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Discrete group type in group theory
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space
Reflection_group
Concept in mathematics
a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial
Complex_reflection_group
Group that admits a formal description in terms of reflections
Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However
Coxeter_group
Geometric symmetry operation
point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry). A point group including a point reflection among
Point_reflection
Group of symmetries of a regular polygon
mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest
Dihedral_group
Isometry of the Eluclidean plane
translations, rotations, reflections, and glide reflections (see below § Classification). The set of Euclidean plane isometries forms a group under composition:
Euclidean_plane_isometry
Group of geometric symmetries with at least one fixed point
or it is a reflection or improper rotation (determinant of M = −1). The geometric symmetries of crystals are described by space groups, which allow
Point_group
Mathematical group
the mathematical theory of reflection groups, the parabolic subgroups are a special kind of subgroup. In the symmetric group of permutations of the set
Parabolic subgroup of a reflection group
Parabolic_subgroup_of_a_reflection_group
Mapping from a Euclidean space to itself
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of
Reflection_(mathematics)
Subgroup of a root system's isometry group
finite 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
Pictorial representation of symmetry
symmetry group of a regular complex polygon is not called a Coxeter group, but instead a Shephard group, a type of Complex reflection group. The order
Coxeter–Dynkin_diagram
Group of symmetries of an n-dimensional hypercube
two-element group generated by the point reflection through the origin. If one divides the cube into chambers by the planes fixed by each of its reflection symmetries
Hyperoctahedral_group
Invariance under a mathematical reflection
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure
Reflection_symmetry
Geometric transformation combining reflection and translation
space groups (which describe e.g. crystal symmetries). Objects with glide-reflection symmetry are in general not symmetrical under reflection alone,
Glide_reflection
Political party in São Tomé and Príncipe
The Democratic Convergence Party-Reflection Group (Portuguese: Partido de Convergência Democrática-Grupo de Reflexão), or PCD-GR, is a political party
Democratic Convergence Party (São Tomé and Príncipe)
Democratic_Convergence_Party_(São_Tomé_and_Príncipe)
Number line and triangular tiling's symmetry mathematical structure
symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their
Affine_symmetric_group
Group of symmetries of the square
other two, the dihedral group of order 8 and the quaternion group, are not. The action of a rotation or diagonal reflection on the corners of a square
Dihedral_group_of_order_8
Topics referred to by the same term
Reflection map may refer to: Reflection mapping in computer graphics A reflection (mathematics), specifically an element of a reflection group an element
Reflection_map
named for Alfred Young. When S n {\displaystyle S_{n}} is viewed as a reflection group, its Young subgroups are precisely its parabolic subgroups. They may
Young_subgroup
Set with associative invertible operation
two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed
Group_(mathematics)
In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by Mitchell (1914). It has the structure
Mitchell's_group
fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular
Vinberg's_algorithm
Classification of a two-dimensional repetitive pattern
wallpaper group of A and B is different from the wallpaper group of C. Another transformation is a glide reflection, a combination of reflection and translation
Wallpaper_group
Type of group in abstract algebra
the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an
Symmetric_group
The reflection group is a hyperbolic reflection group acting on 25-dimensional hyperbolic space. The fundamental domain of the reflection group has 1+23+284
II25,1
Nonabelian group in algebraic group theory
by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3
Binary_tetrahedral_group
Concept in algebraic topology
difference that the Weyl group, rather than being a finite reflection group over the integers, is now a finite p-adic reflection group. They admit a classification
P-compact_group
Pictorial representation of symmetry
algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin
Dynkin_diagram
Non-commutative group with 6 elements
If x is a reflection point (0, 5, 10, 15, 20, or 25), its stabilizer is the group of order two containing the identity and the reflection in x. In other
Dihedral_group_of_order_6
Group realized geometrically by reflections across the sides of a triangle
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle
Triangle_group
Symmetry group in 1D systems
group in 1D is a simple reflection. It can be represented by the simplest Coxeter group, A1, [ ], or Coxeter-Dynkin diagram . Affine symmetry groups represent
One-dimensional symmetry group
One-dimensional_symmetry_group
Vertex algebra acted on by the monster group
torus induced by the Leech lattice and orbifolded by the two-element reflection group. Borcherds, Richard (1986), "Vertex algebras, Kac-Moody algebras, and
Monster_vertex_algebra
Undergraduate mathematics textbook
Mirrors and Reflections: The Geometry of Finite Reflection Groups is an undergraduate-level textbook on the geometry of reflection groups. It was written
Mirrors_and_Reflections
group is a complex reflection group, 3[3]3[3]3 or of order 648, and the product of this with a group of order 2 is another complex reflection group,
Hessian_group
Type of mathematical group
dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along an axis. Every dihedral group is generated
Infinite_dihedral_group
2015 studio album by Fifth Harmony
Reflection is the debut studio album by American girl group Fifth Harmony. It was released on January 30, 2015, by Syco Music and Epic Records. Lyrically
Reflection (Fifth Harmony album)
Reflection_(Fifth_Harmony_album)
Australian mathematician
Robert Howlett), and the determination of the action of a complex reflection group on the cohomology of the complement of its reflecting hyperplanes.
Gustav_Lehrer
Topics referred to by the same term
Look up reflections or reflexions in Wiktionary, the free dictionary. Reflections may refer to: Reflections; or Sentences and Moral Maxims, a series of
Reflections
Digital infrastructure promoting data sharing
within collections, archives, and databases. The e-Infrastructure Reflection Group (e-IRG) has proposed a similar vision. In particular, it envisions
Data_infrastructure
Mathematical group
hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which
(2,3,7)_triangle_group
Solstice-class cruise ship
Celebrity Reflection is the fifth, final, and largest Solstice-class cruise ship, her sister ships being Celebrity Equinox, Celebrity Eclipse, Celebrity
Celebrity_Reflection
European intergovernmental advisory group
The E-Infrastructure Reflection Group (e-IRG) is a European inter-governmental advisory group dealing with policies on electronic infrastructures (e-Infrastructures)
E-Infrastructure Reflection Group
E-Infrastructure_Reflection_Group
Isometry group of Euclidean space
inhomogeneous special orthogonal group. The Euclidean group E(n) comprises all translations, rotations, and reflections of E n {\displaystyle \mathbb {E} ^{n}} ;
Euclidean_group
Usage of reflective programming in C++
Reflection in C++ refers to the usage of reflective programming, introduced to the C++ programming language in C++26. Reflection in C++, unlike other languages
Reflection_(C++)
Root system associated to a symmetric space
associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space
Restricted_root_system
6th President of Latvia
pro-European, as such, in December 2007 she was named vice-chair of the Reflection group on the long-term future of the European Union. She is also known for
Vaira_Vīķe-Freiberga
Type of symmetry group
and reflection in the horizontal axis (isomorphic to C2, the cyclic group of order 2). the groups each consisting of the identity and reflection in a
Frieze_group
Geometry concept
subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations
Point groups in two dimensions
Point_groups_in_two_dimensions
plane symmetry groups: Reflection groups. These are plane symmetry groups that are generated by reflections, possibly limited to reflections in lines through
Plane_symmetry
Spherical triangle that can be used to tile a sphere
angle, the orientation-preserving subgroup of the reflection group of the triangle is a Hecke group. For an ideal triangle in which all angles are zero
Schwarz_triangle
Rotation composed with a reflection
geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space
Improper_rotation
hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which
Hurwitz_surface
Mathematical abelian group
as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation)
Klein_four-group
year. The result was a victory for the Democratic Convergence Party-Reflection Group, which won 33 of the 55 seats, defeating the former sole legal party
1991 São Toméan parliamentary election
1991_São_Toméan_parliamentary_election
2012 single by Marilyn Manson
"No Reflection" is a song by American rock band Marilyn Manson. The track serves as the lead single from the band's eighth studio album, Born Villain.
No_Reflection
Geometric polyhedral group
symmetries double to 24, 48, 120 respectively for the full reflectional groups. The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral
Polyhedral_group
Mathematical function
the ring of invariants of the reflection group. Opdam (1989) gave a uniform proof for all crystallographic reflection groups. Several years later he proved
Selberg_integral
Concept in geometry
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
Coxeter_element
Optical instrument to view patterns due to repeated reflection
displaying short descriptions of redirect targets Reflection group – Discrete group type in group theory Teleidoscope – Optical toy Uniform tilings in
Kaleidoscope
Topics referred to by the same term
subgroup may refer to: a parabolic subgroup of a reflection group a subgroup of an algebraic group that contains a Borel subgroup This disambiguation
Parabolic_subgroup
Manifold of dimension 3 equipped with a hyperbolic metric
/m,m\in \mathbb {N} } ). Such a polytope gives rise to a Kleinian reflection group, which is a discrete subgroup of isometries of hyperbolic space. Taking
Hyperbolic_3-manifold
São Toméan politician
until 2007. Tebus is a member of the Democratic Convergence Party-Reflection Group (PCD-GR). Seibert, Gerhard (1 May 2006). Comrades, Clients and Cousins:
Maria_Tebús
American mathematician (born 1941)
Charles F. (1989), "Differential-difference operators associated to reflection groups", Trans. Amer. Math. Soc., 311 (1): 167–183, doi:10.2307/2001022,
Charles_F._Dunkl
Polish academician and diplomat (born 1938)
foreign minister. While in that position, Rotfeld established the Warsaw Reflection Group on UN Reform and the Transformation of the Euro-Atlantic Security Institutions
Adam_Daniel_Rotfeld
São Toméan politician
3 April 1991. D'Alva co-founded the Democratic Convergence Party – Reflection Group (PCD–RG) in 1991 and subsequently led it for many years. "Leonel Mário
Leonel_Mário_d'Alva
Prime Minister of São Tomé and Príncipe from 1992 to 1994
July 1994. Alegre is a member of the Democratic Convergence Party-Reflection Group (PCD-GR) and is married to former Foreign Minister Alda Bandeira. In
Norberto_Costa_Alegre
Mathematical theorem
a complex reflection group". Shephard and Todd derived a full classification of such groups. Let V be one-dimensional. Then any finite group faithfully
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Day of Reflection group considers how best the conflict can be collectively remembered and reflected upon. The first Day of Private Reflection took place
Healing_Through_Remembering
Real square matrix whose columns and rows are orthogonal unit vectors
bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice
Orthogonal_matrix
Heisenberg group, discrete Heisenberg group Molecular symmetry Nielsen transformation Reflection group Tarski monster group Thompson groups Tietze transformation
List_of_group_theory_topics
24-dimensional repeating pattern of points
two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster
Leech_lattice
American mathematician (born 1949)
Davis–Moussong complex, Davis manifolds, Davis–Januszkiewicz space, and the reflection group trick. Davis attended Princeton University where he earned a bachelor's
Michael_W._Davis
construct analogues of Lie groups over finite fields, called Chevalley groups. complex reflection group complex reflection group coroot coroot Coxeter 1
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
1974 studio album by George Duke
Faces in Reflection is the third studio album by American keyboardist George Duke issued in 1974 on MPS Records. The album reached No. 31 on the Billboard
Faces_in_Reflection
Irish diplomat (born 1954)
School of Civic Education in London. He is a member of the Peacemaking Reflection Group of former senior UN officials under the auspices of the Foundation
Francis_Martin_O'Donnell
under the finite reflection group of a root system. If the reflection group W corresponds to the Weyl group of a compact semisimple group K with maximal
Kostant_polynomial
Groups of point isometries in 3 dimensions
its full symmetry group if and only if the object is chiral. The point groups that are generated purely by a finite set of reflection mirror planes passing
Point groups in three dimensions
Point_groups_in_three_dimensions
rotation axis Cn. The C1 group is covered in the nonaxial groups section. The reflection groups are denoted by Cnh. These groups are characterized by i)
List of character tables for chemically important 3D point groups
List_of_character_tables_for_chemically_important_3D_point_groups
Dialogue among traditionalist Catholics in France
The Group for Reflection among Catholics (French: Groupe de réflexion entre catholiques, GREC) is an informal organization for "meetings and discussions
Group for Reflection among Catholics
Group_for_Reflection_among_Catholics
German mathematician
question of whether every finite complex reflection group is a Weyl group of an object analogous to a finite group of Lie type. They baptized the unknown
Gunter_Malle
Capacity of humans to exercise introspection
Self-reflection is the ability to witness and evaluate one's own cognitive, emotional, and behavioural processes. In psychology, other terms used for this
Self-reflection
Italian professor of economics
Money and Finance” (2003). In 2002 he was part of the Reflection Group on the “Reflection Group on the Spiritual and Cultural Dimension of Europe Archived
Alberto_Quadrio_Curzio
Tiling by polygons whose reflections across edges are other tiles
non-overlapping polygons (a tessellation) with the property that the reflection of any of these polygons across any of its edges is another polygon in
Edge_tessellation
2017 studio album by Silverstein
Dead Reflection is the ninth studio album by the Canadian post-hardcore band Silverstein, released on July 14, 2017 through Rise Records worldwide and
Dead_Reflection
lines through each point. Its complex reflection group is 3[3]3[3]3 or , order 648, also called a Hessian group. It has 27 copies of , order 24, at each
Hessian_polyhedron
2015–16 concert tour by Fifth Harmony
The Reflection Tour was the fourth concert tour by American girl group Fifth Harmony. Visiting Europe, North America and Asia (one date in Adu Dhabi),
The_Reflection_Tour
2013 studio album (Split) by Cough and Windhand
Reflection of the Negative is a split album by American doom metal bands Cough and Windhand. It was released on April 16, 2013 via Relapse Records. In
Reflection_of_the_Negative
Classification system for symmetry groups in geometry
is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing
Coxeter_notation
finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was
Rank_3_permutation_group
Equality of areas of a sliced disk
Yury (2021), "Reflection groups and the pizza theorem", Algebra i Analiz (in Russian), 33 (6): 1–8 Brailov, Yury (2022), "Reflection groups and the pizza
Pizza_theorem
Area of mathematics
Stembridge, John (1989-12-01). "On the eigenvalues of representations of reflection groups and wreath products". Pacific Journal of Mathematics. 140 (2). Mathematical
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Sporadic simple group
In the area of modern algebra known as group theory, the Conway group Co2 is a sporadic simple group of order 42,305,421,312,000 = 218 · 36 · 53 ·
Conway_group_Co2
Group of transformations under which the object is invariant
detail. The isometry groups in one dimension are: the trivial cyclic group C1 the groups of two elements generated by a reflection; they are isomorphic
Symmetry_group
Physical effect used in quantum mechanics
frustrated total internal reflection (FTIR) and is an optical analog of quantum tunneling. Balcou and Dutriaux obtained the group delay from a measurement
Hartman_effect
International cooperative community
this occasion, the Ministerial Council also established a High Level Reflection Group, which was mandated to assess the adequacy of the institutional set
Energy_Community
President of the ICC from 2021 to 2024
2001–2002 he was working as an expert at the Council of Europe in the Reflection Group on developments in international cooperation in criminal matters and
Piotr_Hofmański
Initiation room in Freemasonry
Freemasonry, the Chamber of Reflection, often abbreviated as C.O.R., and alternatively known as the Room of Reflection, Reflection Cabinet, or Meditation Cabinet
Chamber_of_Reflection
1998 song from Disney's Mulan
"Reflection" is a song written and produced by Matthew Wilder and David Zippel for the soundtrack of Disney's 1998 animated film Mulan. In the film, the
Reflection_(song)
Coxeter group is a subalgebra of the integral group ring of the Coxeter group, introduced by Solomon (1976). In the special case of the symmetric group Sn
Descent_algebra
Subgroup of the Clifford algebra associated to a quadratic space
preimage of a reflection squares to ±1 ∈ Ker (Spin(V) → SO(V)), and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in
Pin_group
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