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 ...

, a uniform ''k''₂₁ polytope is a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

in ''k'' + 4 dimensions 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 ref ...

, and having only regular polytope facets. 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 ...

''k''₂₁ by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the ''k''-node sequence.

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...

discovered this family as a part of his 1900 enumeration of the

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

and

semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...

s, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the ''5-ic semiregular figure''.

Family members

The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 6₂₁. It is a tessellation of hyperbolic 9-space constructed of ∞ 9-

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 ...

and ∞ 9-

orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...

facets with all vertices at infinity.) The family starts uniquely as 6-polytopes. The ''triangular prism'' and ''rectified 5-cell'' are included at the beginning for completeness. The ''demipenteract'' also exists in the demihypercube family. They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the ''E''₆ symmetry. The complete family of Gosset semiregular polytopes are: #

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. ...

: −1₂₁ (2

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- colli ...

s and 3

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

faces) # rectified 5-cell: 0₂₁, ''Tetroctahedric'' (5

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 ...

and 5

octahedra In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ...

cells) # demipenteract: 1₂₁, ''5-ic semiregular figure'' (16

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 is ...

and 10 16-cell facets) # 2 21 polytope: 2₂₁, ''6-ic semiregular figure'' (72 5-

and 27 5-

facets) # 3 21 polytope: 3₂₁, ''7-ic semiregular figure'' (576 6-

and 126 6-

facets) # 4 21 polytope: 4₂₁, ''8-ic semiregular figure'' (17280 7-

and 2160 7-

facets) # 5 21 honeycomb: 5₂₁, ''9-ic semiregular check'' tessellates Euclidean 8-space (∞ 8-

and ∞ 8-

facets) # 6 21 honeycomb: 6₂₁, tessellates hyperbolic 9-space (∞ 9-

and ∞ 9-

facets) Each polytope is constructed from (''n'' − 1)-

and (''n'' − 1)-

facets. The orthoplex faces are constructed from the

''D''_''n''−1 and have a

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

of rather than the regular . This construction is an implication of two "facet types". Half the facets around each orthoplex

ridge A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

are attached to another orthoplex, and the others are attached to a simplex. In contrast, every simplex ridge is attached to an orthoplex. 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

as the previous form. For example, the ''rectified 5-cell'' has a vertex figure as a ''triangular prism''.

Elements

References

* T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 * Alicia Boole Stott ''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: 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 * G.Blind and R.Blind, "The semi-regular polyhedra", Commentari Mathematici Helvetici 66 (1991) 150–154 * John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 26. pp. 411–413: The Gosset Series: n₂₁)

External links

PolyGloss v0.05: Gosset figures (Gossetoicosatope)

Regular, SemiRegular, Regular faced and Archimedean polytopes
{{Honeycombs Polytopes Multi-dimensional geometry

Family members

Elements

See also

References

External links