In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, 1_k2 polytope is a uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

 in n-dimensions (n = k+4) constructed from the E_n

En (Lie algebra)

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k=n-4....

 Coxeter group

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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

. The family was named by Coxeter as 1_k2 by its bifurcating Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, with a single ring on the end of the 1-node sequence. It can be named by an expoential Schläfli symbol {3^1,k,2}.

Family members

The family starts uniquely as 6-polytope

6-polytope

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....

s, but can be extended backwards to include the 5-demicube (demipenteract

Demipenteract

In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...

) in 5-dimensions, and the 4-simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

 (5-cell) in 4-dimensions.

Each polytope is constructed from 1_k-1,2 and (n-1)-demicube facets. Each has a vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 of a {3^1,n-2,2} polytope is a birectified n-simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

, t₂{3ⁿ}.

The sequence ends with k=6 (n=10), as an infinite tessellation of 9 dimensional hyperbolic space.

The complete family of 1_k2 polytope polytopes are:

1. 5-cell: 1₀₂, (5 tetrahedral
   Tetrahedron
   In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
   
    cells)
2. 1₁₂ polytope
   Demipenteract
   In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...
   
   , (16 5-cell, and 10 16-cell
   16-cell
   In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....
   
    facets)
3. 1₂₂ polytope, (54 demipenteract
   Demipenteract
   In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...
   
    facets)
4. 1₃₂ polytope, (56 1₂₂ and 126 demihexeract
   Demihexeract
   In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternate vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes....
   
    facets)
5. 1₄₂ polytope, (240 1₃₂ and 2160 demihepteract
   Demihepteract
   In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube with alternated vertices deleted...
   
    facets)
6. 1₅₂ honeycomb, tessellates Euclidean 8-space (∞ 1₄₂ and ∞ demiocteract
   Demiocteract
   In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices deleted...
   
    facets)
7. 1₆₂ honeycomb, tessellates hyperbolic 9-space (∞ 1₅₂ and ∞ demienneract
   Demienneract
   In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes....
   
    facets)

Elements

Gosset 1_k2 figures

n	1k2	Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets... projection	Name Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...	Facets Facet (mathematics) A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...		Elements
				1k-1,2	(n-1)-demicube	Vertices Vertex (geometry) In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...	Edges Edge (geometry) In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....	Faces Face (geometry) In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...	Cells Cell (geometry) In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...	4-faces	5-faces	6-faces	7-faces
4	102		120	--	5 110 Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...	5	10	10	5
5	112		121 Demipenteract In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...	16 120	10 111 16-cell In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....	16	80	160	120	26
6	122		122 Gosset 1 22 polytope In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 ....	27 112 Demipenteract In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...	27 121 Demipenteract In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...	72	720	2160	2160	702	54
7	132		132 Gosset 1 32 polytope In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.Coxeter named it 132 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences....	56 122 Gosset 1 22 polytope In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 ....	126 131 Demihexeract In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternate vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes....	576	10080	40320	50400	23688	4284	182
8	142		142 Gosset 1 42 polytope In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.Coxeter named it 142 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences....	240 132 Gosset 1 32 polytope In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.Coxeter named it 132 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences....	2160 141 Demihepteract In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube with alternated vertices deleted...	17280	483840	2419200	3628800	2298240	725760	106080	2400
9	152		152 Gosset 1 52 honeycomb In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure... (8-space tessellation)	∞ 142 Gosset 1 42 polytope In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.Coxeter named it 142 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences....	∞ 151 Demiocteract In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices deleted...	∞
10	162		162 (9-space hyperbolic tessellation)	∞ 152	∞ 161 Demienneract In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes....	∞

See also

• k₂₁ polytope family
• 2_k1 polytope
  Uniform 2 k1 polytope
  In geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by Coxeter as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence...
  
  family

External links

• PolyGloss v0.05: Gosset figures (Gossetododecatope)