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Simplicial complex
the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are
Coxeter_complex
Pictorial representation of symmetry
a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group
Coxeter–Dynkin_diagram
Mathematical structure
a building Δ is a Coxeter group W, which determines a highly symmetrical simplicial complex Σ = Σ(W, S), called the Coxeter complex. A building Δ is glued
Building_(mathematics)
Canadian geometer (1907–2003)
the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Coxeter was
H._S._M._Coxeter
Group that admits a formal description in terms of reflections
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
Coxeter_group
Concept in mathematics
real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra). A (complex) reflection r (sometimes also
Complex_reflection_group
Generalization of a polytope in real space
symbolic notation developed by Coxeter. Some complex polytopes which are not fully regular have also been described. The complex line C 1 {\displaystyle \mathbb
Complex_polytope
Four-dimensional analogues of the regular polyhedra in three dimensions
Coxeter 1973, § 1.8 Configurations Coxeter, Complex Regular Polytopes, p. 117 Conway, Burgiel & Goodman-Strauss 2008, p. 406, Fig 26.2 Coxeter, Star
Regular_4-polytope
Number of orderings allowing ties
number of faces in the Coxeter complex associated with a Coxeter group of type A n − 1 {\displaystyle A_{n-1}} . Here, a Coxeter group can be thought of
Ordered_Bell_number
Mathematical group
complex reflection group, so the complex reflection groups form another generalization of finite real reflection groups. Suppose that W is a Coxeter group
Parabolic subgroup of a reflection group
Parabolic_subgroup_of_a_reflection_group
Four-dimensional analogue of the cube
1007/s11075-022-01278-y. Coxeter 1973, p. 12, §1.8 Configurations. Coxeter 1973, p. 293. Coxeter, H. S. M., Regular Complex Polytopes, second edition
Tesseract
polyhedron is a regular complex polyhedron 3{3}3{3}3, , in ℂ3. It has 27 vertices, 72 3{} edges, and 27 3{3}3 faces. It is self-dual. Coxeter named it after Ludwig
Hessian_polyhedron
the Coxeter complexes that make them up are colorable. A coloring of a building is associated with a uniform choice of Weyl group for the Coxeter complexes
Weyl_distance_function
Classification system for symmetry groups in geometry
Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter
Coxeter_notation
Polygons which have an accompanying imaginary dimension for each real dimension
regular complex polygons have been completely characterized, and can be described using a symbolic notation developed by Coxeter. A regular complex polygon
Regular_complex_polygon
Graph operation
The Goldberg–Coxeter construction or Goldberg–Coxeter operation (GC construction or GC operation) is a graph operation defined on regular polyhedral graphs
Goldberg–Coxeter_construction
Natural number
hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams
5
Mathematical concept
previous property. Every Coxeter complex, and more generally every building (in the sense of Tits), is shellable. The boundary complex of a (convex) polytope
Shelling_(topology)
dimensional, locally finite, ranked simplicial complex to capture isomorphisms between finite rank Coxeter systems) and asked more related open questions
Isomorphism problem of Coxeter groups
Isomorphism_problem_of_Coxeter_groups
Every vertex is shared by 3 triangular edges. Coxeter named it a Möbius–Kantor polygon for sharing the complex configuration structure as the Möbius–Kantor
Möbius–Kantor_polygon
5-dimensional hypercube
polytopes in n dimensions (n ≥ 5), ISBN 0-486-61480-8 Coxeter, H.S.M. (1991) [1974]. Regular Complex Polytopes. Cambridge University Press. ISBN 0-521-39490-2
5-cube
Subgroup of a root system's isometry group
reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple
Weyl_group
Linear stacking of regular tetrahedra that form helices
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and Arie Hendrick Boerdijk [es], is a linear stacking of regular tetrahedra, arranged so that
Boerdijk–Coxeter_helix
Polytope in 8-dimensional geometry
He called it an 8-ic semi-regular figure. Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end
4_21_polytope
The Coxeter–Todd lattice can be made into a 6-dimensional lattice self dual over the Eisenstein integers. The automorphism group of this complex lattice
Coxeter–Todd_lattice
Coxeter gives it group symbol [1 2 3]3 and Coxeter-Dynkin diagram . Mitchell's group is an index 2 subgroup of the automorphism group of the Coxeter–Todd
Mitchell's_group
Uniform 6-polytope
polytopes, named as V72 (for its 72 vertices). Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end
1_22_polytope
Pictorial representation of symmetry
explicitly so named. Undirected Dynkin diagrams belong to the related family of Coxeter–Dynkin diagrams The fundamental interest in Dynkin diagrams is that they
Dynkin_diagram
Study of complex manifolds and several complex variables
complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry
Complex_geometry
Deformation of the group algebra of a Coxeter group
of the group algebra of a Coxeter group. The Hecke algebra can also be viewed as a q-analog of the group algebra of a Coxeter group. Hecke algebras are
Iwahori–Hecke_algebra
In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.[citation needed] Other configurations in geometry are something
Configuration_(polytope)
Polygon in complex space, or which self-intersects
meshes:Surface (polygonal) Simplification 1997. (retrieved May 2016) Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, 1974. Introduction
Complex_polygon
Four-dimensional analogue of the tetrahedron
pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's α4 polytope), the simplest possible convex 4-polytope, and is analogous
5-cell
Convex polytope, the n-dimensional analogue of a square and a cube
(originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special
Hypercube
Geometric space with four dimensions
posthumously in 1901, and remained largely unknown until publication of H.S.M. Coxeter's Regular Polytopes in 1947. During that interval many others also discovered
Four-dimensional_space
Number line and triangular tiling's symmetry mathematical structure
and certain complex reflection groups. Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups. The
Affine_symmetric_group
Type of geometry
1997, p. 88. Coxeter 2003, p. v. Coxeter 1969, p. 229. Coxeter 2003, p. 14. Coxeter 1969, pp. 93, 261. Coxeter 1969, pp. 234–238. Coxeter 2003, pp. 111–132
Projective_geometry
Geometric object with flat sides
generalised the idea as complex polytopes in complex space, where each real dimension has an imaginary one associated with it. Coxeter developed the theory
Polytope
Polytope with highest degree of symmetry
Coxeter (1973), p. 143. Walter & Deloudi (2009), p. 50. Walter & Deloudi (2009), p. 51. Barnes (2012), p. 46. Coxeter (1973), pp. 120–121. Coxeter (1973)
Regular_polytope
Polyhedral compound
The 30 rhombic faces exist in the planes of the 5 cubes. Coxeter 1973, pp. 49-50. Coxeter 1973, p 98. Cromwell (1997), pp. 360–361. Cromwell, Peter R
Compound_of_five_cubes
Overview of and topical guide to geometry
Sphericon Stereographic projection Stereometry Ball Convex Convex hull Coxeter group Euclidean distance Homothetic center Hyperplane Lattice Ehrhart polynomial
Outline_of_geometry
In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3{3}3{3}3{3}3, and Coxeter diagram . It has 240 vertices
Witting_polytope
Degenerate uniform star polyhedron
hidden, being completely contained inside the first. Small complex icosidodecahedron Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J
Great complex icosidodecahedron
Great_complex_icosidodecahedron
Polygonal chain whose vertices are not all coplanar
(Saddle Polygons)" §2.2 Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover. Coxeter, H.S.M.; Regular complex polytopes (1974). Chapter
Skew_polygon
Euclidean geometry without distance and angles
118 (exercise 3). Coxeter 1955, The Affine Plane, § 2: Affine geometry as an independent system Coxeter 1955, Affine plane, p. 8 Coxeter, Introduction to
Affine_geometry
Group of symmetries of an n-dimensional hypercube
of hyperoctahedral groups forms type B in the classification of finite Coxeter groups. The hyperoctahedral groups were named by Alfred Young in 1930.
Hyperoctahedral_group
Plane figure bounded by line segments
the center of the image, Coxeter, H.S.M.; Regular Polytopes, 3rd Edn, Dover (pbk), 1973, p. 114 Shephard, G.C.; "Regular complex polytopes", Proc. London
Polygon
Uniform 6-polytope
It is also called the Schläfli polytope. Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end
2_21_polytope
vertices, similarly to how its 3D equivalent shares edges. Great complex icosidodecahedron Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J
Small complex icosidodecahedron
Small_complex_icosidodecahedron
Regular tiling of a two-dimensional space
book, Keith Critchlow, pp. 74–75, pattern 2 Coxeter, Regular Complex Polytopes, pp. 111–112, p. 136. Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973)
Hexagonal_tiling
Convex regular 5-polytope in geometry
117. H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973 Coxeter, H.S.M. (1991) [1974]. Regular Complex Polytopes (2nd ed
5-orthoplex
Geometric space with eight dimensions
of reflection, each domain defined by a Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 8-demicube is a unique
Eight-dimensional_space
Dutch graphic artist (1898–1972)
interacted with the mathematicians George Pólya, Roger Penrose, and Donald Coxeter, and the crystallographer Friedrich Haag, and conducted his own research
M._C._Escher
Type of uniform tessellation
8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram. By putting spheres at its vertices
5_21_honeycomb
Regular 5-polytope
Polytopes, three regular polytopes in n dimensions (n ≥ 5) Coxeter, H.S.M. (1991) [1974]. Regular Complex Polytopes (2nd ed.). Cambridge University Press. ISBN 0-521-39490-2
5-demicube
Discrete group type in group theory
group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the
Reflection_group
Honeycombs, p. 224. ISBN 978-1-107-10340-5. Coxeter (1973), p. 120. Coxeter (1973), p. 124. Coxeter, Regular Complex Polytopes, p. 9 Duncan, Hugh (28 September
List_of_regular_polytopes
Relation between sides of a right triangle
Fritz wrote a careful discussion of Hippasus's contributions. For any complex number z=x+iy, the absolute value or modulus is given by r=|z|=x2+y2. So
Pythagorean_theorem
Four-dimensional analog of the octahedron
Macmillan, 1900 H.S.M. Coxeter: Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover. Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed
16-cell
American mathematician (born 1949)
Coxeter groups, Artin groups, and buildings. In his book, The Geometry and Topology of Coxeter Groups, he constructs the Davis complexes for Coxeter groups
Michael_W._Davis
Undergraduate mathematics textbook
dihedral groups, and root systems. Part III of the book concerns Coxeter complexes, and uses them as the basis for some group theory of reflection groups
Mirrors_and_Reflections
German mathematician (1826–1866)
contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis
Bernhard_Riemann
Duocylinder Tesseract Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974). Regular Polytopes, H. S. M. Coxeter, Dover Publications
3-4_duoprism
Skew polygon derived from a polytope
question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, h, is the Coxeter number of the Coxeter group. These polygons
Petrie_polygon
Integral polynomial
George Lusztig (1979). They are indexed by pairs of elements y, w of a Coxeter group W, which can in particular be the Weyl group of a Lie group. In the
Kazhdan–Lusztig_polynomial
Method of drawing geometric objects
us to consider the points as a set of complex numbers. Given any such interpretation of a set of points as complex numbers, the points constructible using
Straightedge and compass construction
Straightedge_and_compass_construction
Matrix equal to its conjugate-transpose
mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose. That is, if
Hermitian_matrix
Regular object in four dimensional geometry
(3rd ed.). New York: Dover. Coxeter, H.S.M. (1991), Regular Complex Polytopes (2nd ed.), Cambridge: Cambridge University Press Coxeter, H.S.M. (1995), Sherk
24-cell
Groups of point isometries in 3 dimensions
"7 The Binary Polyhedral Groups", Regular Complex Polytopes, Cambridge University Press, pp. 73–82. Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators
Point groups in three dimensions
Point_groups_in_three_dimensions
British geometer
in 1936. In 1953 he and Coxeter discovered the Coxeter–Todd lattice. In 1954 he and G. C. Shephard classified the finite complex reflection groups. In March
J._A._Todd
Partial order on a Coxeter group
order, or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties. The
Bruhat_order
Geometric model of the physical space
Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana
Three-dimensional_space
Tiling of a plane by regular hexagons and equilateral triangles
CiteSeerX}}: Cite uses deprecated parameter |citeseerx= (help) Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. pp. 111–2
Trihexagonal_tiling
Family of infinite discrete groups
groups defined by simple presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and
Artin–Tits_group
Four-dimensional analog of the icosahedron
2: 55–65, 110–120 Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover. Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed
600-cell
Representation of cubic graphs
code is a notation devised by Joshua Lederberg, and extended by H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain
LCF_notation
Mathematical group
{\displaystyle Bi=M\wr \mathbb {Z} _{2}.\,} The Bimonster is also a quotient of the Coxeter group corresponding to the Dynkin diagram Y555, a Y-shaped graph with 16
Bimonster_group
Arts complex in Mito, Ibaraki, Japan
acoustical design by Nagata Acoustics. The design is based on the Boerdijk–Coxeter helix. Wikimedia Commons has media related to Art Tower Mito. Kairakuen
Art_Tower_Mito
Regular polytope dual to the hypercube in any number of dimensions
(3): 196–200, doi:10.2307/2975549, JSTOR 2975549. Coxeter, Regular Complex Polytopes, p. 108 Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York:
Cross-polytope
Space with one dimension
one-dimensional space. In particular, if the field is the complex numbers ℂ, then the complex projective line 𝐏1(ℂ) is one-dimensional with respect to
One-dimensional_space
Polytope associated with combinatorial problems
doi:10.4153/CMB-2002-054-1. Carr, Michael; Devadoss, Satyan (2006). "Coxeter complexes and graph-associahedra". Topology and Its Applications. 153 (12):
Cyclohedron
Developer of the Petrie polygon
navigator, Matthew Flinders. He met the geometer Harold Scott MacDonald Coxeter as a student, beginning a lifelong friendship. They collaborated in discovering
John_Flinders_Petrie
Method of describing higher-order polyhedra
of 1. The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the Goldberg-Coxeter construction. The
Conway_polyhedron_notation
Solid with four equal triangular faces
3-demicube, a polyhedron that is by alternating a cube. This form has Coxeter diagram and Schläfli symbol h{4,3}. Let a be the edge length of a regular
Regular_tetrahedron
Algebraic structure in linear algebra
2000, Example 5.13.5, p. 436. Meyer 2000, Exercise 5.13.15–17, p. 442. Coxeter 1987. Anton, Howard; Rorres, Chris (2010), Elementary Linear Algebra: Applications
Vector_space
Spherical polyhedron composed of lunes
in English. MAA. pp. 108–109. ISBN 978-0-88385-511-9. Coxeter, H.S.M. (1974). Regular Complex Polytopes. London: Cambridge University Press. p. 20. ISBN 0-521-20125-X
Hosohedron
Non-Euclidean geometry
proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter: The name "elliptic" is possibly misleading. It does not imply any direct
Elliptic_geometry
Richard. "o3x3o3o3o - rix". Coxeter (1973). Regular Polytopes. Sec 1.8 Configurations. Coxeter, H.S.M. (1975). Regular Complex Polytopes. Cambridge University
Rectified_5-simplexes
Branch of mathematics
mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. As a study of systems of polynomial
Algebraic_geometry
Mathematical space with two coordinates
two-dimensional complex space – such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface – has two complex dimensions
Two-dimensional_space
Property of a mathematical space
sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. A complex number (x+iy) has a real part
Dimension
Matrix equal to its transpose
antimetric) Centrosymmetric matrix Circulant matrix Covariance matrix Coxeter matrix GCD matrix Hankel matrix Hilbert matrix Persymmetric matrix Sylvester's
Symmetric_matrix
Equation for radii of tangent circles
Beecroft and by H. S. M. Coxeter involve four more circles, passing through triples of tangencies of the original three circles; Coxeter also provided a proof
Descartes's_theorem
Geometric space with seven dimensions
of reflection, each domain defined by a Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 7-demicube is a unique
Seven-dimensional_space
Straight figure with zero width and depth
and Winston, p. 114, ISBN 978-0-03-073100-6, LCCN 69-12075, OCLC 47870 Coxeter, H.S.M (1969), Introduction to Geometry (2nd ed.), New York: John Wiley
Line_(geometry)
Straight line segment that passes through the centre of a circle
Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana
Diameter
Branch of differential geometry and differential topology
is a neo-Greek calque of "complex". Previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin com-plexus
Symplectic_geometry
Branch of geometry that studies combinatorial properties and constructive methods
map colourings by Tait, Heawood, and Hadwiger. László Fejes Tóth, H.S.M. Coxeter, and Paul Erdős laid the foundations of discrete geometry. A polytope is
Discrete_geometry
Geometry without using coordinates
), Metaphysics Research Lab, Stanford University, retrieved 2026-10-26 Coxeter, H.S.M.; Greitzer, Samuel L., eds. (1967), "An Introduction to Projective
Synthetic_geometry
Regular tiling of the plane
from list of 107 isohedral tilings, p.473-481 Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136. Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973)
Triangular_tiling
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