In 6-dimensional
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the 1
22 polytope is a
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
, constructed from the
E6 group. It was first published in
's 1912 listing of semiregular polytopes, named as V
72 (for its 72 vertices).
Its
Coxeter symbol
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
is 1
22, describing its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1
22, constructed by positions points on the elements of 1
22. The rectified 1
22 is constructed by points at the mid-edges of the 1
22. The birectified 1
22 is constructed by points at the triangle face centers of the 1
22.
These polytopes are from a family of 39 convex
uniform polytopes in 6-dimensions, made of
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
facets and
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s, defined by all permutations of rings in this
Coxeter-Dynkin diagram: .
122 polytope
The 1
22 polytope contains 72 vertices, and 54
5-demicubic facets. It has a
birectified 5-simplex
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ''rec ...
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
. Its 72 vertices represent the root vectors of the
simple Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
E6.
Alternate names
* Pentacontatetrapeton (Acronym: mo) - 54-facetted polypeton (Jonathan Bowers)
Images
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 6
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
mirrors in 6-dimensional space.
The facet information can be extracted from its
Coxeter-Dynkin diagram, .
Removing the node on either of 2-length branches leaves the
5-demicube
In Five-dimensional space, five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with Alternation (geometry), alternated vertices removed.
It was discovered by Thorold ...
, 1
31, .
The
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
birectified 5-simplex
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ''rec ...
, 0
22, .
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 ref ...
orders.
[Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203]
Related complex polyhedron

The
regular complex polyhedron 332, , in
has a real representation as the ''1
22'' polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 33 faces. Its
complex reflection group
In mathematics, a complex reflection group is a Group (mathematics), finite group acting on a finite-dimensional vector space, finite-dimensional complex numbers, complex vector space that is generated by complex reflections: non-trivial elements t ...
is
3 sub>3
sub>2, order 1296. It has a half-symmetry quasiregular construction as , as a
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
of the
Hessian polyhedron
In geometry, the Hessian polyhedron is a regular complex polytope, regular complex polyhedron 333, , in \mathbb^3. It has 27 vertices, 72 3 edges, and 27 Möbius–Kantor polygon, 33 faces. It is self-dual.
Harold Scott MacDonald Coxeter, Coxete ...
, .
Related polytopes and honeycomb
Along with the semiregular polytope,
221, it is also one of a family of 39 convex
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
s in 6-dimensions, made of
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
facets and
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s, defined by all permutations of rings in this
Coxeter-Dynkin diagram: .
Geometric folding
The 1
22 is related to the
24-cell
In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...
by a geometric
folding
Fold, folding or foldable may refer to:
Arts, entertainment, and media
* ''Fold'' (album), the debut release by Australian rock band Epicure
* Fold (poker), in the game of poker, to discard one's hand and forfeit interest in the current pot
*Abov ...
E6 → F4 of
Coxeter-Dynkin diagrams, E6 corresponding to 1
22 in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the
Coxeter plane
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which hav ...
projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1
22.
Tessellations
This polytope is the
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
for a
uniform tessellation of 6-dimensional space,
222, .
Rectified 122 polytope
The rectified 1
22 polytope (also called 0
221) can tessellate 6-dimensional space as the
Voronoi cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...
of the
E6* honeycomb lattice (dual of E6 lattice).
Alternate names
* Birectified 2
21 polytope
* Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers)
[Klitzing, (o3o3x3o3o *c3o ]
ram
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
Construction
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 short branch leaves the
birectified 5-simplex
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ''rec ...
, .
Removing the ring on either of 2-length branches leaves the
birectified 5-orthoplex in its alternated form: t
2(2
11), .
The
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is determined by removing the ringed node and ringing the neighboring ring. This makes
3-3 duoprism prism, ××, .
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 ref ...
orders.
Truncated 122 polytope
Alternate names
* Truncated 1
22 polytope (Acronym: tim)
Construction
Its construction is based on the
E6 group and information can be extracted from the ringed
Coxeter-Dynkin diagram representing this polytope: .
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
Birectified 122 polytope
Alternate names
* Bicantellated 2
21
* Birectified pentacontatetrapeton (barm) (Jonathan Bowers)
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
Trirectified 122 polytope
Alternate names
* Tricantellated 2
21
* Trirectified pentacontatetrapeton (Acronym: trim, old: cacam, tram, mak) (Jonathan Bowers)
[Klitzing, (x3o3o3o3x *c3o ]
trim
See also
*
List of E6 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
wiley.com
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 1
22)
* o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3o3x3o3o *c3x - tim, o3x3o3x3o *c3o - barm, x3o3o3o3x *c3o - trim
{{Polytopes
6-polytopes