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Generalized manifold
me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word "manifold" already
Orbifold
Classification of a two-dimensional repetitive pattern
wallpaper group; it is called p4m in the IUCr notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2
Wallpaper_group
Notation for 2-dimensional spherical, euclidean and hyperbolic symmetry groups
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing
Orbifold_notation
Topological space
S 1 {\displaystyle S^{1}} -bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact
Seifert_fiber_space
Symmetric subdivision in hyperbolic geometry
Coxeter group [7,3], orbifold (*732) contains these uniform tilings: The (8 3 2) triangle group, Coxeter group [8,3], orbifold (*832) contains these
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
In differential geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic
Euler characteristic of an orbifold
Euler_characteristic_of_an_orbifold
American mathematician (1946–2012)
respectively. In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston
William_Thurston
groupoids as charts. The notion often appears in particular as an inertia orbifold. Let U = ( U 1 ⇉ U 0 ) {\displaystyle U=(U_{1}\rightrightarrows U_{0})}
Inertia_stack
groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups
List of planar symmetry groups
List_of_planar_symmetry_groups
Type of object in algebraic geometry
stack that behaves, in many respects, like an algebraic variety or an orbifold, while still allowing mild stacky phenomena such as finite stabilizer groups
Deligne–Mumford_stack
Symmetry group of a configuration in space
space group + atomic arrangement (motif)). Orbifold notation (2D) Fibrifold notation (3D) Describes the orbifold, given by the quotient of Euclidean space
Space_group
Doughnut-shaped surface of revolution
space of unordered, not necessarily distinct points is accordingly the orbifold Tn / Sn, which is the quotient of the torus by the symmetric group on n
Torus
Object in algebraic geometry
Stacky curves are closely related to 1-dimensional orbifolds and therefore sometimes called orbifold curves or orbicurves. A stacky curve X {\displaystyle
Stacky_curve
2-dimensional integer lattice
symmetry group in IUC notation as p4m, Coxeter notation as [4,4], and orbifold notation as *442. Two orientations of an image of the lattice are by far
Square_lattice
Compact astronomical body
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Black_hole
Computes the Euler characteristic of an orbifold
Kawasaki, is the Riemann–Roch formula for orbifolds. It can compute the Euler characteristic of an orbifold. Kawasaki's original proof made a use of the
Kawasaki's Riemann–Roch formula
Kawasaki's_Riemann–Roch_formula
fundamental domains between two symmetry groups. They are compactly expressed in orbifold notation. These mutations can occur from spherical tilings to Euclidean
Uniform tiling symmetry mutations
Uniform_tiling_symmetry_mutations
resolution conjecture of Ruan (2006) states that the orbifold cohomology of a Gorenstein orbifold is isomorphic to a semiclassical limit of the quantum
Crepant_resolution
Topological invariant in mathematics
manifolds comes from orbifolds (see Euler characteristic of an orbifold). While every manifold has an integer Euler characteristic, an orbifold can have a fractional
Euler_characteristic
Monster and modular connection
which has rank 24. The orbifold construction. In physical terms, this describes a bosonic string propagating on a quotient orbifold. The construction of
Monstrous_moonshine
One of the 7 crystal systems in crystallography
group in Schoenflies notation, Hermann–Mauguin (international) notation, orbifold notation, and Coxeter notation, type descriptors, mineral examples, and
Monoclinic_crystal_system
Geometry formula
equivariantly closed equivariant differential form α {\displaystyle \alpha } on an orbifold M with a torus action and for a sufficient small ξ {\displaystyle \xi }
Localization formula for equivariant cohomology
Localization_formula_for_equivariant_cohomology
Concept in theoretical physics
notion of orbifold, proposed by Augusto Sagnotti in 1987. The novelty is that in the case of string theory the non-trivial element(s) of the orbifold group
Orientifold
Algebraic stack in mathematics
In mathematics, the moduli stack of elliptic curves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M e l l {\displaystyle {\mathcal {M}}_{\mathrm
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Lattice point group
their representations in international notation, Schoenflies notation, orbifold notation, Coxeter notation and mineral examples. There is only one tetragonal
Tetragonal_crystal_system
Part of the mathematical subject of group theory
of groups. Bass–Serre theory has some overlap with orbifold theory, as some one-dimensional orbifolds may be described as graphs of groups. Bass–Serre theory
Bass–Serre_theory
One of the five 2D Bravais lattices
hexagonal lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table
Hexagonal_lattice
Rotation in a discrete subgroup of symmetries of the Euclidean plane
symmetry whose axis passes through the center of rotational symmetry. In the orbifold corresponding to the subgroup, a gyration corresponds to a rotation point
Gyration
This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway used a variation of the Schoenflies notation
List of spherical symmetry groups
List_of_spherical_symmetry_groups
One of the seven crystal systems
point groups, International Tables for Crystallography space group number, orbifold, type, and space groups are listed in the table below. There are a total
Triclinic_crystal_system
Japanese mathematician
often attributed to William Thurston, Satake was the first to introduce orbifold, which he did in the 1950s under the name of V-manifold. In Satake (1956)
Ichirō_Satake
Quadrilateral with only 3 right angles
Lambert quadrilateral fundamental domain in orbifold *p222 *3222 symmetry with 60-degree angle on one of its corners. *4222 symmetry with 45-degree angle
Lambert_quadrilateral
U p {\displaystyle U_{p}} is a smooth orbifold; E p → U p {\displaystyle E_{p}\to U_{p}} is a smooth orbifold vector bundle; S p : U p → E p {\displaystyle
Kuranishi_structure
Topics referred to by the same term
notation (knot theory) Conway polyhedron notation Conway triangle notation Orbifold notation This disambiguation page lists articles associated with the title
Conway_notation
Consequently, a toric variety is a coarse approximation of a toric stack. A toric orbifold is an example of a toric stack. Stanley–Reisner ring Iwanari, Isamu (2009)
Toric_stack
groups Crystal system Polar point groups Schönflies Hermann–Mauguin Orbifold Coxeter Triclinic C1 1 11 [ ]+ Monoclinic C2 Cs 2 m 22 * [2]+ [ ] Orthorhombic
Polar_point_group
Tiling of a plane by regular hexagons and equilateral triangles
the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction
Trihexagonal_tiling
Type of three-dimensional crystal structural geometry
point groups, International Tables for Crystallography space group number, orbifold notation, type, and space groups are listed in the table below. In two
Orthorhombic_crystal_system
Special collection of states in closed string theory
theoretical physics, a twisted sector is a configuration of sectors in orbifold conformal field theories. In the first quantized formalism of string theory
Twisted_sector
Riemannian manifold with SU(n) holonomy
{\displaystyle n=2} one obtains a K3 surface. More generally, Calabi–Yau varieties/orbifolds can be found as weighted complete intersections in a weighted projective
Calabi–Yau_manifold
Japanese mathematician
Bogomolov–Miyaoka–Yau inequality to surfaces with quotient singularities, and in 2008 to orbifold surfaces. Doing so, he obtains sharp bound on the number of quotient singularities
Yoichi_Miyaoka
domains. There are many small index subgroups of p4m, [4,4] symmetry (*442 orbifold notation), that can be seen in relation to the Coxeter diagram, with nodes
Tetrakis_square_tiling
Archimedean solid with 32 faces
the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction
Icosidodecahedron
Division of elementary particles
Additionally, E8 grand unified theories in 10 dimensions compactified on certain orbifolds down to 4 D naturally contain 3 generations of matter. This includes many
Generation_(particle_physics)
Notations Description Example Intl Orbifold Coxeter P.G. p1 ∞∞ [∞]+ C∞ Translations. Abstract group Z, the integers under addition ... --> --> --> -->
Line_group
Mathematical formula of two surfaces
inverse ratio to the degrees of the correspondence. An orbifold covering of degree N between orbifold surfaces S' and S is a branched covering, so the Riemann–Hurwitz
Riemann–Hurwitz_formula
Type of continuous map in topology
slightly weaker conditions) are used in the construction of manifolds, orbifolds, and the morphisms between them. In algebraic topology, covering spaces
Covering_space
American mathematician (1943–2024)
4310/jdg/1214436922. MR 0664497. Zbl 0504.53034. Hamilton, Richard S. (2003). "Three-orbifolds with positive Ricci curvature". In Cao, H. D.; Chow, B.; Chu, S. C.; Yau
Richard_S._Hamilton
Solid with eight equal triangular faces
the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of ∗ n 32 {\displaystyle ^{*}n32} all of these tilings
Regular_octahedron
Geometric shape
a circular arc. Alternatively, a football may refer to a more abstract orbifold, a surface modeled locally on a sphere except at two points. The lemon
Lemon_(geometry)
Groups of point isometries in 3 dimensions
crystallography), Schönflies notation (used to describe molecular symmetry), orbifold notation, and Coxeter notation. The latter three are not only conveniently
Point groups in three dimensions
Point_groups_in_three_dimensions
Quantum mechanical model based on mathematical matrices
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Matrix_theory_(physics)
Mathematical group
non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi orbifold. An exact formula for the volume, in terms of the Dedekind zeta function
Bianchi_group
symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation. Chiral Cn, [n]+, (nn) of order n - n-fold rotational symmetry
Cyclic symmetry in three dimensions
Cyclic_symmetry_in_three_dimensions
Type of mathematical group
Formally, the quotient under aliasing is the orbifold [0, 0.5fs], with a Z/2 action at the endpoints (the orbifold points), corresponding to reflection. The
Infinite_dihedral_group
Concept in differential geometry
and therefore the orbifold charts generate a diffeology on X {\displaystyle X} . This diffeology is uniquely determined by the orbifold structure of X {\displaystyle
Diffeology
Topological space that locally resembles Euclidean space
infinite-dimensional manifolds are studied in functional analysis. Orbifolds An orbifold is a generalization of manifold allowing for certain kinds of "singularities"
Manifold
Invariant action in bosonic string theory
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Nambu–Goto_action
Semiregular tiling of the hyperbolic plane
subsymmetry. The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) symmetry, there are 15 small index subgroup
Truncated order-8 hexagonal tiling
Truncated_order-8_hexagonal_tiling
symmetry. The dual of the tiling represents the fundamental domains of (*∞44) orbifold symmetry. From [(∞,4,4)] (*∞44) symmetry, there are 15 small index subgroup
Truncated infinite-order square tiling
Truncated_infinite-order_square_tiling
Principle in theoretical physics
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Holographic_principle
Chinese mathematician
"The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten Theory of Spin Curves and Orbifolds, Contemporary Mathematics, vol. 403, Providence
Yongbin_Ruan
Rotation composed with a reflection
4 subgroup of [2n,2], , generated as the product of 3 reflections. The Orbifold notation is n×, order 2n. The direct subgroup of S2n is Cn, order n, index
Improper_rotation
Type of Riemannian manifold
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Hyperkähler_manifold
Regular polygonal symmetry
shown below in 3 notations: Schönflies notation, Coxeter notation, and orbifold notation. Chiral Dn, [n,2]+, (22n) of order 2n – dihedral symmetry or para-n-gonal
Dihedral symmetry in three dimensions
Dihedral_symmetry_in_three_dimensions
Type of symmetry group
using Hermann–Mauguin notation, Coxeter notation, Schönflies notation, orbifold notation, nicknames created by mathematician John H. Conway, and finally
Frieze_group
Duality between theories of gravity on anti-de Sitter space and conformal field theories
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
AdS/CFT_correspondence
notes introduced several new ideas into geometric topology, including orbifolds, pleated manifolds, and train tracks. Copies of the original 1980 notes
The geometry and topology of three-manifolds
The_geometry_and_topology_of_three-manifolds
South African theoretical physicist
produced by the collision of a brane in the bulk space with a bounding orbifold plane, beginning from an otherwise cold, vacuous, static universe". Most
Neil_Turok
2-dimensional inclined lattice
oblique lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table
Oblique_lattice
Parametrizes complex structures on a surface
In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex
Teichmüller_space
Harmonic functions as solutions to Laplace's equation
of the conformal group as functions on a multiply connected manifold or orbifold. From the fact that the group of conformal transforms is infinite-dimensional
Potential_theory
simplicial sheaves. See also: simplicial diagram. An effective quotient orbifold, e.g., [ M / G ] {\displaystyle [M/G]} where the G {\displaystyle G} action
Quotient_stack
Algebra combining both supersymmetry and conformal symmetry
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Superconformal_algebra
Hypothetical scenario for the ultimate fate of the universe
Steinhardt, states that the Big Bang could have been caused by two parallel orbifold planes, referred to as branes colliding in a higher-dimensional space.
Big_Crunch
fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by John Horton Conway, Olaf Delgado Friedrichs,
Fibrifold
Regular tiling of the hyperbolic plane
kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections
Order-4_octagonal_tiling
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
List of quantum field theories
List_of_quantum_field_theories
New Zealand physicist (born 1968)
(2023). "Fractional conformal descendants and correlators in general 2D SN orbifold CFTs at large N". Journal of High Energy Physics. 2023 (2): 91. arXiv:2211
A._W._Peet
Concept in mathematics and theoretical physics
classification which is the singularity at the fixed point of the C2/Z2 orbifold where the Z2 group inverts the signs of both complex coordinates in C2
Eguchi–Hanson_space
^{2}} of the hyperbolic plane by Γ {\displaystyle \Gamma } is a hyperbolic orbifold of finite volume. The theorem above implies that every horocycle of H 2
Ratner's_theorems
Chinese-American mathematician (born 1949)
quasi-projective complex varieties. They later extended their work to allow orbifold singularities.[TY91] With Brian Greene, Alfred Shapere, and Cumrun Vafa
Shing-Tung_Yau
Theories in particle physics and cosmology
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Brane_cosmology
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
History_of_string_theory
Mathematical theorem
geometric terms using the compact Riemannian manifold (more generally orbifold) Γ ∖ X {\displaystyle \Gamma \backslash X} . The orbital integrals and
Selberg_trace_formula
German mathematician (1810–1893)
two-dimensional abelian variety by the cyclic group {1, −1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in
Ernst_Kummer
Variant of the notion of the center of a monoid, group, or ring to a category
an orbifold X is the category of sheaves on the inertia orbifold of X. For X being the classifying space of a finite group G, the inertia orbifold is
Center_(category_theory)
Polyhedron with 8 triangles and 6 squares
24 edges can be seen in 4 central hexagons. With octahedral symmetry (orbifold 432), the squares have the 4-fold symmetry, triangles the 3-fold symmetry
Cuboctahedron
Hypothetical Goldstone boson
through a material while all other Standard Model forces are fixed to an orbifold point. Experiments studying double beta decay have set limits on decay
Majoron
Manifold upon which it is possible to perform calculus
use a different notion of chart known as a "plot". Frölicher spaces and orbifolds are other attempts. A rectifiable set generalizes the idea of a piece-wise
Differentiable_manifold
Geometric transformation combining reflection and translation
given a Schoenflies notation as S2∞, Coxeter notation as [∞+,2+], and orbifold notation as ∞×. In the Euclidean plane, reflections and glide reflections
Glide_reflection
Geometry concept
Group Intl Orbifold Coxeter Order Description Cn n n• [n]+ n Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n
Point groups in two dimensions
Point_groups_in_two_dimensions
Uniform tiling of the hyperbolic plane
and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation
Tetrahexagonal_tiling
Hypothetical physical entity
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
String_(physics)
2D conformal field theories
{u}}}_{1}^{N}} , there exist orbifolds of free bosonic CFTs. For example, the Z 2 {\displaystyle \mathbb {Z} _{2}} orbifold of the compactified free boson
Massless free scalar bosons in two dimensions
Massless_free_scalar_bosons_in_two_dimensions
Theory in physics
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Non-critical_string_theory
Crystallographic system where the unit cell is in the shape of a cube
names, point groups (in Schönflies notation, Hermann–Mauguin notation, orbifold, and Coxeter notation), type, examples, international tables for crystallography
Cubic_crystal_system
Periodic set of points
group of the lattice Λ {\displaystyle \Lambda } is given in IUCr notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing
Lattice_(group)
Indian physicist
(2013). "Operator mixing for string states in the D1-D5 CFT near the orbifold point". Phys. Rev. D. 87 (10) 106001. arXiv:1211.6699. Bibcode:2013PhRvD
Samir_D._Mathur
Japanese theoretical physicist
1974 (age 52) Kanagawa, Japan Alma mater University of Tokyo Known for Orbifold GUTs Holographic Higgs Multiverse is the same as quantum many worlds Scientific
Yasunori_Nomura
ORBIFOLD
ORBIFOLD
ORBIFOLD
ORBIFOLD
Boy/Male
Indian, Modern
Intelligence
Boy/Male
Australian, Biblical, Christian, Hebrew
Dwelling; Habitation; To Give Honor to
Boy/Male
Hindu, Indian
Evening
Girl/Female
Indian, Sanskrit
Deer Eyes
Girl/Female
Tamil
Khanjana | காநà¯à®œà®¾à®¨à®¾
Boy/Male
Indian, Punjabi, Sikh
Soft and Peaceful Light
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Unique; Precious; Different from Everyone; Rare
Male
Hebrew
(ש×ָרָר) Hebrew name SHARAR means "enemy" or "to be firm, hard." In the bible, this is the name of the father of Ahiam.
Surname or Lastname
English
English : patronymic from Bond.
Surname or Lastname
English
English : of uncertain origin. Perhaps an altered form of Griswold or Creswell. In the U.S. it is found chiefly in GA.
ORBIFOLD
ORBIFOLD
ORBIFOLD
ORBIFOLD
ORBIFOLD