topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, a branch of mathematics, a lamination is a : * "

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

partitioned into subsets" * decoration (a structure or property at a point) of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

in which some subset of the manifold is partitioned into sheets of some lower dimension, and the sheets are locally

parallel Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science a ...

. A lamination of a surface is a partition of a closed subset of the surface into smooth curves. It may or may not be possible to fill the gaps in a lamination to make a

foliation In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ...

. Oak Ridge National Laboratory

Examples

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

lamination of a 2-dimensional

hyperbolic manifold In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, res ...

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 group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...

and in his theory of earthquake maps. *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.Modeling Julia Sets with Laminations: An Alternative Definition by Debra Mimbs
It is also a topological model of Mandelbrot or

Julia set In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...

Notes

References

Conformal Laminations Thesis by Vineet Gupta, California Institute of Technology Pasadena, California 2004
Topology Manifolds {{topology-stub

Examples

See also

Notes

References