In 8-dimensional
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the 2
41 is a
uniform 8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transiti ...
, constructed within the symmetry of the
E8 group.
Its
Coxeter symbol
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
is 2
41, describing its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.
The rectified 2
41 is constructed by points at the mid-edges of the 2
41. The birectified 2
41 is constructed by points at the triangle face centers of the 2
41, and is the same as the
rectified 142.
These polytopes are part of a family of 255 (2
8 − 1) convex
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
s in 8-dimensions, made of
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
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, 2
41 polytope
, -
, bgcolor=#e7dcc3, Type, , Uniform
8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transitive, ...
, -
, bgcolor=#e7dcc3, Family, ,
2k1 polytope
, -
, bgcolor=#e7dcc3,
Schläfli symbol, , {3,3,3
4,1}
, -
, bgcolor=#e7dcc3,
Coxeter symbol
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, , 2
41
, -
, bgcolor=#e7dcc3,
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, ,
, -
, bgcolor=#e7dcc3, 7-faces, , 17520:
240
23117280
{36}
, -
, bgcolor=#e7dcc3, 6-faces, , 144960:
6720
221138240
{35}
, -
, bgcolor=#e7dcc3, 5-faces, , 544320:
60480
211483840
{34}
, -
, bgcolor=#e7dcc3, 4-faces, , 1209600:
241920
{201967680
{33}
, -
, bgcolor=#e7dcc3, Cells, , 1209600
{32}
, -
, bgcolor=#e7dcc3, Faces, , 483840
{3}
, -
, bgcolor=#e7dcc3, Edges, , 69120
, -
, bgcolor=#e7dcc3, Vertices, , 2160
, -
, bgcolor=#e7dcc3,
Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
, ,
141
, -
, bgcolor=#e7dcc3,
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
, ,
30-gon
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
Regular triacontagon
The '' regular triacontagon'' is a constructible polygon, by an edge- bisection of a regular ...
, -
, bgcolor=#e7dcc3,
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
, ,
E8,
4,2,1">4,2,1, -
, bgcolor=#e7dcc3, Properties, ,
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
The 2
41 is composed of 17,520
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
(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
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
), 483,840
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
(
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s), 69,120
edges, and 2160
vertices. Its
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is a
7-demicube
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. E ...
.
This polytope is a facet in the
uniform tessellation, 251 with
Coxeter-Dynkin diagram:
:
Alternate names
*
E. L. Elte named it V
2160 (for its 2160 vertices) in his 1912 listing of semiregular polytopes.
*It is named 2
41 by
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
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
In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells ''4-faces'', 1792 ''5-faces'', 1024 ''6-faces'', and 256 ''7-faces''.
It has two constr ...
)
: 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
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
...
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
In 7-dimensional geometry, a 7- simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/ ...
: . 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
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
7-demicube
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. E ...
, 1
41, .
Seen in a
configuration matrix, the element counts can be derived by mirror removal and ratios of
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
orders.
{, class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="2",
!
Configuration matrix
, - valign=top
!E
8, , , ,
''k''-face, , f
k , , f
0 , , f
1, , f
2, , f
3, , colspan=2, f
4, , colspan=2, f
5, , colspan=2, f
6, , colspan=2, f
7, ,
''k''-figure, , notes
, - align=right
, D
7 , , , , ( )
!f
0
, BGCOLOR="#e0e0ff" , 2160, , 64, , 672, , 2240, , 560, , 2240, , 280, , 1344, , 84, , 448, , 14, , 64, ,
h{4,3,3,3,3,3} , , E
8/D
7 = 192*10!/64/7! = 2160
, - align=right
, A
6A
1 , , , , { }
!f
1
, , 2, , BGCOLOR="#ffe0ff", 69120, , 21, , 105, , 35, , 140, , 35, , 105, , 21, , 42, , 7, , 7, ,
r{3,3,3,3,3} , , E
8/A
6A
1 = 192*10!/7!/2 = 69120
, - align=right
, A
4A
2A
1 , , , ,
{3}
!f
2
, , 3, , 3, , BGCOLOR="#ffe0e0", 483840, , 10, , 5, , 20, , 10, , 20, , 10, , 10, , 5, , 2, ,
{}x{3,3,3} , , E
8/A
4A
2A
1 = 192*10!/5!/3!/2 = 483840
, - align=right
, A
3A
3 , , , ,
{3,3}
!f
3
, , 4, , 6, , 4, , BGCOLOR="#ffffe0", 1209600, , 1, , 4, , 4, , 6, , 6, , 4, , 4, , 1, ,
{3,3}V( ) , , E
8/A
3A
3 = 192*10!/4!/4! = 1209600
, - align=right
, A
4A
3 , , , , rowspan=2,
{3,3,3}
!rowspan=2, f
4
, , 5, , 10, , 10, , 5, , BGCOLOR="#e0ffe0", 241920, , BGCOLOR="#e0ffe0", *, , 4, , 0, , 6, , 0, , 4, , 0, ,
{3,3} , , E
8/A
4A
3 = 192*10!/5!/4! = 241920
, - align=right
, A
4A
2 , ,
, , 5, , 10, , 10, , 5, , BGCOLOR="#e0ffe0", *, , BGCOLOR="#e0ffe0", 967680, , 1, , 3, , 3, , 3, , 3, , 1, ,
{3}V( ) , , E
8/A
4A
2 = 192*10!/5!/3! = 967680
, - align=right
, D
5A
2 , , , ,
{3,3,31,1}
!rowspan=2, f
5
, , 10, , 40, , 80, , 80, , 16, , 16, , BGCOLOR="#e0ffff", 60480, , BGCOLOR="#e0ffff", *, , 3, , 0, , 3, , 0, ,
{3} , , E
8/D
5A
2 = 192*10!/16/5!/2 = 40480
, - align=right
, A
5A
1 , , , ,
{3,3,3,3}
, , 6, , 15, , 20, , 15, , 0, , 6, , BGCOLOR="#e0ffff", *, , BGCOLOR="#e0ffff", 483840, , 1, , 2, , 2, , 1, ,
{ }V( ) , , E
8/A
5A
1 = 192*10!/6!/2 = 483840
, - align=right
, E
6A
1 , , , ,
{3,3,32,1}
!rowspan=2, f
6
, , 27, , 216, , 720, , 1080, , 216, , 432, , 27, , 72, , BGCOLOR="#e0e0ff", 6720, , BGCOLOR="#e0e0ff", *, , 2, , 0, , rowspan=2, { } , , E
8/E
6A
1 = 192*10!/72/6! = 6720
, - align=right
, A
6 , , , ,
{3,3,3,3,3}
, , 7, , 21, , 35, , 35, , 0, , 21, , 0, , 7, , BGCOLOR="#e0e0ff", *, , BGCOLOR="#e0e0ff", 138240, , 1, , 1, , E
8/A
6 = 192*10!/7! = 138240
, - align=right
, E
7 , , , ,
{3,3,33,1}
!rowspan=2, f
7
, , 126, , 2016, , 10080, , 20160, , 4032, , 12096, , 756, , 4032, , 56, , 576, , BGCOLOR="#ffe0ff", 240, , BGCOLOR="#ffe0ff", *, , rowspan=2, ( ) , , E
8/E
7 = 192*10!/72!/8! = 240
, - align=right
, A
7 , , , ,
{3,3,3,3,3,3}
, , 8, , 28, , 56, , 70, , 0, , 56, , 0, , 28, , 0, , 8, , BGCOLOR="#ffe0ff", *, , BGCOLOR="#ffe0ff", 17280, , E
8/A
7 = 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
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
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 2
41 polytope
, -
, bgcolor=#e7dcc3, Type, , Uniform
8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transitive, ...
, -
, bgcolor=#e7dcc3,
Schläfli symbol, , t
1{3,3,3
4,1}
, -
, bgcolor=#e7dcc3,
Coxeter symbol
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, , t
1(2
41)
, -
, bgcolor=#e7dcc3,
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, ,
, -
, 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
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
, ,
rectified 6-simplex
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''rect ...
prism
, -
, bgcolor=#e7dcc3,
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
, ,
30-gon
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
Regular triacontagon
The '' regular triacontagon'' is a constructible polygon, by an edge- bisection of a regular ...
, -
, bgcolor=#e7dcc3,
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
, ,
E8,
4,2,1">4,2,1, -
, bgcolor=#e7dcc3, Properties, ,
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
The rectified 2
41 is a
rectification of the 2
41 polytope, with vertices positioned at the mid-edges of the 2
41.
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
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
...
upon a set of 8
hyperplane mirrors in 8-dimensional space, defined by root vectors of the
E8 Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
.
The facet information can be extracted from its
Coxeter-Dynkin diagram: .
Removing the node on the short branch leaves the
rectified 7-simplex
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.
There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the ''r ...
: .
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
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. E ...
, 1
41.
The
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
rectified 6-simplex
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''rect ...
prism, .
Visualizations
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
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