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Orbifold

In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.

Definitions of orbifold have been given several times: by Ichirō Satake in the context of automorphic forms in the 1950s under the name V-manifold; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name orbifold, after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name orbihedron.

Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group $$\mathrm{SL}(2,\Z)$$ on the upper half-plane: a version of the Riemann–Roch theorem holds after the quotient is compactified by the addition of two orbifold cusp points. In 3-manifold theory, the theory of Seifert fiber spaces, initiated by Herbert Seifert, can be phrased in terms of 2-dimensional orbifolds. In geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.

In string theory, the word "orbifold" has a slightly different meaning, discussed in detail below. In two-dimensional conformal field theory, it refers to the theory attached to the fixed point subalgebra of a vertex algebra under the action of a finite group of automorphisms.

The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a finite group; thus a manifold with boundary carries a natural orbifold structure, since it is the quotient of its double by an action of $$\Z_2$$.

One topological space can carry different orbifold structures. For example, consider the orbifold O associated with a quotient space of the 2-sphere along a rotation by $$\pi$$; it is homeomorphic to the 2-sphere, but the natural orbifold structure is different. It is possible to adopt most of the characteristics of manifolds to orbifolds and these characteristics are usually different from correspondent characteristics of underlying space. In the above example, the orbifold fundamental group of O is $$\Z_2$$ and its orbifold Euler characteristic is 1.

Definition using orbifold atlas
Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of $$\R^n$$, an orbifold is locally modelled on quotients of open subsets of $$\R^n$$ by finite group actions. The structure of an orbifold encodes not only that of the underlying quotient space, which need not be a manifold, but also that of the isotropy subgroups.

An $$n$$-dimensional orbifold is a Hausdorff topological space $$X$$, called the underlying space, with a covering by a collection of open sets $$U_i$$, closed under finite intersection. For each $$U_i$$, there is
 * an open subset $$V_i$$ of $$\R^n$$, invariant under a faithful linear action of a finite group $$\Gamma_i$$;
 * a continuous map $$\varphi_i$$ of $$V_i$$ onto $$U_i$$ invariant under $$\Gamma_i$$, called an orbifold chart, which defines a homeomorphism between $$V_i/\Gamma_i$$ and $$U_i$$.

The collection of orbifold charts is called an orbifold atlas if the following properties are satisfied:
 * for each inclusion $$U_i\subset U_j$$ there is an injective group homomorphism $$f_{ij}:\Gamma_i\rightarrow\Gamma_j$$.
 * for each inclusion $$U_i\subset U_j$$ there is a $$\Gamma_i$$-equivariant homeomorphism $$\psi_{ij}$$, called a gluing map, of $$V_i$$ onto an open subset of $$V_j$$.
 * the gluing maps are compatible with the charts, i.e. $$\varphi_j\circ\psi_{ij} = \varphi_i$$.
 * the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from $$V_i$$ to $$V_j$$ has the form $$g\circ\psi_{ij}$$ for a unique $$g\in\Gamma_j$$.

As for atlases on manifolds, two orbifold atlases of $$X$$ are equivalent if they can be consistently combined to give a larger orbifold atlas. An orbifold structure is therefore an equivalence class of orbifold atlases.

Note that the orbifold structure determines the isotropy subgroup of any point of the orbifold up to isomorphism: it can be computed as the stabilizer of the point in any orbifold chart. If Ui $$ \subset $$ Uj $$ \subset $$ Uk, then there is a unique transition element gijk in Γk such that

 * gijk·ψik = ψjk·ψij

These transition elements satisfy

 * (Ad gijk)·fik = fjk·fij

as well as the cocycle relation (guaranteeing associativity)

 * fkm(gijk)·gikm  = gijm·gjkm.

More generally, attached to an open covering of an orbifold by orbifold charts, there is the combinatorial data of a so-called complex of groups (see below).

Exactly as in the case of manifolds, differentiability conditions can be imposed on the gluing maps to give a definition of a differentiable orbifold. It will be a Riemannian orbifold if in addition there are invariant Riemannian metrics on the orbifold charts and the gluing maps are isometries.

Definition using Lie groupoids
Recall that a groupoid consists of a set of objects $$G_0$$, a set of arrows $$G_1$$, and structural maps including the source and the target maps $$s, t: G_1 \to G_0$$ and other maps allowing arrows to be composed and inverted. It is called a Lie groupoid if both $$G_0$$ and $$G_1$$ are smooth manifolds, all structural maps are smooth, and both the source and the target maps are submersions. The intersection of the source and the target fiber at a given point $$x \in G_0$$, i.e. the set $$(G_1)_x := s^{-1}(x) \cap t^{-1}(x)$$, is the Lie group called the isotropy group of $$G_1$$ at $$x$$. A Lie groupoid is called proper if the map $$(s,t): G_1 \to G_0 \times G_0$$ is a proper map, and étale if both the source and the target maps are local diffeomorphisms.

An orbifold groupoid is given by one of the following equivalent definitions:

 * a proper étale Lie groupoid;
 * a proper Lie groupoid whose isotropies are discrete spaces.

Since the isotropy groups of proper groupoids are automatically compact, the discreteness condition implies that the isotropies must be actually finite groups.

Orbifold groupoids play the same role as orbifold atlases in the definition above. Indeed, an orbifold structure on a Hausdorff topological space $$X$$ is defined as the Morita equivalence class of an orbifold groupoid $$G \rightrightarrows M$$ together with a homeomorphism $$|M/G| \simeq X$$, where $$|M/G|$$ is the orbit space of the Lie groupoid $$G$$ (i.e. the quotient of $$M$$ by the equivalent relation when $$x \sim y$$ if there is a $$g \in G$$ with $$s(g)=x$$ and $$t(g)=y$$). This definition shows that orbifolds are a particular kind of differentiable stack.

Relation between the two definitions
Given an orbifold atlas on a space $$X$$, one can build a pseudogroup made up by all diffeomorphisms between open sets of $$X$$ which preserve the transition functions $$\varphi_i$$. In turn, the space $$G_X$$ of germs of its elements is an orbifold groupoid. Moreover, since by definition of orbifold atlas each finite group $$\Gamma_i$$ acts faithfully on $$V_i$$, the groupoid $$G_X$$ is automatically effective, i.e. the map $$g \in (G_X)_x \mapsto \mathrm{germ}_x (t \circ s^{-1})$$ is injective for every $$x \in X$$. Two different orbifold atlases give rise to the same orbifold structure if and only if their associated orbifold groupoids are Morita equivalent. Therefore, any orbifold structure according to the first definition (also called a classical orbifold) is a special kind of orbifold structure according to the second definition.

Conversely, given an orbifold groupoid $$G \rightrightarrows M$$, there is a canonical orbifold atlas over its orbit space, whose associated effective orbifold groupoid is Morita equivalent to $$G$$. Since the orbit spaces of Morita equivalent groupoids are homeomorphic, an orbifold structure according to the second definition reduces an orbifold structure according to the first definition in the effective case.

Accordingly, while the notion of orbifold atlas is simpler and more commonly present in the literature, the notion of orbifold groupoid is particularly useful when discussing non-effective orbifolds and maps between orbifolds. For example, a map between orbifolds can be described by a homomorphism between groupoids, which carries more information than the underlying continuous map between the underlying topological spaces.

Examples

 * Any manifold without boundary is trivially an orbifold, where each of the groups Γi is the trivial group. Equivalently, it corresponds to the Morita equivalence class of the unit groupoid.
 * If N is a compact manifold with boundary, its double M can be formed by gluing together a copy of N and its mirror image along their common boundary. There is natural reflection action of Z2 on the manifold M fixing the common boundary; the quotient space can be identified with N, so that N has a natural orbifold structure.
 * If M is a Riemannian n-manifold with a cocompact proper isometric action of a discrete group Γ, then the orbit space X = M/Γ has natural orbifold structure: for each x in X take a representative m in M and an open neighbourhood Vm of m invariant under the stabiliser Γm, identified equivariantly with a Γm-subset of TmM under the exponential map at m; finitely many neighbourhoods cover X and each of their finite intersections, if non-empty, is covered by an intersection of Γ-translates gm·Vm with corresponding group gm Γ gm−1. Orbifolds that arise in this way are called developable or good.
 * A classical theorem of Henri Poincaré constructs Fuchsian groups as hyperbolic reflection groups generated by reflections in the edges of a geodesic triangle in the hyperbolic plane for the Poincaré metric. If the triangle has angles $\pi$/ni for positive integers ni, the triangle is a fundamental domain and naturally a 2-dimensional orbifold. The corresponding group is an example of a hyperbolic triangle group. Poincaré also gave a 3-dimensional version of this result for Kleinian groups: in this case the Kleinian group Γ is generated by hyperbolic reflections and the orbifold is H3 / Γ.
 * If M is a closed 2-manifold, new orbifold structures can be defined on Mi by removing finitely many disjoint closed discs from M and gluing back copies of discs D/ Γi where D is the closed unit disc and Γi is a finite cyclic group of rotations. This generalises Poincaré's construction.

Orbifold fundamental group
There are several ways to define the orbifold fundamental group. More sophisticated approaches use orbifold covering spaces or classifying spaces of groupoids. The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion of loop used in the standard definition of the fundamental group.

An orbifold path is a path in the underlying space provided with an explicit piecewise lift of path segments to orbifold charts and explicit group elements identifying paths in overlapping charts; if the underlying path is a loop, it is called an orbifold loop. Two orbifold paths are identified if they are related through multiplication by group elements in orbifold charts. The orbifold fundamental group is the group formed by homotopy classes of orbifold loops.

If the orbifold arises as the quotient of a simply connected manifold M by a proper rigid action of a discrete group Γ, the orbifold fundamental group can be identified with Γ. In general it is an extension of Γ by π1 M.

The orbifold is said to be developable or good if it arises as the quotient by a group action; otherwise it is called bad. A universal covering orbifold can be constructed for an orbifold by direct analogy with the construction of the universal covering space of a topological space, namely as the space of pairs consisting of points of the orbifold and homotopy classes of orbifold paths joining them to the basepoint. This space is naturally an orbifold.

Note that if an orbifold chart on a contractible open subset corresponds to a group Γ, then there is a natural local homomorphism of Γ into the orbifold fundamental group.

In fact the following conditions are equivalent:
 * The orbifold is developable.
 * The orbifold structure on the universal covering orbifold is trivial.
 * The local homomorphisms are all injective for a covering by contractible open sets.

Orbifolds as diffeologies
Orbifolds can be defined in the general framework of diffeology and have been proved to be equivalent to Ichirô Satake's original definition:

Definition: An orbifold is a diffeological space locally diffeomorphic at each point to some $$\R^n/G$$, where $$n$$ is an integer and $$G$$ is a finite linear group which may change from point to point.

This definition calls a few remarks:

 * This definition mimics the definition of a manifold in diffeology, which is a diffeological space locally diffeomorphic at each point to $$\R^n$$.
 * An orbifold is regarded first as a diffeological space, a set equipped with a diffeology. Then, the diffeology is tested to be locally diffeomorphic at each point to a quotient $$\R^n/G$$ with $$G$$ a finite linear group.
 * This definition is equivalent with Haefliger orbifolds.
 * {Orbifolds} makes a subcategory of the category {Diffeology} whose objects are diffeological spaces and morphisms smooth maps. A smooth map between orbifolds is any map which is smooth for their diffeologies. This resolves, in the context of Satake's definition, his remark: "The notion of $$C^\infty$$-map thus defined is inconvenient in the point that a composite of two $$C^\infty$$-map defined in a different choice of defining families is not always a $$C^\infty$$-map." Indeed, there are smooth maps between orbifolds that do not lift locally as equivariant maps.

Note that the fundamental group of an orbifold as a diffeological space is not the same as the fundamental group as defined above. That last one is related to the structure groupoid and its isotropy groups.

Orbispaces
For applications in geometric group theory, it is often convenient to have a slightly more general notion of orbifold, due to Haefliger. An orbispace is to topological spaces what an orbifold is to manifolds. An orbispace is a topological generalization of the orbifold concept. It is defined by replacing the model for the orbifold charts by a locally compact space with a rigid action of a finite group, i.e. one for which points with trivial isotropy are dense. (This condition is automatically satisfied by faithful linear actions, because the points fixed by any non-trivial group element form a proper linear subspace.) It is also useful to consider metric space structures on an orbispace, given by invariant metrics on the orbispace charts for which the gluing maps preserve distance. In this case each orbispace chart is usually required to be a length space with unique geodesics connecting any two points.

Let X be an orbispace endowed with a metric space structure for which the charts are geodesic length spaces. The preceding definitions and results for orbifolds can be generalized to give definitions of orbispace fundamental group and universal covering orbispace, with analogous criteria for developability. The distance functions on the orbispace charts can be used to define the length of an orbispace path in the universal covering orbispace. If the distance function in each chart is non-positively curved, then the Birkhoff curve shortening argument can be used to prove that any orbispace path with fixed endpoints is homotopic to a unique geodesic. Applying this to constant paths in an orbispace chart, it follows that each local homomorphism is injective and hence:
 * every non-positively curved orbispace is developable (i.e. good).

Complexes of groups
Every orbifold has associated with it an additional combinatorial structure given by a complex of groups.

Definition
A complex of groups (Y,f,g) on an abstract simplicial complex Y is given by
 * a finite group Γσ for each simplex σ of Y
 * an injective homomorphism fστ : Γτ $$\rightarrow$$ Γσ whenever σ $$\subset$$ τ
 * for every inclusion ρ $$\subset$$ σ $$\subset$$ τ, a group element gρστ in Γρ such that (Ad gρστ)·fρτ = fρσ·fστ (here Ad denotes the adjoint action by conjugation)

The group elements must in addition satisfy the cocycle condition

 * fπρ(gρστ) gπρτ = gπστ gπρσ

for every chain of simplices $$\pi \subset \rho\subset \sigma \subset \tau.$$ (This condition is vacuous if Y has dimension 2 or less.)

Any choice of elements hστ in Γσ yields an equivalent complex of groups by defining
 * f'στ = (Ad hστ)·fστ
 * g'ρστ =   hρσ·fρσ(hστ)·gρστ·hρτ−1

A complex of groups is called simple whenever gρστ = 1 everywhere.
 * An easy inductive argument shows that every complex of groups on a simplex is equivalent to a complex of groups with gρστ = 1 everywhere.

It is often more convenient and conceptually appealing to pass to the barycentric subdivision of Y. The vertices of this subdivision correspond to the simplices of Y, so that each vertex has a group attached to it. The edges of the barycentric subdivision are naturally oriented (corresponding to inclusions of simplices) and each directed edge gives an inclusion of groups. Each triangle has a transition element attached to it belonging to the group of exactly one vertex; and the tetrahedra, if there are any, give cocycle relations for the transition elements. Thus a complex of groups involves only the 3-skeleton of the barycentric subdivision; and only the 2-skeleton if it is simple.

Example
If X is an orbifold (or orbispace), choose a covering by open subsets from amongst the orbifold charts fi: Vi $$\rightarrow$$ Ui. Let Y be the abstract simplicial complex given by the nerve of the covering: its vertices are the sets of the cover and its n-simplices correspond to non-empty intersections Uα = Ui 1 $$\cap$$ ··· $$\cap$$ Ui n. For each such simplex there is an associated group Γα and the homomorphisms fij become the homomorphisms fστ. For every triple ρ $$ \subset $$ σ $$ \subset $$ τ corresponding to intersections

 * $$ U_i \supset U_i \cap U_j \supset U_i \cap U_j \cap U_k$$

there are charts φi : Vi $$\rightarrow$$ Ui, φij : Vij $$\rightarrow$$ Ui $$\cap$$ Uj and φijk : Vijk $$\rightarrow$$ Ui $$\cap$$ Uj $$\cap$$ Uk and gluing maps ψ : Vij $$\rightarrow $$ Vi, ψ' : Vijk $$\rightarrow $$ Vij and ψ" : Vijk $$\rightarrow $$ Vi.

There is a unique transition element gρστ in Γi such that gρστ·ψ" = ψ·. The relations satisfied by the transition elements of an orbifold imply those required for a complex of groups. In this way a complex of groups can be canonically associated to the nerve of an open covering by orbifold (or orbispace) charts. In the language of non-commutative sheaf theory and gerbes, the complex of groups in this case arises as a sheaf of groups associated to the covering Ui; the data gρστ is a 2-cocycle in non-commutative sheaf cohomology and the data hστ gives a 2-coboundary perturbation.

Edge-path group
The edge-path group of a complex of groups can be defined as a natural generalisation of the edge path group of a simplicial complex. In the barycentric subdivision of Y, take generators eij corresponding to edges from i to j where i $$\rightarrow $$ j, so that there is an injection ψij : Γi $$ \rightarrow$$ Γj. Let Γ be the group generated by the eij and Γk with relations

 * eij −1 · g · eij = ψij(g)

for g in Γi and

 * eik = ejk·eij·gijk

if i $$\rightarrow $$ j $$\rightarrow$$ k.

For a fixed vertex i0, the edge-path group Γ(i0) is defined to be the subgroup of Γ generated by all products

 * g0 · ei 0 i1 · g1 · ei 1 i2 · ··· · gn · ei ni0

where i0, i1, ..., in, i0 is an edge-path, gk lies in Γi k and eji=eij−1 if i $$\rightarrow $$ j.

Developable complexes
A simplicial proper action of a discrete group Γ on a simplicial complex X with finite quotient is said to be regular if it satisfies one of the following equivalent conditions:
 * X admits a finite subcomplex as fundamental domain;
 * the quotient Y = X/Γ has a natural simplicial structure;
 * the quotient simplicial structure on orbit-representatives of vertices is consistent;
 * if (v0, ..., vk) and (g0·v0, ..., gk·vk) are simplices, then g·vi = gi·vi for some g in Γ.

The fundamental domain and quotient Y = X / Γ can naturally be identified as simplicial complexes in this case, given by the stabilisers of the simplices in the fundamental domain. A complex of groups Y is said to be developable if it arises in this way.
 * A complex of groups is developable if and only if the homomorphisms of Γσ into the edge-path group are injective.
 * A complex of groups is developable if and only if for each simplex σ there is an injective homomorphism θσ from Γσ into a fixed discrete group Γ such that θτ·fστ = θσ. In this case the simplicial complex X is canonically defined: it has k-simplices (σ, xΓσ) where σ is a k-simplex of Y and x runs over Γ / Γσ. Consistency can be checked using the fact that the restriction of the complex of groups to a simplex is equivalent to one with trivial cocycle gρστ.

The action of Γ on the barycentric subdivision X ' of X always satisfies the following condition, weaker than regularity:
 * whenever σ and g·σ are subsimplices of some simplex τ, they are equal, i.e. σ = g·σ

Indeed, simplices in X ' correspond to chains of simplices in X, so that a subsimplices, given by subchains of simplices, is uniquely determined by the sizes of the simplices in the subchain. When an action satisfies this condition, then g necessarily fixes all the vertices of σ. A straightforward inductive argument shows that such an action becomes regular on the barycentric subdivision; in particular
 * the action on the second barycentric subdivision X" is regular;
 * Γ is naturally isomorphic to the edge-path group defined using edge-paths and vertex stabilisers for the barycentric subdivision of the fundamental domain in X".

There is in fact no need to pass to a third barycentric subdivision: as Haefliger observes using the language of category theory, in this case the 3-skeleton of the fundamental domain of X" already carries all the necessary data – including transition elements for triangles – to define an edge-path group isomorphic to Γ.

In two dimensions this is particularly simple to describe. The fundamental domain of X" has the same structure as the barycentric subdivision Y ' of a complex of groups Y, namely:
 * a finite 2-dimensional simplicial complex Z;
 * an orientation for all edges i $$\rightarrow $$ j;
 * if i $$\rightarrow $$ j and j $$\rightarrow $$ k are edges, then i $$\rightarrow $$ k is an edge and (i, j, k) is a triangle;
 * finite groups attached to vertices, inclusions to edges and transition elements, describing compatibility, to triangles.

An edge-path group can then be defined. A similar structure is inherited by the barycentric subdivision Z ' and its edge-path group is isomorphic to that of Z.

Orbihedra
If a countable discrete group acts by a regular simplicial proper action on a simplicial complex, the quotient can be given not only the structure of a complex of groups, but also that of an orbispace. This leads more generally to the definition of "orbihedron", the simplicial analogue of an orbifold.

Definition
Let X be a finite simplicial complex with barycentric subdivision X '. An orbihedron structure consists of:
 * for each vertex i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi.
 * a simplicial map φi of Li' onto the link Li of i in X ', identifying the quotient Li' / Γi with Li.

This action of Γi on Li' extends to a simplicial action on the simplicial cone Ci over Li' (the simplicial join of i and Li'), fixing the centre i of the cone. The map φi extends to a simplicial map of Ci onto the star St(i) of i, carrying the centre onto i; thus φi identifies Ci / Γi, the quotient of the star of i in Ci, with St(i) and gives an orbihedron chart at i.
 * for each directed edge i $$\rightarrow$$ j of X ', an injective homomorphism fij of Γi into Γj.
 * for each directed edge i $$\rightarrow$$ j, a Γi equivariant simplicial gluing map ψij of Ci into Cj.
 * the gluing maps are compatible with the charts, i.e. φj·ψij = φi.
 * the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from Vi to Vj has the form g·ψij for a unique g in Γj.

If i$$ \rightarrow $$ j $$ \rightarrow $$ k, then there is a unique transition element gijk in Γk such that

 * gijk·ψik = ψjk·ψij

These transition elements satisfy

 * (Ad gijk)·fik = fjk·fij

as well as the cocycle relation

 * ψkm(gijk)·gikm  = gijm·gjkm.

Main properties

 * The group theoretic data of an orbihedron gives a complex of groups on X, because the vertices i of the barycentric subdivision X ' correspond to the simplices in X.
 * Every complex of groups on X is associated with an essentially unique orbihedron structure on X. This key fact follows by noting that the star and link of a vertex i of X ', corresponding to a simplex σ of X, have natural decompositions: the star is isomorphic to the abstract simplicial complex given by the join of σ and the barycentric subdivision σ' of σ; and the link is isomorphic to join of the link of σ in X and the link of the barycentre of σ in σ'. Restricting the complex of groups to the link of σ in X, all the groups Γτ come with injective homomorphisms into Γσ. Since the link of i in X ' is canonically covered by a simplicial complex on which Γσ acts, this defines an orbihedron structure on X.
 * The orbihedron fundamental group is (tautologically) just the edge-path group of the associated complex of groups.
 * Every orbihedron is also naturally an orbispace: indeed in the geometric realization of the simplicial complex, orbispace charts can be defined using the interiors of stars.
 * The orbihedron fundamental group can be naturally identified with the orbispace fundamental group of the associated orbispace. This follows by applying the simplicial approximation theorem to segments of an orbispace path lying in an orbispace chart: it is a straightforward variant of the classical proof that the fundamental group of a polyhedron can be identified with its edge-path group.
 * The orbispace associated to an orbihedron has a canonical metric structure, coming locally from the length metric in the standard geometric realization in Euclidean space, with vertices mapped to an orthonormal basis. Other metric structures are also used, involving length metrics obtained by realizing the simplices in hyperbolic space, with simplices identified isometrically along common boundaries.
 * The orbispace associated to an orbihedron is non-positively curved if and only if the link in each orbihedron chart has girth greater than or equal to 6, i.e. any closed circuit in the link has length at least 6. This condition, well known from the theory of Hadamard spaces, depends only on the underlying complex of groups.
 * When the universal covering orbihedron is non-positively curved the fundamental group is infinite and is generated by isomorphic copies of the isotropy groups. This follows from the corresponding result for orbispaces.

Triangles of groups
Historically one of the most important applications of orbifolds in geometric group theory has been to triangles of groups. This is the simplest 2-dimensional example generalising the 1-dimensional "interval of groups" discussed in Serre's lectures on trees, where amalgamated free products are studied in terms of actions on trees. Such triangles of groups arise any time a discrete group acts simply transitively on the triangles in the affine Bruhat–Tits building for SL3(Qp); in 1979 Mumford discovered the first example for p = 2 (see below) as a step in producing an algebraic surface not isomorphic to projective space, but having the same Betti numbers. Triangles of groups were worked out in detail by Gersten and Stallings, while the more general case of complexes of groups, described above, was developed independently by Haefliger. The underlying geometric method of analysing finitely presented groups in terms of metric spaces of non-positive curvature is due to Gromov. In this context triangles of groups correspond to non-positively curved 2-dimensional simplicial complexes with the regular action of a group, transitive on triangles.

A triangle of groups is a simple complex of groups consisting of a triangle with vertices A, B, C. There are groups
 * ΓA, ΓB, ΓC at each vertex
 * ΓBC, ΓCA, ΓAB for each edge
 * ΓABC for the triangle itself.

There is an injective homomorphisms of ΓABC into all the other groups and of an edge group ΓXY into ΓX and ΓY. The three ways of mapping ΓABC into a vertex group all agree. (Often ΓABC is the trivial group.) The Euclidean metric structure on the corresponding orbispace is non-positively curved if and only if the link of each of the vertices in the orbihedron chart has girth at least 6.

This girth at each vertex is always even and, as observed by Stallings, can be described at a vertex A, say, as the length of the smallest word in the kernel of the natural homomorphism into ΓA of the amalgamated free product over ΓABC of the edge groups ΓAB and ΓAC:

 * $$ \Gamma_{AB} \star_{\,\Gamma_{ABC}} \Gamma_{AC} \rightarrow \Gamma_A.$$

The result using the Euclidean metric structure is not optimal. Angles α, β, γ at the vertices A, B and C were defined by Stallings as 2π divided by the girth. In the Euclidean case α, β, γ ≤ π/3. However, if it is only required that α + β + γ ≤ π, it is possible to identify the triangle with the corresponding geodesic triangle in the hyperbolic plane with the Poincaré metric (or the Euclidean plane if equality holds). It is a classical result from hyperbolic geometry that the hyperbolic medians intersect in the hyperbolic barycentre, just as in the familiar Euclidean case. The barycentric subdivision and metric from this model yield a non-positively curved metric structure on the corresponding orbispace. Thus, if α+β+γ≤π,
 * the orbispace of the triangle of groups is developable;
 * the corresponding edge-path group, which can also be described as the colimit of the triangle of groups, is infinite;
 * the homomorphisms of the vertex groups into the edge-path group are injections.

Mumford's example
Let α = $$\sqrt{-7}$$ be given by the binomial expansion of (1 − 8)1/2 in Q2 and set K = Q(α) $$\subset $$ Q2. Let

 * ζ = exp 2πi/7

 * λ = (α − 1)/2 = ζ + ζ2 + ζ4

 * μ = λ/λ*.

Let E = Q(ζ), a 3-dimensional vector space over K with basis 1, ζ, and ζ2. Define K-linear operators on E as follows:
 * σ is the generator of the Galois group of E over K, an element of order 3 given by σ(ζ) = ζ2
 * τ is the operator of multiplication by ζ on E, an element of order 7
 * ρ is the operator given by ρ(ζ) = 1, ρ(ζ2) = ζ and ρ(1) = μ·ζ2, so that ρ3 is scalar multiplication by μ.

The elements ρ, σ, and τ generate a discrete subgroup of GL3(K) which acts properly on the affine Bruhat–Tits building corresponding to SL3(Q2). This group acts transitively on all vertices, edges and triangles in the building. Let

 * σ1 = σ, σ2 = ρσρ−1, σ3 = ρ2σρ−2.

Then
 * σ1, σ2 and σ3 generate a subgroup Γ of SL3(K).
 * Γ is the smallest subgroup generated by σ and τ, invariant under conjugation by ρ.
 * Γ acts simply transitively on the triangles in the building.
 * There is a triangle Δ such that the stabiliser of its edges are the subgroups of order 3 generated by the σi's.
 * The stabiliser of a vertices of Δ is the Frobenius group of order 21 generated by the two order 3 elements stabilising the edges meeting at the vertex.
 * The stabiliser of Δ is trivial.

The elements σ and τ generate the stabiliser of a vertex. The link of this vertex can be identified with the spherical building of SL3(F2) and the stabiliser can be identified with the collineation group of the Fano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ = τ2σ. Identifying F8* with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ(x) = x22 of F8 and τ to be multiplication by any element not in the prime field F2, i.e. an order 7 generator of the cyclic multiplicative group of F8. This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points. The formulas for σ and τ on E thus "lift" the formulas on F8.

Mumford also obtains an action simply transitive on the vertices of the building by passing to a subgroup of Γ1 = <ρ, σ, τ, −I>. The group Γ1 preserves the Q(α)-valued Hermitian form

 * f(x,y) = xy* + σ(xy*) + σ2(xy*)

on Q(ζ) and can be identified with U3(f) $$\cap$$ GL3(S) where S = Z[α,$1⁄2$]. Since S/(α) = F7, there is a homomorphism of the group Γ1 into GL3(F7). This action leaves invariant a 2-dimensional subspace in F73 and hence gives rise to a homomorphism Ψ of Γ1 into SL2(F7), a group of order 16·3·7. On the other hand, the stabiliser of a vertex is a subgroup of order 21 and Ψ is injective on this subgroup. Thus if the congruence subgroup Γ0 is defined as the inverse image under Ψ of the 2-Sylow subgroup of SL2(F7), the action of Γ0 on vertices must be simply transitive.

Generalizations
Other examples of triangles or 2-dimensional complexes of groups can be constructed by variations of the above example.

Cartwright et al. consider actions on buildings that are simply transitive on vertices. Each such action produces a bijection (or modified duality) between the points x and lines x* in the flag complex of a finite projective plane and a collection of oriented triangles of points (x,y,z), invariant under cyclic permutation, such that x lies on z*, y lies on x* and z lies on y* and any two points uniquely determine the third. The groups produced have generators x, labelled by points, and relations xyz = 1 for each triangle. Generically this construction will not correspond to an action on a classical affine building.

More generally, as shown by Ballmann and Brin, similar algebraic data encodes all actions that are simply transitively on the vertices of a non-positively curved 2-dimensional simplicial complex, provided the link of each vertex has girth at least 6. This data consists of:
 * a generating set S containing inverses, but not the identity;
 * a set of relations g h k = 1, invariant under cyclic permutation.

The elements g in S label the vertices g·v in the link of a fixed vertex v; and the relations correspond to edges (g−1·v, h·v) in that link. The graph with vertices S and edges (g, h), for g−1h in S, must have girth at least 6. The original simplicial complex can be reconstructed using complexes of groups and the second barycentric subdivision.

Further examples of non-positively curved 2-dimensional complexes of groups have been constructed by Swiatkowski based on actions simply transitive on oriented edges and inducing a 3-fold symmetry on each triangle; in this case too the complex of groups is obtained from the regular action on the second barycentric subdivision. The simplest example, discovered earlier with Ballmann, starts from a finite group H with a symmetric set of generators S, not containing the identity, such that the corresponding Cayley graph has girth at least 6. The associated group is generated by H and an involution τ subject to (τg)3 = 1 for each g in S.

In fact, if Γ acts in this way, fixing an edge (v, w), there is an involution τ interchanging v and w. The link of v is made up of vertices g·w for g in a symmetric subset S of H = Γv, generating H if the link is connected. The assumption on triangles implies that

 * τ·(g·w) = g−1·w

for g in S. Thus, if σ = τg and u = g−1·w, then

 * σ(v) = w, σ(w) = u, σ(u) = w.

By simple transitivity on the triangle (v, w, u), it follows that σ3 = 1.

The second barycentric subdivision gives a complex of groups consisting of singletons or pairs of barycentrically subdivided triangles joined along their large sides: these pairs are indexed by the quotient space S/~ obtained by identifying inverses in S. The single or "coupled" triangles are in turn joined along one common "spine". All stabilisers of simplices are trivial except for the two vertices at the ends of the spine, with stabilisers H and <τ>, and the remaining vertices of the large triangles, with stabiliser generated by an appropriate σ. Three of the smaller triangles in each large triangle contain transition elements.

When all the elements of S are involutions, none of the triangles need to be doubled. If H is taken to be the dihedral group D7 of order 14, generated by an involution a and an element b of order 7 such that

 * ab= b−1a,

then H is generated by the 3 involutions a, ab and ab5. The link of each vertex is given by the corresponding Cayley graph, so is just the bipartite Heawood graph, i.e. exactly the same as in the affine building for SL3(Q2). This link structure implies that the corresponding simplicial complex is necessarily a Euclidean building. At present, however, it seems to be unknown whether any of these types of action can in fact be realised on a classical affine building: Mumford's group Γ1 (modulo scalars) is only simply transitive on edges, not on oriented edges.

Two-dimensional orbifolds
Two-dimensional orbifolds have the following three types of singular points:

 * A boundary point
 * An elliptic point or gyration point of order n, such as the origin of R2 quotiented out by a cyclic group of order n of rotations.
 * A corner reflector of order n: the origin of R2 quotiented out by a dihedral group of order 2n.

A compact 2-dimensional orbifold has an Euler characteristic $$\chi$$ given by
 * $$ \chi= \chi(X_0) - \sum_{i}(1 - 1/n_i)/2 - \sum_{i} (1 - 1/m_i )$$,

where $$\chi(X_0)$$ is the Euler characteristic of the underlying topological manifold $$X_0$$, and $$n_i$$ are the orders of the corner reflectors, and $$m_i$$ are the orders of the elliptic points.

A 2-dimensional compact connected orbifold has a hyperbolic structure if its Euler characteristic is less than 0, a Euclidean structure if it is 0, and if its Euler characteristic is positive it is either bad or has an elliptic structure (an orbifold is called bad if it does not have a manifold as a covering space). In other words, its universal covering space has a hyperbolic, Euclidean, or spherical structure.

The compact 2-dimensional connected orbifolds that are not hyperbolic are listed in the table below. The 17 parabolic orbifolds are the quotients of the plane by the 17 wallpaper groups.

3-dimensional orbifolds
A 3-manifold is said to be small if it is closed, irreducible and does not contain any incompressible surfaces.

Orbifold Theorem. Let M be a small 3-manifold. Let φ be a non-trivial periodic orientation-preserving diffeomorphism of M. Then M admits a φ-invariant hyperbolic or Seifert fibered structure.

This theorem is a special case of Thurston's orbifold theorem, announced without proof in 1981; it forms part of his geometrization conjecture for 3-manifolds. In particular it implies that if X is a compact, connected, orientable, irreducible, atoroidal 3-orbifold with non-empty singular locus, then M has a geometric structure (in the sense of orbifolds). A complete proof of the theorem was published by Boileau, Leeb & Porti in 2005.

Orbifolds in string theory
In string theory, the word "orbifold" has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion of manifold that allows the presence of the points whose neighborhood is diffeomorphic to a quotient of Rn by a finite group, i.e. Rn/Γ. In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit space M/G where M is a manifold (or a theory), and G is a group of its isometries (or symmetries) — not necessarily all of them. In string theory, these symmetries do not have to have a geometric interpretation.

A quantum field theory defined on an orbifold becomes singular near the fixed points of G. However string theory requires us to add new parts of the closed string Hilbert space — namely the twisted sectors where the fields defined on the closed strings are periodic up to an action from G. Orbifolding is therefore a general procedure of string theory to derive a new string theory from an old string theory in which the elements of G have been identified with the identity. Such a procedure reduces the number of states because the states must be invariant under G, but it also increases the number of states because of the extra twisted sectors. The result is usually a perfectly smooth, new string theory.

D-branes propagating on the orbifolds are described, at low energies, by gauge theories defined by the quiver diagrams. Open strings attached to these D-branes have no twisted sector, and so the number of open string states is reduced by the orbifolding procedure.

More specifically, when the orbifold group G is a discrete subgroup of spacetime isometries, then if it has no fixed point, the result is usually a compact smooth space; the twisted sector consists of closed strings wound around the compact dimension, which are called winding states.

When the orbifold group G is a discrete subgroup of spacetime isometries, and it has fixed points, then these usually have conical singularities, because Rn/Zk has such a singularity at the fixed point of Zk. In string theory, gravitational singularities are usually a sign of extra degrees of freedom which are located at a locus point in spacetime. In the case of the orbifold these degrees of freedom are the twisted states, which are strings "stuck" at the fixed points. When the fields related with these twisted states acquire a non-zero vacuum expectation value, the singularity is deformed, i.e. the metric is changed and becomes regular at this point and around it. An example for a resulting geometry is the Eguchi–Hanson spacetime.

From the point of view of D-branes in the vicinity of the fixed points, the effective theory of the open strings attached to these D-branes is a supersymmetric field theory, whose space of vacua has a singular point, where additional massless degrees of freedom exist. The fields related with the closed string twisted sector couple to the open strings in such a way as to add a Fayet–Iliopoulos term to the supersymmetric field theory Lagrangian, so that when such a field acquires a non-zero vacuum expectation value, the Fayet–Iliopoulos term is non-zero, and thereby deforms the theory (i.e. changes it) so that the singularity no longer exists,.

Calabi–Yau manifolds
In superstring theory, the construction of realistic phenomenological models requires dimensional reduction because the strings naturally propagate in a 10-dimensional space whilst the observed dimension of space-time of the universe is 4. Formal constraints on the theories nevertheless place restrictions on the compactified space in which the extra "hidden" variables live: when looking for realistic 4-dimensional models with supersymmetry, the auxiliary compactified space must be a 6-dimensional Calabi–Yau manifold.

There are a large number of possible Calabi–Yau manifolds (tens of thousands), hence the use of the term "landscape" in the current theoretical physics literature to describe the baffling choice. The general study of Calabi–Yau manifolds is mathematically complex and for a long time examples have been hard to construct explicitly. Orbifolds have therefore proved very useful since they automatically satisfy the constraints imposed by supersymmetry. They provide degenerate examples of Calabi–Yau manifolds due to their singular points, but this is completely acceptable from the point of view of theoretical physics. Such orbifolds are called "supersymmetric": they are technically easier to study than general Calabi–Yau manifolds. It is very often possible to associate a continuous family of non-singular Calabi–Yau manifolds to a singular supersymmetric orbifold. In 4 dimensions this can be illustrated using complex K3 surfaces:

 * Every K3 surface admits 16 cycles of dimension 2 that are topologically equivalent to usual 2-spheres. Making the surface of these spheres tend to zero, the K3 surface develops 16 singularities. This limit represents a point on the boundary of the moduli space of K3 surfaces and corresponds to the orbifold $$T^4/\mathbb{Z}_2\,$$ obtained by taking the quotient of the torus by the symmetry of inversion.

The study of Calabi–Yau manifolds in string theory and the duality between different models of string theory (type IIA and IIB) led to the idea of mirror symmetry in 1988. The role of orbifolds was first pointed out by Dixon, Harvey, Vafa and Witten around the same time.

Music theory
Beyond their manifold and various applications in mathematics and physics, orbifolds have been applied to music theory at least as early as 1985 in the work of Guerino Mazzola and later by Dmitri Tymoczko and collaborators. One of the papers of Tymoczko was the first music theory paper published by the journal Science.  Mazzola and Tymoczko have participated in debate regarding their theories documented in a series of commentaries available at their respective web sites.

[[File:3-orbifold-voronoi-animated.gif|thumbnail|Animated slices of the three-dimensional orbifold $$T^3 / S_3$$.

Slices of cubes standing on end (with their long diagonals perpendicular to the plane of the image) form colored Voronoi regions (colored by chord type) which represent the three-note chords at their centers, with augmented triads at the very center, surrounded by major and minor triads (lime green and navy blue). The white regions are degenerate trichords (one-note repeated three times), with the three lines (representing two note chords) connecting their centers forming the walls of the twisted triangular prism, 2D planes perpendicular to plane of the image acting as mirrors.]]

Tymoczko models musical chords consisting of n notes, which are not necessarily distinct, as points in the orbifold $$T^n/S_n$$ – the space of n unordered points (not necessarily distinct) in the circle, realized as the quotient of the n-torus $$T^n$$ (the space of n ordered points on the circle) by the symmetric group $$S_n$$ (corresponding from moving from an ordered set to an unordered set).

Musically, this is explained as follows:
 * Musical tones depend on the frequency (pitch) of their fundamental, and thus are parametrized by the positive real numbers, R+.
 * Musical tones that differ by an octave (a doubling of frequency) are considered the same tone – this corresponds to taking the logarithm base 2 of frequencies (yielding the real numbers, as $$\mathbf{R} = \log_2 \mathbf{R}^+$$), then quotienting by the integers (corresponding to differing by some number of octaves), yielding a circle (as $$S^1 = \mathbf{R}/\mathbf{Z}$$).
 * Chords correspond to multiple tones without respect to order – thus t notes (with order) correspond to t ordered points on the circle, or equivalently a single point on the t-torus $$T^t := S^1 \times \cdots \times S^1,$$ and omitting order corresponds to taking the quotient by $$S_t,$$ yielding an orbifold.

For dyads (two tones), this yields the closed Möbius strip; for triads (three tones), this yields an orbifold that can be described as a triangular prism with the top and bottom triangular faces identified with a 120° twist (a $1⁄3$ twist) – equivalently, as a solid torus in 3 dimensions with a cross-section an equilateral triangle and such a twist.

The resulting orbifold is naturally stratified by repeated tones (properly, by integer partitions of t) – the open set consists of distinct tones (the partition $$t = 1 + 1 + \cdots + 1$$), while there is a 1-dimensional singular set consisting of all tones being the same (the partition $$t = t$$), which topologically is a circle, and various intermediate partitions. There is also a notable circle which runs through the center of the open set consisting of equally spaced points. In the case of triads, the three side faces of the prism correspond to two tones being the same and the third different (the partition $$3 = 2 + 1$$), while the three edges of the prism correspond to the 1-dimensional singular set. The top and bottom faces are part of the open set, and only appear because the orbifold has been cut – if viewed as a triangular torus with a twist, these artifacts disappear.

Tymoczko argues that chords close to the center (with tones equally or almost equally spaced) form the basis of much of traditional Western harmony, and that visualizing them in this way assists in analysis. There are 4 chords on the center (equally spaced under equal temperament – spacing of 4/4/4 between tones), corresponding to the augmented triads (thought of as musical sets) C♯FA, DF♯A♯, D♯GB, and EG♯C (then they cycle: FAC♯ = C♯FA), with the 12 major chords and 12 minor chords being the points next to but not on the center – almost evenly spaced but not quite. Major chords correspond to 4/3/5 (or equivalently, 5/4/3) spacing, while minor chords correspond to 3/4/5 spacing. Key changes then correspond to movement between these points in the orbifold, with smoother changes effected by movement between nearby points.

title: Orbifold


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Hangul consonant and vowel tables

The following tables of consonants and vowels (jamo) of the Korean alphabet (Hangul) display (in blue) the basic forms in the first row and their derivatives in the following row(s). They are divided into initials (leading consonants), vowels (middle), and finals tables (trailing consonants).

The jamo shown below are individually romanized according to the Revised Romanization of Hangeul (RR Transliteration), which is a system of transliteration rules between the Korean and Roman alphabets, originating from South Korea. However, the tables below are not sufficient for normal transcription of the Korean language as the overarching Revised Romanization of Korean system takes contextual sound changes into account.

Leading consonants
Called choseong, or "initials", there are 19 initial consonants, whereof one (ㅇ) is silent, and five (ㄲ, ㄸ, ㅃ, ㅆ, ㅉ) are doubled:

Medial vowels
Called jungseong, or "vowels", there are 21 medial vowels:

Trailing consonants
Called jongseong, or "finals", there are 27 final consonants; with the additional case of no final consonant, there is a total of 28 possibilities:

Collation
Several collation sequences are used to order words (like alphabetical sorting). The North and South differ on (a) the treatment of composite jamo consonants in syllable-leading (choseong) and -trailing (jongseong) position, and (b) on the treatment of composite jamo vowels in syllable-medial (jungseong) position.

This first sequence is official in South Korea (and is the basic binary order of codepoints in Unicode):

Sequences of this second type are common in North Korea:

Consonants
* Consonant names in the 15th century seem to have ended in a vowel (without adding the last consonant repeating a shortened version of the initial), judging from 1451 Hunmin Jeongeum Eonhae's forms such as "ㄱ는", which may have been pronounced geuneun.

Vowels
The "names" of the vowels are given according to the sound they make (their pronunciation). To be technical, the silent consonant would be added before the sound (e.g., ㅏ becomes 아).

Hangul syllables
With 19 possible initial consonants, 21 possible medial (one- or two-letter) vowels, and 28 possible final consonants (of which one corresponds to the case of no final consonant), there are a total of $19 × 21 × 28 = 11,172$ theoretically possible "Korean syllable letters" which are contiguously encoded in the 11,172 Unicode code points from U+AC00 (Decimal: ) through U+D7A3 (Decimal: = 44,032 + 11,171) within the Hangul Syllables Unicode block. However, the majority of these theoretically possible syllables do not correspond to syllables found in actual Korean words or proper names.

Jump to tables with initial letter:

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Hangul consonant and vowel tables


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Sukhavati

Sukhavati (IAST: Sukhāvatī; "Blissful") is the pure land of Amitābha in Mahayana Buddhism. It is also called the Land of Bliss or Western Pure Land and is the most well-known of Buddhist pure lands due to the popularity of Pure Land Buddhism in East Asia.

Etymology and names
The word is the feminine form of the Buddhist Hybrid Sanskrit word sukhāvat ("full of joy; blissful"), from sukha ("delight, joy") and -vat ("full of").

Sukhavati is known by different names in other languages. East Asian names are based on Chinese translations, and longer names may consist of the words "Western", "Blissful" and "Pure Land" in various combinations. Some names and combinations are more popular in certain countries. Due to its importance, Sukhavati is often simply called "The Pure Land" without distinguishing it from other pure lands.

* Only common in Chinese.

Nine levels of birth
In the final part of the Amitāyurdhyāna Sūtra, Gautama Buddha discusses the nine levels into which those born into the pure land are categorized. These are as follows:

 * 1) The highest level of the highest grade of rebirth: devoted followers of the Mahayana who have deep faith and engage in virtuous actions. Upon their death, they are welcomed by Amitābha and a multitude of sages. Once they are born in Sukhavati, they are immediately able to see Amitābha and hear his teachings, and immediately attain several high spiritual states.
 * 2) The middle level of the highest grade: those who have faith in the law of karma and are able to understand the Mahayana teachings. Upon their death, they are welcomed by Amitābha and a multitude of sages. Seven days after their rebirth in the pure land, they gain the stage of non-retrogression.
 * 3) The lower level of the highest grade: those who believe in the law of karma and develop bodhicitta. Upon their death, they are welcomed by Amitābha and a multitude of sages. They are enclosed in lotus buds for a single day, and after seven days they can see Amitābha and gradually gain enlightenment.
 * 4) The highest level of the middle grade: those who observe the five precepts and other precepts. Upon their death, they are welcomed by Amitābha and a multitude of sages. Once their lotus buds open, they are able to hear the Dharma and attain arhatship.
 * 5) The middle level of the middle grade: those who follow various Buddhist precepts, even for a single day. Upon their death, they are welcomed by Amitābha and a multitude of sages. Their lotus buds open after seven days, immediately becoming stream-enterers and attaining arhatship half a kalpa later.
 * 6) The lower level of the middle grade: those who do good deeds in their current life, such as being dutiful to their parents. Upon their death, they are welcomed by Amitābha and a multitude of sages. Their lotus buds open after seven days, and attain arhatship after a small kalpa.
 * 7) The highest level of the lower grade: those who commit various evil actions, but hear Mahayana sutras before they die and are instructed to recite the name of Amitābha. Upon their death, they are escorted by the transformed body of Amitābha and the transformed bodies of several bodhisattvas. They are confined in lotus buds for seven weeks, and when they are freed they hear the dharma from the two attendant bodhisattvas of Amitābha, Mahāsthāmaprāpta and Avalokiteśvara. After ten small kalpas, they attain the first bhūmi as bodhisattvas.
 * 8) The middle level of the lower grade: those who commit various offenses against the Dharma and the sangha, and break various precepts. When they are about to die and be reborn in the hell realms, they hear about Amitābha from a virtuous teacher. After they are reborn in the pure land, they are confined in lotus buds for six kalpas, and once they are freed they can finally hear the Mahayana teachings.
 * 9) The lower level of the lower grade: those who are true evildoers and commit the gravest offenses, which would inevitably let them be reborn in the lowest levels of hell. Before they die they meet a good teacher who encourages them to repeat the name of Amitābha. Once they have repeated his name ten times, their evil karma is extinguished, and they are able to see golden lotus pods at death. After twelve long kalpas, their lotus buds open, and they can finally hear the Mahayana teachings.

Buddhist funerals
In Tibetan Buddhism, the world of Sukhavati is invoked during Buddhist funerals as a favorable destination for the deceased. Such rituals are often accompanied with the tantric technique of phowa ("transference of consciousness") to the pure land of Amitābha, performed by a lama on the behalf of the departed. Halkias (2013:148) explains that "Sukhavati features in funeral rites and scriptures dedicated to the ritual care of the dead ('das-mchod). The structure and performance of Tibetan death ceremonies varies according to a set sequence of events...For the duration of these rites, the consciousness of the dead is coaxed into increasing levels of clarity until the time for the ritual transference to Sukhavati."

Raigō (来迎) in Japanese Buddhism is the appearance of the Amida on a "purple" cloud (紫雲) at the time of one's death. The most popular belief is that the soul would then depart to the Western Paradise. A number of hanging scroll paintings depict the western paradise.

Namesakes
A number of temples are named after Sukhāvatī:
 * Kek Lok Si, Malaysia
 * Kek Look Seah, Malaysia
 * Jile Temple, Harbin, China
 * Shinshōgokuraku-ji, Kyoto, Japan

title: Sukhavati


text:
D'Alembert operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: $$\Box$$), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (cf. nabla symbol) is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert.

In Minkowski space, in standard coordinates $(t, x, y, z)$, it has the form

\begin{align} \Box & = \partial^\mu \partial_\mu = \eta^{\mu\nu} \partial_\nu \partial_\mu = \frac{1}{c^{2}} \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} \\ & = \frac{1}{c^2} {\partial^2 \over \partial t^2} - \nabla^2 = \frac{1}{c^2}{\partial^2 \over \partial t^2} - \Delta. \end{align} $$

Here $$ \nabla^2 := \Delta $$ is the 3-dimensional Laplacian and $η^{μν}$  is the inverse Minkowski metric with
 * $$\eta_{00} = 1$$, $$\eta_{11} = \eta_{22} = \eta_{33} = -1$$, $$\eta_{\mu\nu} = 0$$ for $$\mu \neq \nu$$.

Note that the $μ$ and $ν$ summation indices range from 0 to 3: see Einstein notation.

(Some authors alternatively use the negative metric signature of (− + + +), with $$\eta_{00} = -1,\; \eta_{11} = \eta_{22} = \eta_{33} = 1$$.)

Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian yields a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.

The box symbol and alternate notations
There are a variety of notations for the d'Alembertian. The most common are the box symbol $$\Box$$ (Unicode: ) whose four sides represent the four dimensions of space-time and the box-squared symbol $$\Box^2$$ which emphasizes the scalar property through the squared term (much like the Laplacian). In keeping with the triangular notation for the Laplacian, sometimes $$\Delta_M$$ is used.

Another way to write the d'Alembertian in flat standard coordinates is $$\partial^2$$. This notation is used extensively in quantum field theory, where partial derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian.

Sometimes the box symbol is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol $$\nabla$$ is then used to represent the space derivatives, but this is coordinate chart dependent.

Applications
The wave equation for small vibrations is of the form
 * $$ \Box_{c} u\left(x,t\right) \equiv u_{tt} - c^2u_{xx} = 0~,$$

where $u(x, t)$ is the displacement.

The wave equation for the electromagnetic field in vacuum is
 * $$ \Box A^{\mu} = 0 $$

where $A^{μ}$ is the electromagnetic four-potential in Lorenz gauge.

The Klein–Gordon equation has the form
 * $$\left(\Box + \frac{m^2c^2}{\hbar^2}\right) \psi = 0~.$$

Green's function
The Green's function, $$G\left(\tilde{x} - \tilde{x}'\right)$$, for the d'Alembertian is defined by the equation
 * $$ \Box G\left(\tilde{x} - \tilde{x}'\right) = \delta\left(\tilde{x} - \tilde{x}'\right)$$

where $$\delta\left(\tilde{x} - \tilde{x}'\right)$$ is the multidimensional Dirac delta function and $$\tilde{x}$$ and $$\tilde{x}'$$ are two points in Minkowski space.

A special solution is given by the retarded Green's function which corresponds to signal propagation only forward in time
 * $$G\left(\vec{r}, t\right) = \frac{1}{4\pi r} \Theta(t) \delta\left(t - \frac{r}{c}\right)$$

where $$\Theta$$ is the Heaviside step function.

title: D'Alembert operator


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Moral clarity

Moral clarity is a catchphrase associated with American political conservatives. Popularized by William J. Bennett's Why We Fight: Moral Clarity and the War on Terrorism, the phrase was first used in its current context during the 1980s, in reference to the politics of Ronald Reagan.

The phrase moral clarity encodes a complex political argument that includes all of the following claims:

 * The War on Terrorism, like some previous wars involving the United States (particularly World War II and the Cold War), is a conflict between good and evil.
 * Traditional American values like democracy and freedom are universal human rights, worth promoting and defending through military intervention.
 * Attempts to understand or explain the actions of anti-Western terrorists as justifiable responses to actions of the United States or Israel are a sign of moral weakness at best, and sympathy for the terrorists at worst, and will hamper efforts to defeat them.
 * Though the actions of the United States and its allies may lead to civilian deaths or other forms of collateral damage, may require the use of means such as torture that would be condemned in other contexts, and may involve temporary alliances with undemocratic regimes, these actions are justified by the greater moral necessity of defeating terrorism and thus promoting American values and ensuring long-term U.S. security.
 * Opponents of action against terrorists are guilty of promoting moral relativism or moral equivalence, in which the allegedly similar means of both anti-terrorists and terrorists are used to blur the moral differences between good and evil.

Opposing views
For opponents of the notion of "moral clarity", dividing the world into good and evil does not lend itself to a workable foreign policy. For example, if Iraq was invaded for reasons of "moral clarity", it follows that other "rogue states" (like North Korea) would also be similarly attacked. This apparent contradiction is used to argue that proponents of "moral clarity" are guilty of hypocrisy or special pleading, and that the slogan moral clarity masks less exalted reasons for military intervention, particularly economic motives.

Opponents of the notion of "moral clarity" also note that organizations such as al Qaida and neo-Nazis also believe that their own point of view is one of "moral clarity", which justifies them to take whatever action they feel necessary against what they see as the "forces of evil". Thus, opponents of the notion of "moral clarity" assert that, but for the accident of their country and culture of birth, the proponents of "moral clarity" might well be on the opposite side, holding the same views. Proponents of this view argue that understanding the motivations and values of other cultures, rather than labeling them "evil", is a better first step in eliminating violence.

Critics also argue that "moral clarity" promotes a dangerous view expressed in the famous phrase "My country, right or wrong". The idea that the United States is always "good" is actually an argument of moral relativism, they say, since it makes no distinction between right and wrong actions. Furthermore, it is argued that "moral clarity" is used to discredit those who want to hold the United States to a higher moral standard than "My country, right or wrong", a standard expressed by U.S. Brigadier General Carl Schurz: "Our country right or wrong. When right, to be kept right; when wrong, to be put right."

title: Moral clarity


text:
Bad Boys

Bad Boys may refer to:

Film

 * Bad Boys (1961 film), a Japanese film by Susumu Hani
 * Bad Boys (1983 film), an American crime drama starring Sean Penn
 * Bad Boys (franchise), the American action-comedy series starring Will Smith and Martin Lawrence, it consists of:
 * Bad Boys (1995 film), an American action film
 * Bad Boys II, a 2003 sequel to the 1995 film
 * Bad Boys for Life, a 2020 sequel to the 2003 film
 * Bad Boys: Ride or Die, a 2024 sequel to the 2020 film
 * Bad Boys (soundtrack), a soundtrack album from the 1995 film
 * Bad Boys: Miami Takedown, a 2004 video game based on the sequel
 * Bad Boys (2003 film), a Finnish crime drama starring Peter Franzén
 * Bad Boys, a 2003 Indian Hindi film starring Rakhi Sawant

Television

 * Bad Boyes, a 1980s British children's television series
 * "Bad Boys" (The IT Crowd), a 2010 episode of series 4 of The IT Crowd

Albums

 * Bad Boys (Baccara album), or the title song, 1981
 * Bad Boys (DeBarge album), 1987
 * Bad Boys (Haywire album), 1986

Songs

 * "Bad Boys" (Alexandra Burke song), 2009
 * "Bad Boys" (Inner Circle song), 1987, used as the theme song of the TV show Cops and in the 1995 film Bad Boys
 * "Bad Boys" (Roxus song), 1991
 * "Bad Boys" (Wham! song), 1983
 * "Bad Boys" (Zara Larsson song), 2013.
 * "Bad Boys", a song by Shyne from Shyne
 * "Bad Boys", a song by Whitesnake from Whitesnake
 * "Bad Boys", a song by Victoria Justice from Victorious 3.0
 * "Bad Boys", a song by Geetha Madhuri and Priya Himesh from Businessman
 * "Bad Bad Boys", a song by Midi, Maxi & Efti

Other uses

 * Bad Boys (basketball), a nickname for the 1980-1994 Detroit Pistons of the NBA
 * Bad Boys (manga), a manga series by Hiroshi Tanaka
 * Bad Boys (poetry collection), a 1980 book by Sandra Cisneros.
 * Bad Boys, a Japanese owarai duo who host the variety show AKBingo!

title: Bad Boys


text:
Poros

Poros (Πόρος; ) is a small Greek island-pair in the southern part of the Saronic Gulf, about 58 km south of the port of Piraeus and separated from the Peloponnese by a 200 m wide sea channel, with the town of Galatas on the mainland across the strait. Its surface area is about 31 km2 and it has 3,261 inhabitants (2021). The ancient name of Poros was Pogon. Like other ports in the Saronic, it is a popular weekend destination for Athenian travellers.

Poros consists of two islands: Sphaeria (Σφαιρία, ), the southern part, which is of volcanic origin, where today's city is located, and Kalaureia (Καλαυρία, ), also Kalavria or Calauria (meaning 'gentle breeze'), the northern and largest part. A bridge connects the two islands over a narrow strait.

Poros is an island with rich vegetation. Much of the northern and far eastern/western sides of the island are bushy, whereas large areas of old pine forest are found in the south and center of the island. It has a good road network and adequate tourist infrastructure, which makes it a popular resort for short holidays.

Geography
The land area of the municipality is 49.582 km². This includes the islands and part of the adjacent Peloponnese coast, adjacent to Galatas, Thermisia and the island of Hydra. The landscape is very hilly and mountainous. The highest peak is the Vigla (358 m) in the west-central part. Following the island's topography and geology, small creeks and seasonal streams flow through steep valleys of the southern and northeastern part. The western and northern part of the island feature smooth hills and shallow valleys. Sandy beaches are restricted to the southern shore of the island, except for a bay in the northern part called Vayionia.

Poros contains the main village Poros and the smaller villages Ágios Nektários and Kyaní Aktí.

Geology
The geology of the island comprises Mesozoic to Cenozoic sedimentary rocks (limestone, and Flysch-type sandstone) and ophiolites, as well as Neogene volcanic rocks on Sferia. The island is tectonically dissected and part of a Tertiary tectonic mélange. There are karstic sinkholes in the island's central limestone massif, and limestone caves with stalactites. Visible marine fossils are mainly found in the limestone, no occurrences of precious stones or ore deposits are known.

Limestone from Poros, an example of the poros stone, was an important component in the construction of Mycenaean tholos tombs in the Late Bronze Age, being used particularly for monuments at Mycenae, including the Tomb of Aegisthus.

History

In the northeastern part of the island, in a location called “Kavos Vasili”, the archaeologists have discovered the ruins of a settlement of the Early Bronze Age. This settlement is the oldest of the wider area of Trizinia, Peloponnese, and is believed to be interrelated with the wreck found on the nearby Dokos island which dates to the same period.

Antiquity
Poros was divided into two islands during antiquity: Sphairia (also known by its modern Greek form Sfairia) and Calauria (also known by its ancient Greek form Kalaureia and its modern Greek form Kalavria). Sphairia consisted of the area of the modern island which includes its current capital. Calauria was the larger part of the island to its north. During the period of Mycenaean dominance (1400-1100 BC) Calauria was quite powerful and the most important naval base of region was located on islet Monti or Liontari on its eastern coast. The city-state of Calauria was home to an asylum dedicated to Poseidon, the ruins of which are still accessible on a hilltop close to the town. This asylum may have been linked to the sanctuaries at Geraistos and Tainaros. Ancient historians stated that Calauria was the center of an amphictyony, a religious alliance between its city-state and those of Athens, Poros, Aegina, Epidaurus, Hermione, Troezen, Nauplio, Orchomenus, and Prasaiai. Modern archaeology has discovered no evidence for its actual existence, however, and now believe the "Calaurian League" to have been a later Hellenistic invention. An enormous feast was found dating to the Hellenistic period in the ruins of the Calaurian asylum, along with a plaque celebrating the "revival" of its amphictyony.

During the 5th century, the Persian Empire annexed the Greek cities of western Anatolia. When the Athenians helped these cities in a revolt against Persian rule, Poros, along with the other Greek states of mainland Greece and the Aegean islands, was drawn into the Greco-Persian Wars. At the beginning of the Peloponnesian War, Troezen and Calauria offered asylum to an anti-Macedonian politician who eventually became the tyrant of the region. After the death of Alexander the Great in 323 BC, the Ptolemies of Egypt occupied Calauria. Around the same time, the Athenian orator Demosthenes came to the island, where some report his suicide. In 273 BC, the last explosion of the Methana volcano dramatically changed the morphology of Poros and the wider region.

The Sanctuary of Poseidon has been excavated by Swedish archaeologists. The 1894 field season is considered to be the first Swedish archaeological campaign in Greece. Swedes have continued excavations from 1997 in collaboration with the Greek National Heritage Board.

Roman and Byzantine times
During the Roman period (86 BC to 395 AD) Poros was part of the Roman Empire along with Trizina, to which it was a tributary. In Byzantine times, Poros and other islands were often raided by the pirates that dominated the Aegean Sea.

Venetian period
In 1484 the Venetians occupied Poros and used it as a strategic port in their sea battles with the Ottomans. Poros was the most powerful city of the wider area, also governing Methana Island, Epidaurus, Damalas (Trizina), Fanari and Valario. During that time, the island had about 15,000 inhabitants, making it one of the largest cities in Greece. Arvanites were recorded among the inhabitants of the island in 1688. They lived in poverty as most of the Albanians in Greece at the time. Venetian rule ended in 1715.

Ottoman period
The Ottoman Period began in 1715, much later in Poros than in the rest of Greece. Shipping and commerce were the inhabitants' main activities, but Poros' fleet wasn't as famous as Hydra's or Spetses' fleet, and did not participate in many sea battles.

The Greek Revolution
Poros had an important role during the Greek Revolution in 1821, due to its strategic position. The Greek revolutionary leaders, often met in Poros to discuss and plan their future actions. The first Greek naval base was established in Poros in 1828 and remained there until 1878. In September 1828, the ambassadors of England, France and Russia met in Poros with Ioannis Kapodistrias in order to determine the borders of the future Greek state, which was established two years later, in 1830.

Russian naval base

With the Treaty of Kuchuk Kainarji, Russia secured free shipping for its navy, war and merchant alike, throughout the waters of the Ottoman Empire. As Russian naval activity grew, need arose for a supply station, and land was acquired at the edge of Poros town. Extensive materiel, coal, and food storage facilities were built, as well as a hardtack baking factory. After Greek independence, Governor Capodistrias requisitioned the facilities for use of the Greek war navy, and offered the Russians an alternative location in a nearby cove. The new facilities were far larger, and were used by Russian ships throughout the 19th century. The number of Russian residents of Poros increased and even a Russian school was established. Then as Russian naval activity declined, so did the base and by the early 20th century only a single Russian watchman was left guarding it. It was then granted to the Greek Navy by the Czar but was never put to actual use, and the abandoned buildings were left to decay. The ruins, in elaborately carved stone, were listed as protected architectural monuments in 1989.

Twentieth century
In the beginning of the 20th century, among the activities of the Poros' inhabitants were agriculture (mainly wheat, grapevines and olives), livestock, fishing and shipping.

Main sights

 * The town of Poros, with its neoclassical edifices, is built amphitheatrically on the slopes of a hill. Its most famous landmark is a clock tower built in 1927.
 * The Archaeological Museum of Poros, at Korizis Square, houses findings from the Sanctuary of Poseidon, from ancient Troizen, and from other archaeological sites nearby.
 * In the northern part of the island are the remains of the Sanctuary of Poseidon, the centre of the Kalaureian amphictyony. The exact date it was built is not known, although researchers estimate it to have been around 520 BC. The dimensions of the temple, which is of the Doric order, are 27.4×14.4 m. There are six columns on each short side and twelve on each long side. It was here that Demosthenes, the famous orator, poisoned himself with hemlock in 322 BC fleeing from the Macedonian Governor Antipatros.
 * Poros was the site of the first naval base in modern Greece, established in 1827 during the Greek War of Independence. Most of the activities of Poros naval base were moved to Salamis Naval Base in 1881. The site is still used today by the Hellenic Navy as a training centre for naval personnel.
 * Bourtzi castle. It was built by von Heideck at around 1828, at the place where an old church of St. Constantine had stood. The island was given to the Poros monastery from 1770 on, by Antonios Serfos. The stand-off between Hydra island and Capodistrias was the cause for heavy fighting here in 1831.
 * Poros Shell Museum. Founded in 2006, a permanent exhibition, Shells from the Greek Seas, is displayed at the Chatzopouleios Library. More info can be found on the webpage.
 * The Holy Monastery of Zoodochos Pigi, founded in 1720 AD, is located 4 km east of the main city of Poros island. It is built on the slope of a pine forest. In 1814, a group of monks from Mount Athos took refuge in this monastery. A few years later, these monks founded the Zoodochos Pigi Monastery at Longovarda on Paros. Katholiko is a basilica with a dome and a tower-like belfry being the main church of the monastery.

Notable people

 * Alexandros Koryzis (1885–1941), Prime Minister of Greece. A museum is Poros is dedicated to his life.
 * Minos Kyriakou (1942–2017), Greek billionaire businessman
 * Henry Miller (1891–1980), American writer wrote about Poros in The Colossus of Maroussi: "I don't know which affected me more deeply-the story of the lemon groves just opposite us or the sight of Poros itself when suddenly I realized we were sailing through the streets. If there is one dream which I like above all others it is that of sailing on land.  Coming into Poros gives the illusion of the deep dream.  Suddenly the land converges on all sides and the boat is squeezed into a narrow strait from which there is no egress.  The men and women of Poros are hanging out of the windows just above your head.  You pull in right under their frosty nostrils, as though for a shave and a haircut en route.  The loungers on the quay are walking with the same speed as the boat.  They can walk faster than the boat if they choose to quicken their pace."
 * Other prominent literary people have written about the beauty of Poros such as the Nobel laureate Giorgos Seferis, Kosmas Politis, Kostis Palamas, and Ιoulia Dragoumi.

Transportation
Poros is linked to mainland Greece by a car ferry to Galatas. Poros has no airport but is easily accessible from Athens by ferry or hydrofoil.

There is local bus service on the island from Poros harbor to Neorio and Monastiri.

A bike sharing system named Cyclopolis is also available, with one station and a capacity of 13 bikes. Users can register for a small fee and rent bikes for free for up to 3 hours per rental.

title: Poros


text:
Beetlejuice

Beetlejuice is a 1988 American fantasy comedy horror film directed by Tim Burton from a screenplay by Michael McDowell and Warren Skaaren based on a story by McDowell and Larry Wilson. The film stars Alec Baldwin, Geena Davis, Jeffrey Jones, Catherine O'Hara, Winona Ryder and Michael Keaton as the titular character. The first installment of the Beetlejuice franchise, the plot revolves around a recently deceased couple. As ghosts, they are not allowed to leave their house. They contact Betelgeuse, a charismatic "bio-exorcist" to scare the house's new inhabitants away.

Beetlejuice was released in the United States on March 30, 1988, by Warner Bros. The film was a critical and commercial success, grossing $74.7 million on a $15 million budget. It won the Academy Award for Best Makeup and a trio of Saturn Awards: Best Horror Film, Best Makeup and Best Supporting Actress for Sylvia Sidney. The film's success spawned an animated television series, video games and a 2018 stage musical.

A sequel, Beetlejuice Beetlejuice, is scheduled for theatrical release on September 6, 2024.

Plot
In Winter River, Connecticut, married couple Barbara and Adam Maitland decide to spend their vacation decorating their idyllic country home. Attempting to avoid a dog while driving home from a trip into town, their car crashes through the side of its covered bridge and plunges into the namesake river below. After mysteriously returning home, Adam attempts to leave the house, but ends up in a strange and otherworldly desert-like landscape populated by enormous "sandworms". The encounter lasts only a few seconds as Barbara rescues him, stating he was gone for two hours. They then discover that they now lack reflections and have received a Handbook for the Recently Deceased, which leads them to realize they drowned in the river and are now ghosts confined to their house.

It is later sold to the Deetz family, who hail from New York City. Its patriarch, Charles, is a former real estate developer; his second wife, Delia, is a sculptor and conceptual artist; and his teenage goth daughter, Lydia, from his first marriage, is an aspiring photographer. Under the guidance of interior designer Otho Fenlock, the family renovates the house into a new-wave work of postmodern art.

Consulting the Handbook, the Maitlands travel to an otherworldly waiting room populated by other distressed souls, discovering that the afterlife is structured according to a complex bureaucracy involving vouchers and caseworkers. The Maitlands' own caseworker, Juno, tells them they must remain in the house for the next 125 years on pain of a dire fate. If they want the Deetzes out of the house, it is up to the Maitlands to scare them away. Although they are invisible to most of the Deetzes, Lydia is able to see the Maitlands; she attributes her paranormal intuition to her "strange and unusual" nature. Against Juno's advice, the Maitlands contact the miscreant Betelgeuse, her former assistant and a now-freelance "bio-exorcist", to scare the Deetzes away. However, he quickly offends the couple with his crude and morbid demeanor and they reconsider hiring him, but too late to stop him from tormenting the Deetzes.

Winter River's charm and the supernatural events inspire Charles to pitch his boss, Maxie Dean, on transforming the town into a tourist hot spot, but Maxie wants proof of the ghosts. Using the Handbook for the Recently Deceased, Otho conducts what appears to be a séance and summons the Maitlands using their wedding clothes, but they begin to age and decay, as an exorcism has been inadvertently performed instead. Horrified, Lydia summons Betelgeuse for help, but he will only help her if she marries him; marrying a human would enable Betelgeuse to freely cause chaos in the mortal world. Betelgeuse saves the Maitlands, disposes of Maxie, his wife and Otho, then prepares his wedding with Lydia. The Maitlands intervene before the ceremony is completed, with Barbara riding a sandworm that swallows Betelgeuse whole through the house.

The Deetzes and Maitlands agree to live in harmony within the house, with the latter forming a stronger bond with Lydia, who occasionally persuades them to unleash their ghostly powers. Elsewhere, Betelgeuse is now stuck in the afterlife waiting room; he runs afoul of a witch doctor, who shrinks his head in retaliation.

Writing
After the financial success of Pee-wee's Big Adventure (1985), Burton became a "bankable" director and began working on a script for Batman with Sam Hamm. While Warner Bros. was willing to pay for the script's development, it was less willing to green-light Batman. Burton had become disheartened by the lack of imagination and originality in the scripts he had been sent, particularly Hot to Trot.

Michael McDowell and Larry Wilson formed a partnership (Pecos Productions) with entertainment attorney Michael Bender, and Beetlejuice was their first original project. After developing the story, McDowell and Wilson decided they would write the first draft of the screenplay together, while Wilson would only take 'Story By' credit, as well as his 'Producer' credit.

Burton had gotten to know and worked with McDowell and Wilson (who co-wrote the script for "The Jar", an episode of Alfred Hitchcock Presents that Burton directed). Burton read their first draft of Beetlejuice, liked it but had other projects that kept him from becoming involved at that time.

The original script is far less comedic and much darker; the Maitlands' car crash is depicted graphically, with Barbara's arm crushed and the couple screaming for help as they slowly drown. A reference to this remains: Barbara remarks that her arm feels frozen upon returning home as a ghost. Instead of possessing the Deetzes and forcing them to dance during dinner, the Maitlands cause a vine-patterned carpet to come to life and attack them by tangling them to their chairs.

The character of Betelgeuse—envisioned in the first draft as a winged demon who takes on the form of a short Middle Eastern man—is also intent on killing the Deetzes rather than scaring them and wants sex from Lydia instead of marriage. In this version of the script, Betelgeuse need only be exhumed from his grave to be summoned, after which he is free to wreak havoc; he can be summoned, but not controlled, by saying his name three times and wanders the world freely, tormenting different characters in different manifestations.

In another version of the script, the film concludes with the Maitlands, Deetzes, and Otho conducting an exorcism ritual that destroys Betelgeuse and the Maitlands, transforming into miniature versions of themselves and moving into Adam's model of their home, which they refurbish to look like their house before the Deetzes moved in.

Co-author and producer Larry Wilson has talked about the reaction to the first draft by a prominent executive at Universal, where Wilson was employed at the time:

"I won't name names here, but I worked at Universal Studios at the time. I was director of development for the director Walter Hill. I had a very good relationship with a very prominent executive at Universal. He liked me, and he liked what I was doing with Walter, and the material I was bringing in.

I gave him Beetlejuice to read, and I gave it to him on a Friday, and on Monday his assistant called me and said 'well, he wants to meet with you'. My initial reaction was 'wow! He'd read it. He must have loved it or he wouldn't have wanted to see me so soon.' But I went into his office, and he literally said, 'what are you doing with your career?'

'This piece of weirdness, this is what you're going to go out into the world with? You're developing into a very good executive. You've got great taste in material. Why are you going to squander all that for this piece of shit' was basically what he was saying. It goes to show, right? Shortly after that, we sold it to the Geffen Company."

Skaaren's rewrite shifted the film's tone, eliminating the graphic nature of the Maitlands' deaths and further developing the concept created by McDowell and Wilson that the Afterlife is a complex bureaucracy. Skaaren's rewrite also added to McDowell and Wilson's depiction of the limbo that keeps Barbara and Adam trapped inside their home; in the original script, it takes the form of a massive void filled with giant clock gears that shred the fabric of time and space as they move. Skaaren had Barbara and Adam encounter different limbos every time they leave their home, including the "clock world" and the sandworm world, identified as Saturn's moon Titan. Skaaren also introduced the leitmotif of music accompanying Barbara and Adam's ghostly hijinks, although his script specified R&B tunes instead of Harry Belafonte and was to have concluded with Lydia dancing to "When a Man Loves a Woman".

Skaaren's first draft retained some of McDowell's Betelgeuse's more sinister characteristics but toned the character down to make him a troublesome pervert rather than blatantly murderous. Betelgeuse's true form was that of the Middle Eastern man, and much of his dialogue was written in African-American Vernacular English. This version concluded with the Deetzes returning to New York and leaving Lydia in the care of the Maitlands, who, with Lydia's help, transform their home's exterior into a stereotypical haunted house while returning the interior to its previous state. It also featured deleted scenes such as the real estate agent, Jane, trying to convince the Deetzes to allow her to sell the house for them (having sold it to them in the first place—Charles and Delia decline) and a revelation of how Betelgeuse had died centuries earlier (he attempted to hang himself while drunk—having been rejected by a woman—only to mess it up and die slowly by choking to death rather than quickly by snapping his neck) and wound up working for Juno before striking out on his own as a "freelance bio-exorcist".

Retrospectively, McDowell was impressed with how many people made the connection between the film's title and the star Betelgeuse.

Casting
Burton's original choice for Betelgeuse was Sammy Davis Jr. The producers also considered Dudley Moore and Sam Kinison for the role, but Geffen suggested Keaton. Burton was unfamiliar with Keaton's work, but was quickly convinced. Several actresses auditioned for the role of Lydia Deetz, including Sarah Jessica Parker, Brooke Shields, Lori Loughlin, Diane Lane, Justine Bateman, Molly Ringwald, Juliette Lewis, and Jennifer Connelly. Alyssa Milano was the runner-up for the role. Burton cast Ryder upon seeing her in Lucas. Anjelica Huston was originally cast as Delia Deetz but dropped out because of illness. O'Hara quickly signed on, while Burton claimed it took a lot of time to convince other cast members to sign, as "they didn't know what to think of the weird script".

Filming
Beetlejuice's budget was $15 million, with just $1 million given over to visual effects work. Considering the scale and scope of the effects, which included stop motion, replacement animation, prosthetic makeup, puppetry and blue screen, it was always Burton's intention to make the style similar to that of the B movies he grew up with as a child. He said that he wanted to make the effects look cheap and purposely fake-looking. Burton wanted to hire Anton Furst as production designer after being impressed with his work on The Company of Wolves (1984) and Full Metal Jacket (1987), but Furst was committed to High Spirits, a choice he later regretted. He hired Bo Welch, his future collaborator on Edward Scissorhands and Batman Returns. The test screenings were met with positive feedback and prompted Burton to film an epilogue featuring Betelgeuse foolishly angering a witch doctor. Warner Bros. disliked the title Beetlejuice and wanted to call the film House Ghosts. As a joke, Burton suggested the name Scared Sheetless and was horrified when the studio actually considered using it. While the setting is the fictional village of Winter River, Connecticut, all outdoor scenes were filmed in East Corinth, a village in the town of Corinth, Vermont. Interiors were filmed at The Culver Studios in Culver City, California. Principal photography took place from March 11 to June 11, 1987.

Soundtrack
The Beetlejuice soundtrack, first released in 1988 on LP, CD, and cassette tape, features most of the film's score, written and arranged by Danny Elfman. Geffen reissued the original 1988 soundtrack on vinyl in 2015, which was remastered and pressed to vinyl by Waxwork Records in 2019 for the film's 30th anniversary. The soundtrack features two original recordings performed by Harry Belafonte used in the film: "Day-O (The Banana Boat Song)" and "Jump in the Line (Shake, Senora)". Two other vintage Belafonte recordings that appear in the film are absent from the soundtrack: "Man Smart, Woman Smarter" and "Sweetheart from Venezuela". The soundtrack entered the Billboard 200 albums chart the week ending June 25, 1988, at No. 145, peaking two weeks later at No. 118 and spending a total of six weeks on the chart. This was after the film had already fallen out of the top 10 and before the video release in October. "Day-O" received a fair amount of airplay at the time in support of the soundtrack.

The complete score (with the Belafonte tracks included) was released in both the DVD and the Blu-ray as an isolated music track in the audio settings menu; this version of the audio track consists entirely of "clean" musical cues, uninterrupted by dialogue or sound effects.

Box office
Beetlejuice opened theatrically in the United States on March 30, 1988, earning $8,030,897 its opening weekend. The film eventually grossed $74,664,632 in North America. Beetlejuice was a financial success, recouping its $15 million budget, and the 10th-highest grossing film of 1988.

Critical response
Beetlejuice was met with a mostly positive response. Based on reviews collected by Rotten Tomatoes, Beetlejuice received an  overall approval rating with a weighted average of. The website's critical consensus reads, "Brilliantly bizarre and overflowing with ideas, Beetlejuice offers some of Michael Keaton's most deliciously manic work—and creepy, funny fun for the whole family." On Metacritic, the film has a weighted average score of 70 out of 100, based on 18 reviews. Audiences surveyed by CinemaScore gave the film a B on a grade scale of A to F.

Pauline Kael called the film a "comedy classic", while Jonathan Rosenbaum of Chicago Reader gave a highly positive review. Rosenbaum felt Beetlejuice had originality and creativity absent from other films. Vincent Canby of The New York Times called it "a farce for our time" and wished Keaton had more screen time. Desson Howe of the Washington Post felt Beetlejuice had the "perfect" balance of bizarreness, comedy and horror.

Janet Maslin of the New York Times gave the film a negative review, writing that the film "tries anything and everything for effect, and only occasionally manages something marginally funny" and "is about as funny as a shrunken head". Roger Ebert gave the film two out of four stars, writing that he "would have been more interested if the screenplay had preserved their [Alec Baldwin and Geena Davis] sweet romanticism and cut back on the slapstick". Ebert called Keaton "unrecognizable behind pounds of makeup" and said "his scenes don't seem to fit with the other action".

In his book Comedy-Horror Films: A Chronological History, 1914-2008, Bruce G. Hallenbeck praised the film's lively script, assured direction, offbeat casting, and "delightfully off-kilter, Edward Gorey-like look", citing the explorer with the shrunken head and the animated sandworm as particularly memorable visuals.

Accolades
At the 61st Academy Awards, Beetlejuice won the Academy Award for Best Makeup (Steve La Porte, Ve Neill, and Robert Short), while the British Academy of Film and Television Arts nominated the film for Best Visual Effects and Makeup at the 42nd British Academy Film Awards.

Beetlejuice won Best Horror Film and Best Make-up at the 1988 Saturn Awards. Sidney also won the Saturn for Best Supporting Actress, and the film received five other nominations: Direction for Burton, Writing for McDowell and Skaaren, Best Supporting Actor for Keaton, Music for Elfman, and Special Effects. Beetlejuice was nominated for the Hugo Award for Best Dramatic Presentation. Beetlejuice was 88th in the American Film Institute's list of Best Comedies.

Sequel
A sequel, Beetlejuice Beetlejuice, produced by Brad Pitt's studio Plan B Entertainment alongside Warner Bros., with Michael Keaton, Winona Ryder, and Catherine O'Hara reprising their roles, is scheduled for release on September 6, 2024.

Video rental
On March 10, 1998, Beetlejuice became the first of more than 5.2 billion DVDs shipped by Netflix, which launched as a mail-based rental business.

title: Beetlejuice


text:
Talk:Information technology

Remove the History of Computer Technology section
The very first section of the article following the abstract description is "1 History of Computer Technology". I question the relevance to Information Technology and think it should belong in a subsection of Computing instead.

At the very least, the content should be reduced to a summary and focus on the relationship between Computer Science and IT, while directing the reader to the other article. For those readers who are clearly unable to tell the difference between IT and Computer Science or even Human–computer interaction, a redirecting About-reference should be introduced.

Personally, I think the section should be removed entirely as Information Technology is something else entirely separate from Computer Technology. Vitdom (talk) 11:40, 19 October 2022 (UTC)

 * That was my first thought when reading this article. Frankly, the whole article may need to be trashed; it's incoherent, and doesn't seem to be directly related to information technology. The EDP section is not awful, but we never explain how it fits in the topic, and much of the information likely belongs in the main article instead. DFlhb (talk) 18:25, 3 January 2023 (UTC)
 * The world needs to know it's Information Science and Computation Technology, not the Information Technology and Computer Science. Science allow us build, Technology make us build more, build faster and build better. We are the same, what we build is never. Science is the cause like the Information and Computing is the need like the Technology. Information is the cause why we need computers.[ Do not delete without informing me] 157.34.92.47 (talk) 05:51, 18 November 2023 (UTC)

Proposed merge of IT infrastructure into Information technology
wide overlap. fgnievinski (talk) 06:06, 28 March 2023 (UTC)
 * Oppose: there is some overlap, but the importance of the topic of information technology warrants coverage over several pages with daughter articles. IT infrastructure can be viewed as one such article. Klbrain (talk) 11:57, 16 December 2023 (UTC)

"👩💻" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=%F0%9F%91%A9%E2%80%8D%F0%9F%92%BB&redirect=no 👩💻] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. Hey man im josh (talk) 12:58, 25 July 2023 (UTC)

"👨💻" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=%F0%9F%91%A8%E2%80%8D%F0%9F%92%BB&redirect=no 👨💻] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. Edward-Woodrow :) [ talk ] 20:46, 22 August 2023 (UTC)

Wiki Education assignment: Research Process and Methodology - FA23 - Sect 202 - Thu
— Assignment last updated by EdereOmnes (talk) 18:22, 17 October 2023 (UTC)

Wiki Education assignment: Research Process and Methodology - SP24 - Sect 201 - Thu
— Assignment last updated by Xl112233 (talk) 12:49, 25 April 2024 (UTC)

title: Talk:Information technology