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Generalised concept of incidence structure of polygons
special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). Many generalized polygons arise from groups of Lie type
Generalized_polygon
Property of geometry, also used to generalize the notion of "distance" in metric spaces
geometric progression and let the sides be a, ar, ar2, ar3. Then the generalized polygon inequality requires that 0 < a < a r + a r 2 + a r 3 0 < a r < a
Triangle_inequality
Mathematical term in geometry
In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but they can also
Polygram_(geometry)
Plane figure bounded by line segments
around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external
Polygon
Polygon with 2 sides and 2 vertices
be viewed as a representation of a graph with two vertices, see "Generalized polygon". A regular digon has both angles equal and both sides equal and
Digon
Concave polygon Constructible polygon Convex polygon Cyclic polygon Equiangular polygon Equilateral polygon Penrose tile Polyform Regular polygon Simple
List of two-dimensional geometric shapes
List_of_two-dimensional_geometric_shapes
Field of mathematics which studies incidence structures
4-gon is a generalized quadrangle (possibly degenerate). Every finite generalized polygon except the projective planes is a near polygon. Any connected
Incidence_geometry
Family of cubic graphs formed from regular and star polygons
vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph
Generalized_Petersen_graph
Positive integer of the form (2^(2^n))+1
current top 20 generalized Fermat primes and the current top 100 generalized Fermat primes. Constructible polygon: which regular polygons are constructible
Fermat_number
Polygon with an infinite number of sides
tiling Apeirogonal prism Apeirogonal antiprism Teragon, a fractal generalized polygon that also has infinitely many sides McMullen & Schulte (2002) provide
Apeirogon
Generalization of a polytope in real space
regular polygon, while there are no faces. 2D orthogonal projections of complex polygons 2{r}q Polygons of the form 2{4}q are called generalized orthoplexes
Complex_polytope
Concept in incidence geometry
geometries which they called near polygons. These structures generalise the notion of generalized polygon as every generalized 2n-gon is a near 2n-gon of a
Near_polygon
Mathematical structure
2 are precisely the generalized polygons, and a plethora of examples exist. (There are free constructions of infinite generalized n-gons for every n ≥
Building_(mathematics)
Conic solid with a polygonal base
A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral
Pyramid_(geometry)
Belgian mathematician (1930–2021)
yielding the groups directly. In the rank-2 case spherical building are generalized n-gons, and in joint work with Richard Weiss he classified these when
Jacques_Tits
Geometric system with a finite number of points
projective plane. Discrete space Finite space Generalized polygon Incidence geometry Linear space (geometry) Near polygon Partial geometry Polar space Laywine
Finite_geometry
Polygon intersected up to twice by lines orthogonal to a given line
directions in which a given simple polygon is monotone. It was generalized to report all ways to decompose a simple polygon into two monotone chains (possibly
Monotone_polygon
Regular polytope dual to the hypercube in any number of dimensions
{C} } ^{n}} . A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets. Generalized orthoplexes make
Cross-polytope
Path that surrounds an area
} An equilateral polygon is a polygon which has all sides of the same length (for example, a rhombus is a 4-sided equilateral polygon). To calculate the
Perimeter
Formula for area of a grid polygon
has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. Suppose that a polygon has integer coordinates for all
Pick's_theorem
American mathematician
Fermat theorem on polygonal numbers" in the Annals of Mathematics, "Representation by Extended Polygonal Numbers and by Generalized Polygonal Numbers" and
Lois_Wilfred_Griffiths
Polygonal chain whose vertices are not all coplanar
polygon is a closed polygonal chain in Euclidean space. It is a figure similar to a polygon except its vertices are not all coplanar. While a polygon
Skew_polygon
Type of polygon
polygon must be a generalized 3-gon, 4-gon, 6-gon, or 8-gon, so the purpose of the aforementioned book was to analyze these four cases. A generalized
Moufang_polygon
Polygons which have an accompanying imaginary dimension for each real dimension
visualized. A complex polygon is generalized as a complex polytope in C n {\displaystyle \mathbb {C} ^{n}} . A complex polygon may be understood as a
Regular_complex_polygon
Polygon with 12 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278) The Lighter Side of Mathematics:
Dodecagon
Skew polygon derived from a polytope
York: Dover, 1973. (Sec. 2.6 Petrie Polygons pp. 24–25, and Chapter 12, pp. 213–235, The generalized Petrie polygon) Coxeter, H.S.M. (1974) Regular complex
Petrie_polygon
Polygon in which all sides have equal length
geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also
Equilateral_polygon
Regular graph with fewest possible nodes for its girth
generalized polygons. The numbers of vertices in the known (r,g) cages, for values of r > 2 and g > 2, other than projective planes and generalized polygons
Cage_(graph_theory)
Geometric concept of a 2D space with "points at infinity" adjoined
projective plane. Combinatorial design Difference set Incidence structure Generalized polygon Projective geometry Non-Desarguesian plane Smooth projective plane
Projective_plane
Convex polytope, the n-dimensional analogue of a square and a cube
\mathbb {C} ^{n}} . The facets are generalized (n−1)-cube and the vertex figure are regular simplexes. The regular polygon perimeter seen in these orthogonal
Hypercube
Theorem on equilateral triangles
above for the equilateral triangle generalizes to any (convex) equilateral polygon. In other words, if a polygon has all sides equal, the sum of the
Viviani's_theorem
Type of incidence structure
{L}}|} . Projective space Affine space Polar space Generalized quadrangle Generalized polygon Near polygon Shult, Ernest E. (2011), Points and Lines, Universitext
Partial_linear_space
Concept in geometry
Buekenhout, Francis (2000), Prehistory and History of Polar Spaces and of Generalized Polygons (PDF) Buekenhout, Francis; Cohen, Arjeh M. (2013), Diagram Geometry:
Polar_space
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
English mathematician
Feit, Walter; Higman, Graham (1964). "The nonexistence of certain generalized polygons". Journal of Algebra. 1 (2): 114–131. doi:10.1016/0021-8693(64)90028-6
Graham_Higman
Regular graph with girth more than twice its diameter
Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons. Some examples are the even cycles C2n, the complete bipartite graphs
Moore_graph
Form of abstraction
establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among all parts.
Generalization
Method in geometry for representing a polygon by a topological skeleton
Straight skeletons were first defined for simple polygons by Aichholzer et al. (1995), and generalized to planar straight-line graphs (PSLG) by Aichholzer
Straight_skeleton
Shape with six sides
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278) Dunajski, Maciej (2022). Geometry:
Hexagon
Smallest convex set containing a given set
arbitrary real vector spaces or affine spaces; convex hulls may also be generalized in a more abstract way, to oriented matroids. It is not obvious that
Convex_hull
Discrete dynamical system on polygons in the projective plane and on their moduli space
map is a discrete dynamical system acting on polygons in the projective plane. It defines a new polygon whose vertices are obtained as the intersection
Pentagram_map
Partition of a polygon into triangles of equal area
certain results can be easily generalized. All results stated for a regular polygon also hold for affine-regular polygons; in particular, results concerning
Equidissection
Shape with nine sides
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) TMBW.net Properties of a Nonagon
Nonagon
Mathematical algorithm for calculating area of a simple polygon
formula, is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane
Shoelace_formula
Unrelated vertices in graphs
The nine blue vertices form a maximum independent set for the Generalized Petersen graph GP(12,4).
Independent set (graph theory)
Independent_set_(graph_theory)
Polygon shape with eight sides
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Look up octagon in Wiktionary, the
Octagon
Type of center of a polygon
these operations has the same circumcenter of mass as the original polygon. The generalized Euler line makes other appearances in the theory of integrable
Circumcenter_of_mass
Algorithm in computer graphics
self-intersecting and non-convex polygons. It can be trivially generalized to compute other Boolean operations on polygons, such as union and difference
Greiner–Hormann clipping algorithm
Greiner–Hormann_clipping_algorithm
Size of a geometric arrangement of points
for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral
Figurate_number
Number of windings of a polytope around its center of symmetry
of a polygon is the number of times that the polygonal boundary winds around its center. For convex polygons, and more generally simple polygons (not
Density_(polytope)
Polygon with 13 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278) Colin R. Bruce, II, George Cuhaj
Tridecagon
Polygon with 20 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Coxeter, Mathematical recreations
Icosagon
Study of triangles in other spaces than the Euclidean plane
is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid
Generalized_trigonometry
Geometric figure which is "snugly enclosed" by another figure
polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon
Inscribed_figure
Polygon with 14 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Coxeter, Mathematical recreations
Tetradecagon
Prime number of the form 2^u × 3^v + 1
studied the same class of regular polygons; his work is what gives the name to the Pierpont primes. Pierpont generalized compass and straightedge constructions
Pierpont_prime
Shape with ten sides
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Coxeter, Mathematical recreations
Decagon
Shape with five sides
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Weisstein, Eric W. "Cyclic Pentagon
Pentagon
Curve traced by a vertex of a polygon as it rolls
In geometry, a cyclogon is the curve traced by a vertex of a regular polygon that rolls without slipping along a straight line. In the limit, as the number
Cyclogon
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
Polygon with 18 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Hirschhorn & Hunt 1985. Coxeter
Octadecagon
Convex polyhedron with regular faces
polygons, no two in the same plane; those polygons are called the faces. A Johnson solid is a convex polyhedron whose faces are all regular polygons,
Johnson_solid
Polygon with 24 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Coxeter, Mathematical recreations
Icositetragon
Theorem in topology
required a proof. It is easy to establish this result for polygons, but the problem came in generalizing it to all kinds of badly behaved curves, which include
Jordan_curve_theorem
Belgian mathematician
University Press 1995: "Projective geometry over a finite field" and "Generalized Polygons" in F. Buekenhout, Handbook of Incidence Geometry, North-Holland
Joseph_A._Thas
Polygon with 17 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278) Dunham, William (September 1996)
Heptadecagon
Extends the Jordan curve theorem to characterize the inner and outer regions
Moise (1977) and Thomassen (1992). The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity
Schoenflies_problem
Flat-sided three-dimensional shape
Under any definition, polyhedra are typically understood to generalize two-dimensional polygons and to be the three-dimensional specialization of polytopes
Polyhedron
NP-hard problem in combinatorial optimization
for retooling the robot (single-machine job sequencing problem). The generalized travelling salesman problem, also known as the "travelling politician
Travelling_salesman_problem
Shape with three sides
A triangle or trigon is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional
Triangle
Summation method for divergent series
Mittag-Leffler summation with α = 1. (wB) can be seen as the limiting case of generalized Euler summation method (E,q) in the sense that as q → ∞ the domain of
Borel_summation
Operation that cuts polytope vertices, creating a new facet in place of each vertex
A truncated n-sided polygon will have 2n sides (edges). A regular polygon uniformly truncated will become another regular polygon: t{n} is {2n}. A complete
Truncation_(geometry)
Algorithm for finding a zero of a function
polynomial. The bisection method has been generalized to multi-dimensional functions. Such methods are called generalized bisection methods. Some of these methods
Bisection_method
In geometry, a weakly simple polygon is a generalization of a simple polygon, allowing the polygon sides to touch each other in limited ways. Different
Weakly_simple_polygon
Construct in computational geometry
triangulation satisfying these properties always exists. Jonathan Shewchuk has generalized this definition to constrained Delaunay triangulations of three-dimensional
Constrained Delaunay triangulation
Constrained_Delaunay_triangulation
Polygon with 16 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Coxeter, Mathematical recreations
Hexadecagon
resulting generalized algorithm is not linear time, however: its time complexity depends on the depth of nesting of certain features of one polygon within
Relative_convex_hull
Assignment of colors to edges of a graph
such as the generalized Petersen graphs G(6n + 3, 2) for n ≥ 2. The only known nonplanar uniquely 3-colorable graph is the generalized Petersen graph
Edge_coloring
Polygon with 15 edges
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) The Lighter Side of Mathematics:
Pentadecagon
Numerical method for solving physical or engineering problems
15-sided polygonal region Ω {\displaystyle \Omega } in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear
Finite_element_method
Symmetric bipartite cubic graph with 16 vertices and 24 edges
August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph G(8,3): that is, it is formed by the vertices of an octagon
Möbius–Kantor_graph
Every positive integer is a sum of at most n+2 centered n-gonal numbers
In additive number theory, the centered polygonal number theorem states that every positive integer is a sum of at most n+2 centered n-gonal numbers.
Centered polygonal number theorem
Centered_polygonal_number_theorem
Shape with eleven sides
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Mossinghoff, Michael J. (2006),
Hendecagon
Planar surface that forms part of the boundary of a solid object
solid volume; the faces are the two-dimensional polygons of these definitions. Other names for a polygonal face include polyhedron side and Euclidean plane
Face_(geometry)
Mathematical connection between field theory and group theory
regular polygons that are constructible (this characterization was previously given by Gauss but without the proof that the list of constructible polygons was
Galois_theory
Coordinate system that is defined by points instead of vectors
corresponds to a point not having unique generalized barycentric coordinates except when P is a simplex. Dual to generalized barycentric coordinates are slack
Barycentric_coordinate_system
Straight line segment that passes through the centre of a circle
Line Line segment Ray Curve Length Two-dimensional Surface Plane Area Polygon Triangle Centers Altitude Hypotenuse Pythagorean theorem Circular Hyperbolic
Diameter
Infinite polyhedron with non-planar faces
to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra (apeirohedra)
Skew_apeirohedron
Circle that passes through the vertices of a triangle
generally, an n-sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special
Circumcircle
Straight path on a curved surface or a Riemannian manifold
great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one
Geodesic
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
Smallest convex polygon containing a given polygon
can be generalized to the convex hull of any polygonal chain, and the algorithm for simple polygons can be started at any vertex of the polygon rather
Convex hull of a simple polygon
Convex_hull_of_a_simple_polygon
Shape with seven sides
Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) Salthouse, J.A; Ware, M.J. (1972)
Heptagon
Graph drawn with all edges intersecting
significant relaxation of the standard thrackle definition is the generalized thrackle. In a generalized thrackle drawing, any pair of edges is required to intersect
Thrackle
Notation for polytopes and tessellations
named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their
Schläfli_symbol
On continuous motion of a simple polygon to convex
Panina & Streinu (2010) for spherical polygons of edge length smaller than 2π. John Pardon (2009) generalized the Carpenter's rule problem to rectifiable
Carpenter's_rule_problem
Vertex-transitive tiling of the plane by regular polygons
hollow tilings, using the first two expansions above: star polygon faces and generalized vertex figures. H. S. M. Coxeter, M. S. Longuet-Higgins, and
Uniform_tiling
Developable roller constructed from a cone
family generalizes the sphericon. It was discovered by the Israeli inventor David Hirsch in 2017. Two adjacent edges of an even sided regular polygon are
Polycon
Curved curface derived from a coarse polygon mesh
Subsurf) is a curved surface represented by the specification of a coarser polygon mesh and produced by a recursive algorithmic method. The curved surface
Subdivision_surface
Half of the sum of side lengths of a polygon
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears
Semiperimeter
GENERALIZED POLYGON
GENERALIZED POLYGON
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
GENERALIZED POLYGON
GENERALIZED POLYGON
Female
English
 English variant spelling of Italian Arianna, ARIANA means "utterly pure." Compare with another form of Ariana.
Boy/Male
Hindu
Name of the emperor, With beautiful banner
Boy/Male
Indian, Punjabi, Sikh
Victory with Lord's Love
Girl/Female
Arabic, Muslim
Official; Formal
Boy/Male
Tamil
Sentiment of Love and affection
Girl/Female
Arabic, Muslim
Moon-like (Face)
Surname or Lastname
English
English : unexplained. Perhaps a patronymic from Enoch or a variant of Irish Ennis.
Girl/Female
Hindu
Forehead, Intelligence
Girl/Female
Indian
Mother of Lord Hanuman, Illusion (Maya), Hotness
Boy/Male
Tamil
Baladitya | பாலாதிதà¯à®¯
Young Sun, Young Man, The newly risen Sun
GENERALIZED POLYGON
GENERALIZED POLYGON
GENERALIZED POLYGON
GENERALIZED POLYGON
GENERALIZED POLYGON
p. pr. & vb. n.
of Generalize
v. t.
To derive or deduce (a general conception, or a general principle) from particulars.
v. i.
To form into a genus; to view objects in their relations to a genus or class; to take general or comprehensive views.
n.
A generalized concept of magnitude.
v. t.
To generalize or conclude as an inference from all the particulars; -- the opposite of deduce.
a.
Comprising structural characters which are separated in more specialized forms; synthetic; as, a generalized type.
n.
A fishlike creature (Amphioxus lanceolatus), two or three inches long, found in temperature seas; -- also called the lancelet. Its body is pointed at both ends. It is the lowest and most generalized of the vertebrates, having neither brain, skull, vertebrae, nor red blood. It forms the type of the group Acrania, Leptocardia, etc.
v. t.
To make universal; to generalize.
imp. & p. p.
of Generalize
a.
Capable of being generalized, or reduced to a general form of statement, or brought under a general rule.
n.
The act or process of centralizing, or the state of being centralized; the act or process of combining or reducing several parts into a whole; as, the centralization of power in the general government; the centralization of commerce in a city.
imp. & p. p.
of Centralize
v. t.
To apply to other genera or classes; to use with a more extensive application; to extend so as to include all special cases; to make universal in application, as a formula or rule.
imp. & p. p.
of Federalize
n.
The system by which power is centralized, as in a government.
v. t.
To impregnate with a mineral; as, mineralized water.
imp. & p. p.
of Mineralize
v. t.
To bring under a genus or under genera; to view in relation to a genus or to genera.
n.
Any plant of the genus Polygonum.
n.
One who takes general or comprehensive views.