, f₆, , colspan=2, f₇, , ''k''-figure, , notes
, - align=right
, D₇ , , , , ( )
!f₀
, BGCOLOR="#e0e0ff" , 2160, , 64, , 672, , 2240, , 560, , 2240, , 280, , 1344, , 84, , 448, , 14, , 64, , h{4,3,3,3,3,3} , , E₈/D₇ = 192*10!/64/7! = 2160
, - align=right
, A₆A₁ , , , , { }
!f₁
, , 2, , BGCOLOR="#ffe0ff", 69120, , 21, , 105, , 35, , 140, , 35, , 105, , 21, , 42, , 7, , 7, , r{3,3,3,3,3} , , E₈/A₆A₁ = 192*10!/7!/2 = 69120
, - align=right
, A₄A₂A₁ , , , , {3}
!f₂
, , 3, , 3, , BGCOLOR="#ffe0e0", 483840, , 10, , 5, , 20, , 10, , 20, , 10, , 10, , 5, , 2, , {}x{3,3,3} , , E₈/A₄A₂A₁ = 192*10!/5!/3!/2 = 483840
, - align=right
, A₃A₃ , , , , {3,3}
!f₃
, , 4, , 6, , 4, , BGCOLOR="#ffffe0", 1209600, , 1, , 4, , 4, , 6, , 6, , 4, , 4, , 1, , {3,3}V( ) , , E₈/A₃A₃ = 192*10!/4!/4! = 1209600
, - align=right
, A₄A₃ , , , , rowspan=2, {3,3,3}
!rowspan=2, f₄
, , 5, , 10, , 10, , 5, , BGCOLOR="#e0ffe0", 241920, , BGCOLOR="#e0ffe0", *, , 4, , 0, , 6, , 0, , 4, , 0, , {3,3} , , E₈/A₄A₃ = 192*10!/5!/4! = 241920
, - align=right
, A₄A₂ , ,
, , 5, , 10, , 10, , 5, , BGCOLOR="#e0ffe0", *, , BGCOLOR="#e0ffe0", 967680, , 1, , 3, , 3, , 3, , 3, , 1, , {3}V( ) , , E₈/A₄A₂ = 192*10!/5!/3! = 967680
, - align=right
, D₅A₂ , , , , {3,3,3^1,1}
!rowspan=2, f₅
, , 10, , 40, , 80, , 80, , 16, , 16, , BGCOLOR="#e0ffff", 60480, , BGCOLOR="#e0ffff", *, , 3, , 0, , 3, , 0, , {3} , , E₈/D₅A₂ = 192*10!/16/5!/2 = 40480
, - align=right
, A₅A₁ , , , , {3,3,3,3}
, , 6, , 15, , 20, , 15, , 0, , 6, , BGCOLOR="#e0ffff", *, , BGCOLOR="#e0ffff", 483840, , 1, , 2, , 2, , 1, , { }V( ) , , E₈/A₅A₁ = 192*10!/6!/2 = 483840
, - align=right
, E₆A₁ , , , , {3,3,3^2,1}
!rowspan=2, f₆
, , 27, , 216, , 720, , 1080, , 216, , 432, , 27, , 72, , BGCOLOR="#e0e0ff", 6720, , BGCOLOR="#e0e0ff", *, , 2, , 0, , rowspan=2, { } , , E₈/E₆A₁ = 192*10!/72/6! = 6720
, - align=right
, A₆ , , , , {3,3,3,3,3}
, , 7, , 21, , 35, , 35, , 0, , 21, , 0, , 7, , BGCOLOR="#e0e0ff", *, , BGCOLOR="#e0e0ff", 138240, , 1, , 1, , E₈/A₆ = 192*10!/7! = 138240
, - align=right
, E₇ , , , , {3,3,3^3,1}
!rowspan=2, f₇
, , 126, , 2016, , 10080, , 20160, , 4032, , 12096, , 756, , 4032, , 56, , 576, , BGCOLOR="#ffe0ff", 240, , BGCOLOR="#ffe0ff", *, , rowspan=2, ( ) , , E₈/E₇ = 192*10!/72!/8! = 240
, - align=right
, A₇ , , , , {3,3,3,3,3,3}
, , 8, , 28, , 56, , 70, , 0, , 56, , 0, , 28, , 0, , 8, , BGCOLOR="#ffe0ff", *, , BGCOLOR="#ffe0ff", 17280, , E₈/A₇ = 192*10!/8! = 17280

Visualizations

{, class=wikitable width=600
!E8
0! 0! 4, - align=center
,
(1)
,
,
, - align=center
!E7
8!E6
2! , - align=center
,
,
(1,8,24,32)
,

Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

{, class=wikitable width=600
, - align=center
!D3 / B2 / A3
!D4 / B3 / A2
!D5 / B4
, - align=center
,
,
,
, - align=center
!D6 / B5 / A4
0!D7 / B6
2!D8 / B7 / A6
4, - align=center
,
,
(1,3,9,12,18,21,36)
,
, - align=center
!B8
6/2!A5
!A7
, - align=center
,
,
,

Related polytopes and honeycombs

Rectified 2_41 polytope

{, class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2, Rectified 2₄₁ polytope
, -
, bgcolor=#e7dcc3, Type, , Uniform 8-polytope
, -
, bgcolor=#e7dcc3, Schläfli symbol, , t₁{3,3,3^4,1}
, -
, bgcolor=#e7dcc3, Coxeter symbol, , t₁(2₄₁)
, -
, bgcolor=#e7dcc3, Coxeter diagram, ,
, -
, bgcolor=#e7dcc3, 7-faces, , 19680 total:
240 t₁(2₂₁)

17280 t₁{3⁶}

2160 1₄₁
, -
, bgcolor=#e7dcc3, 6-faces, , 313440
, -
, bgcolor=#e7dcc3, 5-faces, , 1693440
, -
, bgcolor=#e7dcc3, 4-faces, , 4717440
, -
, bgcolor=#e7dcc3, Cells, , 7257600
, -
, bgcolor=#e7dcc3, Faces, , 5322240
, -
, bgcolor=#e7dcc3, Edges, , 19680
, -
, bgcolor=#e7dcc3, Vertices, , 69120
, -
, bgcolor=#e7dcc3, Vertex figure, , rectified 6-simplex prism
, -
, bgcolor=#e7dcc3, Petrie polygon, , 30-gon
, -
, bgcolor=#e7dcc3, Coxeter group, , E₈, ^4,2,1, -
, bgcolor=#e7dcc3, Properties, , convex

The rectified 2₄₁ is a rectification of the 2₄₁ polytope, with vertices positioned at the mid-edges of the 2₄₁.

Alternate names

* Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (known as robay for short)Klitzing, (o3x3o3o *c3o3o3o3o - robay)

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E₈ Coxeter group.

The facet information can be extracted from its Coxeter-Dynkin diagram: .

Removing the node on the short branch leaves the rectified 7-simplex: .

Removing the node on the end of the 4-length branch leaves the rectified 2₃₁, .

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1₄₁.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, .

Visualizations

Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

{, class=wikitable width=600
!E8
0! 0! 4, - align=center
,
(1)
,
,
, - align=center
!E7
8!E6
2! , - align=center
,
,
(1,8,24,32)
,

{, class=wikitable width=600
, - align=center
!D3 / B2 / A3
!D4 / B3 / A2
!D5 / B4
, - align=center
,
,
,
, - align=center
!D6 / B5 / A4
0!D7 / B6
2!D8 / B7 / A6
4, - align=center
,
,
(1,3,9,12,18,21,36)
,
, - align=center
!B8
6/2!A5
!A7
, - align=center
,
,
,

See also

* List of E8 polytopes

Notes

References

*
* H. S. M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay

{{Polytopes

8-polytopes