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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
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
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
Concept in differential geometry
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
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
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
American mathematician (1946–2012)
prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around
William_Thurston
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
2-dimensional integer lattice
group, denoted in IUC notation as p4m, Coxeter notation as [4,4], and orbifold notation as *442. A pattern with this lattice of translational symmetry
Square_lattice
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
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
Colombian mathematician
from the University of Wisconsin-Madison with thesis Twisted K-Theory and Orbifold Cohomology of the Symmetric Product under the supervision of Alejandro
Bernardo_Uribe
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Notations Description Example Intl Orbifold Coxeter P.G. p1 ∞∞ [∞]+ C∞ Translations. Abstract group Z, the integers under addition ... --> --> --> -->
Line_group
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
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
Semiregular tiling of the hyperbolic plane
subsymmetry. The dual of the tiling represents the fundamental domains of (*642) orbifold symmetry. From [6,4] symmetry, there are 15 small index subgroup by mirror
Truncated tetrahexagonal tiling
Truncated_tetrahexagonal_tiling
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
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
Mathematical group
is a non-compact, hyperbolic 3-fold of finite volume, called a Bianchi orbifold. An exact formula for its volume in terms of the Dedekind zeta function
Bianchi_group
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
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
Uniform tiling of the hyperbolic plane
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
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
one considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that
Torus_action
Pattern in hyperbolic geometry
(*662). The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup
Truncated order-4 hexagonal tiling
Truncated_order-4_hexagonal_tiling
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)
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)
2-dimensional lattice
rectangular lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table
Rectangular_lattice
Abstract mathematical spacetime
first described by Charles W. Misner. It is also known as the Lorentzian orbifold R 1 , 1 / boost {\displaystyle \mathbb {R} ^{1,1}/{\text{boost}}} . It
Misner_space
Type of smooth complex surface of kodaira dimension 0
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
K3_surface
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
Concept in mathematics
necessarily distinct[clarification needed] unordered points is instead an orbifold. A configuration space is a type of classifying space or (fine) moduli
Configuration space (mathematics)
Configuration_space_(mathematics)
Origin and evolution of the symbols used to write equations and formulas
operator-based approach of Sin-Itiro Tomonaga and Julian Schwinger. The orbifold notation system, invented by William Thurston, has been developed for representing
History of mathematical notation
History_of_mathematical_notation
Uniform tiling of the hyperbolic plane
with gyration points removed, becoming orbifold (*22222), and its direct subgroup [5*,4]+, index 20, becomes orbifold (22222). Wikimedia Commons has media
Truncated tetrapentagonal tiling
Truncated_tetrapentagonal_tiling
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
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
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
Portuguese mathematician (born 1968)
completing her Ph.D. in 1996. Her dissertation, Multiplicity Formulas for Orbifolds, was supervised by Victor Guillemin. After a temporary position as Morrey
Ana_Cannas_da_Silva
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
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
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
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
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
Geometry term
quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries
Hexaoctagonal_tiling
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
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
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)
Hypothetical elementary particle that mediates gravity
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
Graviton
Prize awarded by the International Centre for Theoretical Physics
contributions to superstring theory. Their studies range from early work on orbifold compactifications, physics and mathematics of mirror symmetry, D-branes
Dirac_Medal_(ICTP)
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
Theory of subatomic structure
supersymmetry, while Lance Dixon and others worked out the physical properties of orbifolds, distinctive geometrical singularities allowed in string theory. Cumrun
String_theory
Covering by shapes without overlaps or gaps
categorized by the seven frieze groups describing the possible frieze patterns. Orbifold notation can be used to describe wallpaper groups of the Euclidean plane
Tessellation
Geometric space
{\displaystyle {\mathcal {M}}_{g}} is irreducible. Properness, or compactness for orbifolds, follows from a theorem on stable reduction on curves. This can be found
Moduli_of_algebraic_curves
Collection of possible string theory vacua
manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics)
String_theory_landscape
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
hyperbolic triangle groups are notable NEC groups. Others are listed in Orbifold notation. Non-Euclidean geometry Isometry group Fuchsian group Uniform
Non-Euclidean crystallographic group
Non-Euclidean_crystallographic_group
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
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
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
Japanese theoretical physicist
1974 (age 52) Kanagawa, Japan Education University of Tokyo Known for Orbifold GUTs Holographic Higgs Multiverse is the same as quantum many worlds Scientific
Yasunori_Nomura
Non-orientable surface with one edge
and generalizations to more points, is a significant application of orbifolds to music theory. Modern musical groups taking their name from the Möbius
Möbius_strip
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
In mathematics, a Riemann surface
Vincent; Rito, Carlos; Roulleau, Xavier (2021). "The Bolza curve and some orbifold ball quotient surfaces". Michigan Mathematical Journal. 70 (2): 423-448
Bolza_surface
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