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 c ...

, 1_k2 polytope is a

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 ...

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

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 refl ...

. The family was named by their

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 ...

1_k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol .

Family members

The family starts uniquely as

6-polytope In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. Definition A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A ...

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

demicube In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex ( ...

(

demipenteract In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...

) in 5-dimensions, and the 4-

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

(

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...

) in 4-dimensions. Each polytope is constructed from 1_k-1,2 and (n-1)-

facets. Each has a

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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

of a ' polytope is a birectified n-

, ''t₂''. 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: #

: 1₀₂, (5

tetrahedral 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 ...

cells) # 1₁₂ polytope, (16

, and 10

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...

facets) # 1₂₂ polytope, (54

facets) # 1₃₂ polytope, (56 1₂₂ and 126

demihexeract In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte ...

facets) # 1₄₂ polytope, (240 1₃₂ and 2160

demihepteract 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. El ...

facets) # 1₅₂ honeycomb, tessellates Euclidean 8-space (∞ 1₄₂ and ∞

demiocteract In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte ...

facets) # 1₆₂ honeycomb, tessellates hyperbolic 9-space (∞ 1₅₂ and ∞

demienneract In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it ...

facets)

Elements

References

Alicia Boole Stott Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Groni ...

''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 ** Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910. ** Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910. ** Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam * Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, ''Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam'' (eerstie sectie), vol 11.5, 1913. *

H. S. M. 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 t ...

: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940 * N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985 * H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988

External links

PolyGloss v0.05: Gosset figures (Gossetododecatope)
{{Honeycombs Multi-dimensional geometry Polytopes

Family members

Elements

See also

References

External links