Arc (projective geometry)
An (simple) arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of curved figures in continuous geometries. Loosely speaking, they are sets of points t
De Bruijn–Erdős theorem (incidence geometry)
In incidence geometry, the De Bruijn–Erdős theorem, originally published by Nicolaas Govert de Bruijn and Paul Erdős, states a lower bound on the number of lines determined by n points in a projective
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that in general the number of
Lie sphere geometry
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main
Near polygon
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 l
Generalized quadrangle
In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar spa
Ternary equivalence relation
In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive. The cla
Problem of Apollonius
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 BC – c. 190 BC) posed and solved th
Generalized polygon
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and g
Fano plane
In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines t
Moulton plane
In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues's theorem does not hold. It is named after the American astronomer Forest Ray Moulton. The points of the Mo
Buekenhout geometry
In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by .
Concyclic points
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the pla
Loomis–Whitney inequality
In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a -dimensional set by the sizes of its -dimensional projections.
Intersection theorem
In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their i
Flag (geometry)
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. More formally, a flag ψ of an n-polytope is a set {F–1, F0
Incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean
Incidence geometry
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, contin
Abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points an
Moufang polygon
In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of .In a book on th
Linear space (geometry)
A linear space is a basic structure in incidence geometry. A linear space consists of a set of elements called points, and a set of elements called lines. Each line is a distinct subset of the points.
Metasymplectic space
In mathematics, a metasymplectic space, introduced by Freudenthal and , 10.13), is a Tits building of type F4 (a specific generalized incidence structure).The four types of vertices are called points,
Bundle theorem
In Euclidean geometry, the bundle theorem is a statement about six circles and eight points in the Euclidean plane. In general incidence geometry, it is a similar property that a Möbius plane may or m
Ovoid (projective geometry)
In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential geom
Partial geometry
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:
* For any pair o
Möbius plane
In mathematics, a Möbius plane (named after August Ferdinand Möbius) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of line
Oval (projective plane)
In projective geometry an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate conics. However, a conic is only defined in a pappian plan
Unital (geometry)
In geometry, a unital is a set of n3 + 1 points arranged into subsets of size n + 1 so that every pair of distinct points of the set are contained in exactly one subset. n ≥ 3 is required by some auth
Qvist's theorem
In projective geometry, Qvist's theorem, named after the Finnish mathematician , is a statement on ovals in finite projective planes. Standard examples of ovals are non-degenerate (projective) conic s
Segre's theorem
In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement:
* Any oval in a finite pappian projective plane of odd order is a nondegenerate proje
Collinearity
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater gen
Ovoid (polar space)
In mathematics, an ovoid O of a (finite) polar space of rank r is a set of points, such that every subspace of rank intersects O in exactly one point.
Affine plane (incidence geometry)
In geometry, an affine plane is a system of points and lines that satisfy the following axioms:
* Any two distinct points lie on a unique line.
* Given any line and any point not on that line there
Similarity system of triangles
A similarity system of triangles is a specific configuration involving a set of triangles. A set of triangles is considered a configuration when all of the triangles share a minimum of one incidence r
Projective plane
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 pa
PG(3,2)
In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane.It has 15 points, 35 lines, and 15 planes. It also has the follow
Topological geometry
Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations