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Finite Subdivision Rules
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule. Definition A subdivision rule takes a tiling of the plane by polygons and turns it into a new tiling by subdividing each polygon into smaller polygons. It is finite if there are only finitely many ways that every polygon can subdivide. Each way of subdividing a tile is called a tile type. Each tile type is represented by a label (usually a letter). Every tile type subdivides into small ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Orthogonal Dodecahedral Honeycomb
Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola * of or pertaining to hyperbole, the use of exaggeration as a rhetorical device or figure of speech * Hyperbolic (album), ''Hyperbolic'' (album), by Pnau, 2024 See also * Exaggeration * Hyperboloid {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penrose Dart 2
Penrose may refer to: Places United States * Penrose, Arlington, Virginia, a neighborhood * Penrose, Colorado, a town * Penrose, St. Louis, Missouri, a neighborhood * Penrose, Philadelphia, Pennsylvania, a neighborhood * Penrose, North Carolina, an unincorporated community * Penrose, Utah, an unincorporated community * Penrose, Virginia, an historic district in Arlington County Elsewhere * Penrose, New South Wales (Wingecarribee), Australia * Penrose, New South Wales (Wollongong), New South Wales, Australia * Penrose, Cornwall, a country house and National Trust estate in England * Penrose, New Zealand * Penrose Peak (other) * Penrose railway station (other) People * Penrose Stout (1887–1934), American architect * Penrose Hallowell (c. 1928–2021), Pennsylvania secretary of agriculture * Penrose (surname), including a list of people with the name Other uses * Penrose (brand), a brand name owned by ConAgra Foods, Inc. * '' The Penrose Annual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any Loop (topology), loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the 3-sphere). Let ''N'' be a tubular neighborhood of ''K''; so ''N'' is a solid torus. The knot complement is then the complement of ''N'', :X_K = M - \mbox(N). The knot complement ''XK'' is a compact 3-manifold; the boundary of ''XK'' and the boundary of the neighborhood ''N'' are homeomorphic to a two-torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu .... Sometimes the ambient manifold ''M'' is understood to be the 3-sphere. Context is needed to determine the usage. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattès Map
In mathematics, a Lattès map is a rational map ''f'' = Θ''L''Θ−1 from the complex sphere to itself such that Θ is a holomorphic map from a complex torus In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ... to the complex sphere and ''L'' is an affine map ''z'' → ''az'' + ''b'' from the complex torus to itself. Lattès maps are named after French mathematician Samuel Lattès, who wrote about them in 1918. References * * {{DEFAULTSORT:Lattes map Dynamical systems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are Irreducible component, irreducible. Definition Formal definition Formally, a rational map f \colon V \to W between two varieties is an equivalence class of pairs (f_U, U) in which f_U is a morphism of varieties from a Empty set, non-empty open set U\subset V to W, and two such pairs (f_U, U) and (_, U') are considered equivalent if f_U and _ coincide on the intersection U \cap U' (this is, in particular, vacuous truth, vacuously true if the intersection is empty, but since V is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma: * If two morphisms of varieties are equal on some non-empty open set, then they are equal. f is said to be dominant if one (equivalently, every) representative f_U in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penrose Star 3
Penrose may refer to: Places United States * Penrose, Arlington, Virginia, a neighborhood * Penrose, Colorado, a town * Penrose, St. Louis, Missouri, a neighborhood * Penrose, Philadelphia, Pennsylvania, a neighborhood * Penrose, North Carolina, an unincorporated community * Penrose, Utah, an unincorporated community * Penrose, Virginia, an historic district in Arlington County Elsewhere * Penrose, New South Wales (Wingecarribee), Australia * Penrose, New South Wales (Wollongong), New South Wales, Australia * Penrose, Cornwall, a country house and National Trust estate in England * Penrose, New Zealand * Penrose Peak (other) * Penrose railway station (other) People * Penrose Stout (1887–1934), American architect * Penrose Hallowell (c. 1928–2021), Pennsylvania secretary of agriculture * Penrose (surname), including a list of people with the name Other uses * Penrose (brand), a brand name owned by ConAgra Foods, Inc. * '' The Penrose Annual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
