Lamination (topology)
Encyclopedia
In topology
, a branch of mathematics, a lamination is a :
Lamination of surface is a partition
of closed subset of surface into unions of smooth curves
It may or may not be possible to fill the gaps in a lamination to make a foliation
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics, a lamination is a :
- "A topological spaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
partitionedPartition-Computing:* Partition , the division of a database* Disk partitioning, the division of a hard disk drive* Logical partition , a subset of a computer's resources, virtualized as a separate computer-Mathematics:...
into subsets" - decoration (a structure or property at a point) of a manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
in which some subset of the manifold is partitioned into sheets of some lower dimension, and the sheets are locally parallelParallel (geometry)Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
.
Lamination of surface is a partition
Partition
-Computing:* Partition , the division of a database* Disk partitioning, the division of a hard disk drive* Logical partition , a subset of a computer's resources, virtualized as a separate computer-Mathematics:...
of closed subset of surface into unions of smooth curves
It may or may not be possible to fill the gaps in a lamination to make a foliation
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
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Examples
- A geodesic lamination of a 2-dimensional hyperbolic manifoldHyperbolic manifoldIn mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
is a closed subset together with a foliation of this closed subset by geodesics. These are used in Thurston's classification of elements of the mapping class groupMapping class groupIn mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
and in his theory of earthquake mapEarthquake mapIn hyperbolic geometry, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by .-Earthquake maps:Given a simple geodesic on an oriented hyperbolic manifold and a real number t, one can cut the manifold along the geodesic, slide the edges a distance t to the...
s. - Quadratic laminations, which remain invariant under the angle doubling map. These laminations are associated with quadratic maps . It is a closed collection of chords in the unit disc . It is also topological model of Mandelbrot or Julia set.