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Polygon associated with a compact Riemann surface
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only information about
Fundamental_polygon
Algebraic surface
Riemann surface by associating sides of a hyperbolic icosagon (see fundamental polygon). The identification pattern is given in the adjoining diagram. The
Bring's_curve
Plane figure bounded by line segments
solid polygons, a polygon may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygons and
Polygon
Natural number
sporadic groups. An icosagon is a polygon with 20 edges. Bring's curve is a Riemann surface, whose fundamental polygon is a regular hyperbolic icosagon
20_(number)
Term in geometric topology
simple closed curve c on S. Consider the torus represented by a fundamental polygon with edges a and b T 2 ≅ R 2 / Z 2 . {\displaystyle \mathbb {T} ^{2}\cong
Dehn_twist
Topological space
over the fundamental domain, when integrating functions with poles and zeros on the boundary of the domain. Free regular set Fundamental polygon Brillouin
Fundamental_domain
Non-orientable mathematical surface
1882 by the mathematician Felix Klein. The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding
Klein_bottle
Two-dimensional manifold
from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge
Surface_(topology)
Compact non-orientable two-dimensional manifold
two projective planes creates the Klein bottle. The article on the fundamental polygon describes the higher non-orientable surfaces. Real projective space
Real_projective_plane
Natural number
20-sided icosagon. By the Gauss-Bonnet theorem, the area of this fundamental polygon is equal to 12 π {\displaystyle 12\pi } . Twelve is the smallest
12_(number)
Shape with equal angles and equal sides
polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may
Regular_polygon
Describes the fundamental group in terms of a cover by two open path-connected subspaces
expresses the structure of the fundamental group of a topological space X {\displaystyle X} in terms of the fundamental groups of two open, path-connected
Seifert–Van_Kampen_theorem
Type of topological space
petals, namely the boundary of a fundamental polygon. A rose can have infinitely many petals, leading to a fundamental group which is free on infinitely
Rose_(topology)
Geometrical object in four-dimensional space
Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together.
Clifford_torus
Connected series of line segments
together, the control points form a polygonal chain called a control polygon. Polygonal chains are also a fundamental data type in computational geometry
Polygonal_chain
of the fundamental polygon found using the Gauss–Bonnet theorem. We can say the area of the whole triangle is equal to the number of fundamental triangles
Coxeter decompositions of hyperbolic polygons
Coxeter_decompositions_of_hyperbolic_polygons
Algebraic tool for computing topological spaces' invariants
Kampen theorem for the fundamental group, and a precise relation exists for homology of dimension one. Similar to the fundamental group or the higher homotopy
Mayer–Vietoris_sequence
Vertex-transitive tiling of the plane by regular polygons
singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by its group
Uniform_tiling
Doughnut-shaped surface of revolution
together, described as a fundamental polygon ABA−1B−1. The fundamental group of the torus is just the direct product of the fundamental group of the circle
Torus
Number of "holes" of a surface
construction of surfaces of the genus g is given in the article on the fundamental polygon. Genus of orientable surfaces Planar graph: genus 0 Toroidal graph:
Genus_(mathematics)
Group representation of a Riemann surface
relation. the Kleinian model, an analogous construction for 3-manifolds Fundamental polygon Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and Kleinian groups
Fuchsian_model
Representation of mathematical space
be triangulated and then using the triangulation to construct a fundamental polygon for the surface. Giving spaces simplicial structures can help to
Triangulation_(topology)
Skew polygon derived from a polytope
the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive
Petrie_polygon
Path that surrounds an area
Euclidean spaces, is described by the theory of Caccioppoli sets. Polygons are fundamental to determining perimeters, not only because they are the simplest
Perimeter
Cellular automaton traffic flow model
The fundamental polygon of the torus, on which the cars move
Biham–Middleton–Levine traffic model
Biham–Middleton–Levine_traffic_model
Polygon with 1 million edges
A megagon or 1,000,000-gon (million-gon) is a circle-like polygon with one million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix
Megagon
Shape with five sides
Greek πέντε (pente) 'five' and γωνία (gonia) 'angle') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
Pentagon
Fundamental unit of a texture map
computer graphics, a texel, texture element, or texture pixel is the fundamental unit of a texture map. Textures are represented by arrays of texels representing
Texel_(graphics)
Set of basic shapes which assemble into a polygon
as polygons, and one approach to preparation for electron-beam lithography is to decompose these polygon regions into fundamental figures. Polygon decomposition
Polygon_partition
Type of plane partition
Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons, after Alfred H. Thiessen. Voronoi diagrams have practical and theoretical
Voronoi_diagram
Basic shapes represented in vector graphics
prefer to consider triangles primitives, because every polygon can be constructed from triangles (polygon triangulation). All other graphic elements are built
Geometric_primitive
Extremes of a linear function over a convex polygonal region occur at the region's corners
the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region
Fundamental theorem of linear programming
Fundamental_theorem_of_linear_programming
Algebraic structure associated with a topological space
torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist
Homology_(mathematics)
Theory in algebraic topology
homology of the genus g surface Σ g {\displaystyle \Sigma _{g}} . The fundamental polygon of Σ g {\displaystyle \Sigma _{g}} is a 4 g {\displaystyle 4g} -gon
Cellular_homology
Shape with six sides
meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting)
Hexagon
Point where two or more curves, lines, or edges meet
point where two lines meet to form an angle and the point where edges of polygons and polyhedra meet are vertices. The term vertex is also used for a different
Vertex_(geometry)
Isogonal polyhedron with regular faces
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows
Uniform_polyhedron
examples. Vertex the ridge or (n−2)-face of the polygon Edge the facet or (n−1)-face of the polygon Vertex the peak or (n−3)-face of the polyhedron Edge
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Polygons which have an accompanying imaginary dimension for each real dimension
create starry groups with overlapping fundamental domains: , , , , , and . The dual polygon of p{q}r is r{q}p. A polygon of the form p{q}p is self-dual. Groups
Regular_complex_polygon
Symmetrical calligraphic or typographic visual pun
geometry. Word patterns in tessellations can start from 35 different fundamental polygons, such as the rhombus, the isosceles right triangle, or the parallelogram
Ambigram
Smallest convex set containing a given set
Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. They can be solved in time O ( n
Convex_hull
Self-intersecting uniform polyhedron
imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic
Uniform_star_polyhedron
GIS analysis operation on vector data
(GIS) for integrating two or more vector spatial data sets. Terms such as polygon overlay, map overlay, and topological overlay are often used synonymously
Vector_overlay
Natural number
alongside a regular triangle and a 42-sided polygon (3.7.42). Otherwise, for any regular n-sided polygon, the maximum number of intersecting diagonals
7
Class of topological space
construct a planar regular p sided polygon. Put two points n and s directly above and below the center of the polygon. Construct the bipyramid by joining
Lens_space
Covering by shapes without overlaps or gaps
point is identical; that is, the arrangement of polygons about each vertex is the same. The fundamental region is a shape such as a rectangle that is repeated
Tessellation
Property of objects which are scaled or mirrored versions of each other
similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. Two congruent shapes
Similarity_(geometry)
Topics referred to by the same term
functions Kernel (geometry), the set of points within a polygon from which the whole polygon boundary is visible Kernel (statistics), a weighting function
Kernel
Number of times a curve wraps around a point in the plane
divided by 2π. In polygons, the turning number is referred to as the polygon density. For convex polygons, and more generally simple polygons (not self-intersecting)
Winding_number
Spherical triangle that can be used to tile a sphere
fundamental domain obtained by reflecting two sides of the geodesic triangle to form a geodesic parallelogram (a special case of Poincaré's polygon theorem)
Schwarz_triangle
Concept in geometry
Verdenduin, P. (1983), "Chapter 8: Polygons and Polyhedra", in H. Behnke; F. Bachmann; K. Fladt; H. Kunle (eds.), Fundamentals of Mathematics, Volume II: Geometry
Area_of_a_circle
Natural number
sides is called a pentagon. The equilateral pentagon is the first regular polygon that does not tile the plane with copies of itself. The pentagon solid
5
Convex polygon which can tile the plane by itself
geometry, a planigon is a convex polygon that can fill the plane with only copies of itself (isotopic to the fundamental units of monohedral tessellations)
Planigon
Natural number
{\displaystyle E_{7}} has a fundamental representation of dimension 56. Plutarch states that the Pythagoreans associated a polygon of 56 sides with Typhon
56_(number)
Geometry of figures on the surface of a sphere
quaternion methods, and the use of numerical methods. A spherical polygon is a polygon on the surface of the sphere. Its sides are arcs of great circles—the
Spherical_trigonometry
Geometric object with flat sides
an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In
Polytope
Polyhedron with 2 faces
which is the fundamental domain of a lens space L(p,q). Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons. As a spherical
Dihedron
regular polygon. The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented
List_of_regular_polytopes
Point on a line segment which is equidistant from both endpoints
between opposite vertices is the polygon's center. The midpoint-stretching polygon of a cyclic polygon P (a polygon whose vertices all fall on the same
Midpoint
Natural number
representation of powers of 11. An 11-sided polygon is called a hendecagon, or undecagon. A regular hendecagon is the polygon with the fewest number of sides that
11_(number)
Shortest paths on a bounded deformed sphere-like quadric surface
geodesic polygon is a polygon whose sides are geodesics. It is analogous to a spherical polygon, whose sides are great circles. The area of such a polygon may
Geodesics_on_an_ellipsoid
Flat-sided three-dimensional shape
ἕδρον (-hedron) 'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron"
Polyhedron
Branch of computer science
or found. Some fundamental problems of this type are: Convex hull: Given a set of points, find the smallest convex polyhedron/polygon containing all the
Computational_geometry
Number, approximately 3.14
Wei mathematician Liu Hui created a polygon-based iterative algorithm, with which he constructed a 3,072-sided polygon to approximate π as 3.1416. Liu later
Pi
r, while Schwarz triangles allow rational numbers p,q,r and allow star polygon faces, and have overlapping elements. The seven generator points with each
List of uniform tilings on the sphere, plane, and hyperbolic plane
List_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
Mathematical model of the physical space
icosahedron — each a convex polyhedron whose faces are congruent regular polygons meeting in the same arrangement at every vertex. Propositions 13–17 construct
Euclidean_geometry
2013 video game
design. Joystiq felt creativity and methodical approaches were encouraged. Polygon likened rapid character switching during heist missions to "film editing
Grand_Theft_Auto_V
Symmetric subdivision in hyperbolic geometry
fundamental triangle, the Wythoff constructions are slightly different. For instance in the (4,3,3) triangle family, the snub form has six polygons around
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Four-dimensional analogue of the cube
the tesseract possible. The characteristic 5-cell of the 4-cube is a fundamental region of the tesseract's defining symmetry group, the group which generates
Tesseract
Polytope with highest degree of symmetry
polytopes are the generalised analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra
Regular_polytope
Concept in geometry
generators are the fundamental invariants; their degrees are given in the table above. Notice that if m is a degree of a fundamental invariant then so
Coxeter_element
Type of program in computer graphics
Fluid animation Lighting Global illumination Hidden-surface determination Polygon mesh (Triangle mesh) Shading Deferred Surface triangulation Wire-frame
Shader
Video game
February 2021). "What is Valheim, the Viking game blowing up on Steam?". Polygon. Archived from the original on 9 February 2021. Retrieved 10 February 2021
Valheim
Season of television series
2026). "Critical Role's Schemer table is redefining a fundamental aspect of Dungeons & Dragons". Polygon. Retrieved March 22, 2026. Hart, Aimee (May 22, 2026)
Critical_Role_campaign_four
Graphics that use a three-dimensional representation of geometric data
that define the shape and form polygons. A polygon is an area formed from at least three vertices (a triangle). A polygon of n points is an n-gon. The overall
3D_computer_graphics
Process supporting machine learning
machines to interpret the data in line with its intended use. Data is a fundamental component in the development of artificial intelligence (AI). Data annotation
Data_annotation
2017 browser-based video game
best games of 2017". Polygon. 18 December 2017. Archived from the original on 31 March 2019. Retrieved 27 December 2017. Staff, Polygon (2019-11-04). "The
Universal_Paperclips
Figure formed by two rays meeting at a common point
is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior
Angle
Property of a mathematical space
the intervening shape of the line as straight- or curved-line segments. Polygon (2-dimensional) usually represented as a line that closes at its endpoints
Dimension
2015 video game
all the way back to 2006". Polygon. Archived from the original on December 21, 2016. Retrieved November 23, 2019. Polygon staff (November 4, 2019). "The
Undertale
Polyhedron with four faces
tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the
Tetrahedron
Shape with four equal sides and angles
equivalent ways. If a polygon in the Euclidean plane satisfies any one of the following criteria, it satisfies all of them: A square is a polygon with four equal
Square
Generalization of a polytope in real space
create starry groups with overlapping fundamental domains: , , , , , and . The dual polygon of p{q}r is r{q}p. A polygon of the form p{q}p is self-dual. Groups
Complex_polytope
Topics referred to by the same term
in Wiktionary, the free dictionary. A quadrilateral, in geometry, is a polygon with 4 sides. Quadrilateral may also refer to: "Quadrilateral", an alternative
Quadrilateral (disambiguation)
Quadrilateral_(disambiguation)
Shape with three equal sides
with three sides of equal length and three equal angles. It is a regular polygon, occasionally known as the regular triangle. It is a special case of an
Equilateral_triangle
Violations of the convexity assumptions of elementary economics
L'economie Publique [Fundamentals of public economics]. MIT. pp. 63–65. ISBN 0-262-12127-1. Starrett, David A. (1972). "Fundamental nonconvexities in the
Non-convexity_(economics)
American actor (born 1964)
5, 2023). "Dead by Daylight adds Nicolas Cage as a playable Survivor". Polygon. Archived from the original on December 22, 2023. Retrieved January 10
Nicolas_Cage
Nintendo Switch game controller
pre-launch software on review units. A Nintendo spokesperson stated to Polygon that the company would "continue to monitor the performance of Nintendo
Joy-Con
Mathematical theorem
case of the conjecture, in which each cell is required to be a convex polygon. The full conjecture was proven in 1999 by mathematician Thomas C. Hales
Honeycomb_theorem
Part of the mathematical subject of group theory
free product with amalgamation and HNN extension, via the notion of the fundamental group of a graph of groups. Bass–Serre theory has some overlap with orbifold
Bass–Serre_theory
Element representing a value on a grid in three dimensional space
polygons are often explicitly represented by the coordinates of their vertices (as points). A direct consequence of this difference is that polygons can
Voxel
German polymath and scholar (1777–1855)
proved the construction of the heptadecagon, the first progress in regular polygon construction in over 2000 years. He also introduced the concept of Gaussian
Carl_Friedrich_Gauss
American adult animated science fiction sitcom
season 7 will replace Justin Roiland's characters with 'soundalikes'". Polygon. Archived from the original on September 27, 2023. Retrieved July 22, 2023
Rick_and_Morty
Home video game console
received favorable reviews from critics and reviewers. In its launch review, Polygon gave the Xbox One a score of 8/10. Its design was described as "inoffensive"
Xbox_One
2017 puzzle-solving horror game
GamesRadar+. Retrieved 25 March 2018. Polygon staff (18 December 2017). "The 50 best games of 2017". Polygon. Retrieved 12 February 2018. "Best of 2017
Little_Nightmares
integral transforms and infinite series, or by employing appropriate fundamental solutions. For example, the Dirichlet problem of the heat equation on
Fokas_method
Field of mathematics which studies incidence structures
generalized polygon except the projective planes is a near polygon. Any connected bipartite graph is a near polygon and any near polygon with precisely
Incidence_geometry
Mathematical space with two coordinates
represented using numbers rather than geometric axioms. One of the most fundamental two-dimensional spaces is the real coordinate space, denoted R 2 , {\displaystyle
Two-dimensional_space
2016 video game
including journalists at Kotaku and Polygon, streamed their starting playthroughs of the game starting from 5 August 2016. Polygon noted that Sony had not issued
No_Man's_Sky
Graphical perspective in video games
reality First-person (video games) Rollings, Andrew; Ernest Adams (2006). Fundamentals of Game Design. Prentice Hall. ISBN 9780131687479. Archived from the
Third-person_(video_games)
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FUNDAMENTAL POLYGON
FUNDAMENTAL POLYGON
Boy/Male
Arabic, Muslim
Testimony; Evidence; Fundamental Belief in Islam; Witness
Surname or Lastname
English
English : occupational name for a merchant or trader, Middle English chapman, Old English cēapmann, a compound of cēap ‘barter’, ‘bargain’, ‘price’, ‘property’ + mann ‘man’.This name was brought independently to North America from England by numerous different bearers from the 17th century onward. John Chapmen (sic) was one of the free planters who assented to the ‘Fundamental Agreement’ of the New Haven Colony on June 4, 1639.
Male
Turkish
Turkish name TEMEL means "basic, fundamental."
Surname or Lastname
English
English : from an Old English byname, Budde, which was applied to a thickset or plump person. By the Middle English period it had become a common personal name, with derivatives formed with hypocoristic suffixes, Budecok and Budekin. Reaney derives it from Old English budda ‘beetle’.Shortened form of German Budde.John Budd was one of the free planters who assented to the ‘Fundamental Agreement’ of the New Haven Colony on June 4, 1639.
Surname or Lastname
English
English : from the Middle English and Old English personal name Brūning, originally a patronymic from the byname Brūn (see Brown).This name was brought independently to North America from England by numerous different bearers from the 17th century onward. William Browning was one of the free planters who assented to the ‘Fundamental Agreement’ of the New Haven Colony on June 4, 1639.
Surname or Lastname
English
English : habitational name from a place in Berkshire named with the Old English personal name Benna + Old English hamm ‘river meadow’.John Benham was one of the free planters who assented to the ‘Fundamental Agreement’ of the New Haven Colony on June 4, 1639.
Surname or Lastname
English
English : from a pet form of the Middle English personal name Boye.Jarvis Boykin was one of the free planters who assented to the ‘Fundamental Agreement’ of the New Haven Colony on June 4, 1639.
Surname or Lastname
English
English : habitational name from any of the various places, in Kent, Oxfordshire, and Sussex, named Beckley, from the Old English byname Becca (see Beck 4) + Old English lēah ‘woodland clearing’.Altered spelling of the South German and Swiss topographic names Bächle, Bächli (see Bach 1).Richard Beckley was one of the free planters who assented to the ‘Fundamental Agreement’ of the New Haven Colony on June 4, 1639.
FUNDAMENTAL POLYGON
FUNDAMENTAL POLYGON
FUNDAMENTAL POLYGON
FUNDAMENTAL POLYGON
FUNDAMENTAL POLYGON
FUNDAMENTAL POLYGON
FUNDAMENTAL POLYGON
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