In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect in one and only one point.
In geometry, a secant of a curve is a line that intersects the curve in at least two (distinct) points. The word secant comes from the Latin word secare, meaning to cut. In the case of a circle, a secant will intersect the circle in exactly two points and a chord is the line segment determined by these two points, that is the interval on a secant whose endpoints are these points.
A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In modern mathematics, a point refers usually to an element of some set called a space.
In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it into tetrahedra packed together.
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles. A generalized quadrangle is by definition a polar space of rank two. They are the generalized n-gons with n = 4 and near 2n-gons with n = 2. They are also precisely the partial geometries pg(s,t,α) with α = 1.
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes and generalized quadrangles. Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss. Every generalized n-gon with n even is also a near polygon.
An incidence structure consists of points , lines , and flags where a point is said to be incident with a line if . It is a (finite) partial geometry if there are integers such that:
In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension.
In mathematics, a Laguerre plane is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane, named after the French mathematician Edmond Nicolas Laguerre.
In mathematics, a Möbius plane is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real affine plane.
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by W. T. Tutte (1963), states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex. It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.
In mathematics, a near polygon is an incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in 1980. Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line 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 particular kind. Near polygons were extensively studied and connection between them and dual polar spaces was shown in 1980s and early 1990s. Some sporadic simple groups, for example the Hall-Janko group and the Mathieu groups, act as automorphism groups of near polygons.
