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Encyclopedia > Fundamental polygon

In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics, a surface is a two-dimensional manifold. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... Wiktionary has a definition of: Polygon A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ...


This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon.

Contents

Examples

For other uses, see sphere (disambiguation). ... In mathematics, a projective plane consists of a set of lines and a set of points with the following properties: Given any two distinct points, there is exactly one line incident with both of them. ... Two views of a Klein bottle immersed in three-dimensional space. ... Geometry In geometry, a torus (pl. ...

Standard fundamental polygons

An orientable closed surface of genus n has the following standard fundamental polygon:

A non-orientable closed surface of (non-orientable) genus n has the following standard fundamental polygon:

Fundamental polygon of a compact Riemann surface

The fundamental polygon of a (hyperbolic) compact Riemann surface has a number of important properties that relate the surface to its Fuchsian model. That is, a hyperbolic compact Riemann surface has the upper half-plane as the universal cover, and can be represented as a quotient manifold H/Γ where Γ is a non-Abelian group isomorphic to the deck transformation group of the surface. The cosets of the quotient space have the standard fundamental polygon as a representative element. In the following, note that all Riemann surfaces are orientable. In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...


Metric fundamental polygon

Given a point z0 in the upper half-plane H, and a discrete subgroup Γ of PSL(2,R) that acts freely discontinuously on the upper half-plane, then one can define the metric fundamental polygon as the set of points In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation...

Here, d is a hyperbolic metric on the upper half-plane. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

  • This fundamental polygon is a fundamental domain.
  • This fundamental polygon is convex in that the geodesic joining any two points of the polygon is contained entirely inside the polygon.
  • The diameter of F is less than or equal to the diameter of H/Γ. In particular, the closure of F is compact.
  • If Γ has no fixed points in H and H/Γ is compact, then F will have finitely many sides.
  • Each side of the polygon is a geodesic arc.
  • For every side s of the polygon, there is precisely one other side s' such that gs=s' for some g in Γ. Thus, this polygon will have an even number of sides.
  • The set of group elements g that join sides to each other are generators of Γ, and there is no smaller set that will generate Γ.
  • The upper half-plane is tiled by the closure of F under the action of Γ. That is, where is the closure of F.

In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... For the geometric term, see diameter. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

Standard fundamental polygon

Given any metric fundamental polygon F, one can construct, with a finite number of steps, another fundamental polygon, the standard fundamental polygon, which has an additional set of noteworthy properties:

  • The vertices of the standard polygon are all equivalent. By vertex is meant the point where two sides meet. By equivalent, it is meant that each vertex can be carried to any of the other vertices by some g in Γ.
  • The number of sides is divisible by four.
  • A given element g of Γ will carry at most one side of the polygon to another. Thus, the sides can be marked off in pairs. Since the action of Γ is orientation-preserving, if one side is called A, then the other of the pair can be marked with the opposite orientation A − 1.
  • The edges of the standard polygon can be arranged so that the list of adjecent sides takes the form . That is, pairs of sides can be arranged so that they interleave in this way.
  • The standard polygon is convex.
  • The sides can be arranged to be geodesic arcs.

The above construction is sufficient to gaurentee that each side of the polygon is a closed (non-trivial) loop in the manifold H/Γ. As such, each side can thus an element of the fundamental group . In particular, the fundamental group has 2n generators , with exactly one defining constraint, In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

.

The genus of the resulting manifold H/Γ is n.


Example

Note that the metric fundamental polygon and the standard fundamental polygon will usually have a different number of sides. Thus, for example, the standard fundamental polygon on a torus is a fundamental parallelogram. By contrast, the metric fundamental polygon is six-sided, a hexagon. This can be most easily seen by noting that the sides of the hexagon are perpendicular bisectors of the edges of the paralleogram. That is, one picks a point in the lattice, and then considers the set of straight lines joing this point to nearby neighbors. Bisecting each such line by another perpendicular line, the smallest space walled off by this second set of lines is a hexagon. Geometry In geometry, a torus (pl. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ...


In fact, this last construction works in generality: picking a point x, one then considers the geodesics between x and gx for g in Γ. Bisecting these geodesics is another set of curves, the locus of points equidistant between x and gx. The smallest region enclosed by this second set of lines is the metric fundamental polygon. In mathematics, a locus (plural loci) is a collection of points which share a common property. ...


Area

The area of the standard fundamental polygon is 4π(n − 1) where n is the genus of the Riemann surface (equivalently, where 4n is the number of the sides of the polygon). Since the standard polygon is a representive of H/Γ, the total area of the Riemann surface is equal to the area of the standard polygon. The area formula follows from the Gauss-Bonnet theorem and is in a certain sense generalized through the Riemann-Hurwitz formula. The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ... In mathematics, the Riemann-Hurwitz formula describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. ...


References

  • Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4.
  • Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X.

  Results from FactBites:
 
GameDev.net - The Polygon Primeval (2500 words)
Filling a polygon is fundamentally a rasterization process involving drawing all of the horizontal lines within the polygon's boundaries.
Rasterization of convex polygons is easily done by starting at the top of the polygon and tracing down the left and right sides, one scan line (one vertical pixel) at a time, filling the extent between the two edges on each scan line, until the bottom of the polygon is reached.
Polygons are skewed toward the top and left edges, which not only introduces drawing error relative to the ideal polygon but also means that a filled polygon won't match the same polygon drawn unfilled.
  More results at FactBites »

 
 

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