Geometry
Projecting asphere to aplane
Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry
Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry
Zero-dimensional Point
One-dimensional Line segment ray Length
Two-dimensional Plane Area Polygon Triangle Altitude Hypotenuse Pythagorean theorem Parallelogram Square Rectangle Rhombus Rhomboid Quadrilateral Trapezoid Kite Circle Diameter Circumference Area
Three-dimensional Volume Cube cuboid Cylinder Dodecahedron Icosahedron Octahedron Pyramid Platonic Solid Sphere Tetrahedron
Four-/other-dimensional Tesseract Hypersphere
Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers
by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Chern Present day Atiyah Gromov
v t e

Afinite geometry is anygeometric system that has only afinite number ofpoints.The familiarEuclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where thepixels 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 finiteprojective andaffine spaces because of their regularity and simplicity. Other significant types of finite geometry are finiteMöbius or inversive planes andLaguerre planes, which are examples of a general type calledBenz planes, and their higher-dimensional analogs such as higher finiteinversive geometries.

Finite geometries may be constructed vialinear algebra, starting fromvector spaces over afinite field; the affine andprojective planes so constructed are calledGalois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finiteprojective space of dimension three or greater isisomorphic to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely thenon-Desarguesian planes. Similar results hold for other kinds of finite geometries.

The following remarks apply only to finiteplanes.There are two main kinds of finite plane geometry:affine andprojective.In anaffine plane, the normal sense ofparallel lines applies.In aprojective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simpleaxioms.

An affine plane geometry is a nonempty setX (whose elements are called "points"), along with a nonempty collectionL of subsets ofX (whose elements are called "lines"), such that:

1. For every two distinct points, there is exactly one line that contains both points.
2. Playfair's axiom: Given a line${\displaystyle \ell }$ and a point${\displaystyle p}$ not on${\displaystyle \ell }$, there exists exactly one line${\displaystyle {\ell }^{\prime }}$ containing${\displaystyle p}$ such that${\displaystyle \ell \cap {\ell }^{\prime }=\varnothing .}$
3. There exists a set of four points, no three of which belong to the same line.

The last axiom ensures that the geometry is nottrivial (eitherempty or too simple to be of interest, such as a single line with an arbitrary number of points on it), while the first two specify the nature of the geometry.

The simplest affine plane contains only four points; it is called theaffine plane of order 2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel".

The affine plane of order 3 is known as theHesse configuration.

More generally, a finite affine plane of ordern hasn² points andn² +n lines; each line containsn points, and each point is onn + 1 lines.

A projective plane geometry is a nonempty setX (whose elements are called "points"), along with a nonempty collectionL of subsets ofX (whose elements are called "lines"), such that:

1. For every two distinct points, there is exactly one line that contains both points.
2. The intersection of any two distinct lines contains exactly one point.
3. There exists a set of four points, no three of which belong to the same line.

An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged.This suggests the principle ofduality for projective plane geometries, meaning that any true statement valid in all these geometries remains true if we exchange points for lines and lines for points.The smallest geometry satisfying all three axioms contains seven points. In this simplest of the projective planes, there are also seven lines; each point is on three lines, and each line contains three points.

This particular projective plane is sometimes called theFano plane.If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2.The Fano plane is calledtheprojective plane of order 2 because it is unique (up to isomorphism).In general, the projective plane of ordern hasn² + n + 1 points and the same number of lines; each line containsn + 1 points, and each point is onn + 1 lines.

A permutation of the Fano plane's seven points that carriescollinear points (points on the same line) to collinear points is called acollineation of the plane. The fullcollineation group is of order 168 and is isomorphic to the groupPSL(2,7) ≈ PSL(3,2), which in this special case is also isomorphic to thegeneral linear groupGL(3,2) ≈ PGL(3,2).

A finite plane ofordern is one such that each line hasn points (for an affine plane), or such that each line hasn + 1 points (for a projective plane). One major open question in finite geometry is:

Is the order of a finite plane always a prime power?

This is conjectured to be true.

Affine and projective planes of ordern exist whenevern is aprime power (aprime number raised to apositiveintegerexponent), by using affine and projective planes over the finite field withn =p^k elements. Planes not derived from finite fields also exist (e.g. for${\displaystyle n=9}$), but all known examples have order a prime power.^[1]

The best general result to date is theBruck–Ryser theorem of 1949, which states:

Ifn is apositive integer of the form4k + 1 or4k + 2 andn is not equal to the sum of two integersquares, thenn does not occur as the order of a finite plane.

The smallest integer that is not a prime power and not covered by the Bruck–Ryser theorem is 10; 10 is of the form4k + 2, but it is equal to the sum of squares1² + 3². The non-existence of a finite plane of order 10 was proven in acomputer-assisted proof that finished in 1989 – see (Lam 1991) for details.

The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.

Individual examples can be found in the work ofThomas Penyngton Kirkman (1847) and the systematic development of finite projective geometry given byvon Staudt (1856).

The first axiomatic treatment of finite projective geometry was developed by theItalian mathematicianGino Fano. In his work^[2] on proving the independence of the set of axioms forprojectiven-space that he developed,^[3] he considered a finite three dimensional space with 15 points, 35 lines and 15 planes (see diagram), in which each line had only three points on it.^[4]

In 1906Oswald Veblen and W. H. Bussey describedprojective geometry usinghomogeneous coordinates with entries from theGalois field GF(q). Whenn + 1 coordinates are used, then-dimensional finite geometry is denoted PG(n, q).^[5] It arises insynthetic geometry and has an associated transformationgroup.

For some important differences between finiteplane geometry and the geometry of higher-dimensional finite spaces, seeaxiomatic projective space. For a discussion of higher-dimensional finite spaces in general, see, for instance, the works ofJ.W.P. Hirschfeld. The study of these higher-dimensional spaces (n ≥ 3) has many important applications in advanced mathematical theories.

Aprojective spaceS can be defined axiomatically as a setP (the set of points), together with a setL of subsets ofP (the set of lines), satisfying these axioms :^[6]

• Each two distinct pointsp andq are in exactly one line.
• Veblen's axiom:^[7] Ifa,b,c,d are distinct points and the lines throughab andcd meet, then so do the lines throughac andbd.
• Any line has at least 3 points on it.

The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as anincidence structure(P,L,I) consisting of a setP of points, a setL of lines, and anincidence relationI stating which points lie on which lines.

Obtaining afinite projective space requires one more axiom:

• The set of pointsP is a finite set.

In any finite projective space, each line contains the same number of points and theorder of the space is defined as one less than this common number.

A subspace of the projective space is a subsetX, such that any line containing two points ofX is a subset ofX (that is, completely contained inX). The full space and the empty space are always subspaces.

Thegeometric dimension of the space is said to ben if that is the largest number for which there is a strictly ascending chain of subspaces of this form:

${\displaystyle \varnothing =X_{-1}\subset X_0\subset \cdots \subset X_n=P.}$

A standard algebraic construction of systems satisfies these axioms. For adivision ringD construct an(n + 1)-dimensional vector space overD (vector space dimension is the number of elements in a basis). LetP be the 1-dimensional (single generator) subspaces andL the 2-dimensional (two independent generators) subspaces (closed under vector addition) of this vector space. Incidence is containment. IfD is finite then it must be afinite field GF(q), since byWedderburn's little theorem all finite division rings are fields. In this case, this construction produces a finite projective space. Furthermore, if the geometric dimension of a projective space is at least three then there is a division ring from which the space can be constructed in this manner. Consequently, all finite projective spaces of geometric dimension at least three are defined over finite fields. A finite projective space defined over such a finite field hasq + 1 points on a line, so the two concepts of order coincide. Such a finite projective space is denoted byPG(n,q), where PG stands for projective geometry,n is the geometric dimension of the geometry andq is the size (order) of the finite field used to construct the geometry.

In general, the number ofk-dimensional subspaces ofPG(n,q) is given by the product:^[8]

${\displaystyle {(\genfrac{}{}{0ex}{}{n+1}{k+1})}_q=\prod _{i=0}^k\frac{q^{n+1-i}-1}{q^{i+1}-1},}$

which is aGaussian binomial coefficient, aq analogue of abinomial coefficient.

• Dimension 0 (no lines): The space is a single point and is so degenerate that it is usually ignored.
• Dimension 1 (exactly one line): All points lie on the unique line, called aprojective line.
• Dimension 2: There are at least 2 lines, and any two lines meet. A projective space forn = 2 is aprojective plane. These are much harder to classify, as not all of them are isomorphic with aPG(d,q). TheDesarguesian planes (those that are isomorphic with aPG(2,q)) satisfyDesargues's theorem and are projective planes over finite fields, but there are manynon-Desarguesian planes.
• Dimension at least 3: Two non-intersecting lines exist. TheVeblen–Young theorem states in the finite case that every projective space of geometric dimensionn ≥ 3 is isomorphic with aPG(n,q), then-dimensional projective space over some finite field GF(q).

The smallest 3-dimensional projective space is over the fieldGF(2) and is denoted byPG(3,2). It has 15 points, 35 lines, and 15 planes. Each plane contains 7 points and 7 lines. Each line contains 3 points. As geometries, these planes areisomorphic to theFano plane.

Every point is contained in 7 lines. Every pair of distinct points are contained in exactly one line and every pair of distinct planes intersects in exactly one line.

In 1892,Gino Fano was the first to consider such a finite geometry.

PG(3,2) arises as the background for a solution ofKirkman's schoolgirl problem, which states: "Fifteen schoolgirls walk each day in five groups of three. Arrange the girls’ walk for a week so that in that time, each pair of girls walks together in a group just once." There are 35 different combinations for the girls to walk together. There are also 7 days of the week, and 3 girls in each group. Two of the seven non-isomorphic solutions to this problem can be stated in terms of structures in the Fano 3-space, PG(3,2), known aspackings. Aspread of a projective space is apartition of its points into disjoint lines, and a packing is a partition of the lines into disjoint spreads. In PG(3,2), a spread would be a partition of the 15 points into 5 disjoint lines (with 3 points on each line), thus corresponding to the arrangement of schoolgirls on a particular day. A packing of PG(3,2) consists of seven disjoint spreads and so corresponds to a full week of arrangements.

• Block design – a generalization of a finite projective plane.
• Discrete space
• Finite space
• Generalized polygon
• Incidence geometry
• Linear space (geometry)
• Near polygon
• Partial geometry
• Polar space

• Batten, Lynn Margaret (1997),Combinatorics of Finite Geometries, Cambridge University Press,ISBN 0521590140
• Beutelspacher, Albrecht; Rosenbaum, Ute (1998),Projective geometry: from foundations to applications,Cambridge University Press,ISBN 978-0-521-48364-3,MR 1629468
• Collino, Alberto; Conte, Alberto; Verra, Alessandro (2013). "On the life and scientific work of Gino Fano".arXiv:1311.7177 [math.HO].
• Dembowski, Peter (1968),Finite geometries,Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York:Springer-Verlag,ISBN 3-540-61786-8,MR 0233275
• Eves, Howard (1963),A Survey of Geometry: Volume One, Boston: Allyn and Bacon Inc.
• Hall, Marshall (1943), "Projective planes",Transactions of the American Mathematical Society,54 (2), American Mathematical Society:229–277,doi:10.2307/1990331,ISSN 0002-9947,JSTOR 1990331,MR 0008892
• Lam, C. W. H. (1991),"The Search for a Finite Projective Plane of Order 10",American Mathematical Monthly,98 (4):305–318,doi:10.2307/2323798,JSTOR 2323798

• Malkevitch, Joe."Finite Geometries?". RetrievedDec 2, 2013.
• Meserve, Bruce E. (1983),Fundamental Concepts of Geometry, New York: Dover Publications
• Polster, Burkard (1999). "Yea why try her raw wet hat: A tour of the smallest projective space".The Mathematical Intelligencer.21 (2):38–43.doi:10.1007/BF03024845.S2CID 122352568.
• Segre, Beniamino (1960),On Galois Geometries(PDF), New York: Cambridge university Press, pp. 488–499, archived fromthe original(PDF) on 2015-03-30, retrieved2015-07-02

• Shult, Ernest E. (2011),Points and Lines, Universitext, Springer,doi:10.1007/978-3-642-15627-4,ISBN 978-3-642-15626-7

• Ball, Simeon (2015),Finite Geometry and Combinatorial Applications, London Mathematical Society Student Texts, Cambridge University Press,ISBN 978-1107518438.

• Weisstein, Eric W."finite geometry".MathWorld.
• Incidence Geometry by Eric Moorhouse
• Algebraic Combinatorial Geometry byTerence Tao
• Essay on Finite Geometry by Michael Greenberg
• Finite geometry (Script)
• Finite Geometry ResourcesArchived 2011-09-27 at theWayback Machine
• J. W. P. Hirschfeld, researcher on finite geometries
  • Books by Hirschfeld on finite geometry
• AMS Column: Finite Geometries?
• Galois Geometry and Generalized Polygons, intensive course in 1998
• Carnahan, Scott (2007-10-27),"Small finite sets",Secret Blogging Seminar, notes on a talk byJean-Pierre Serre on canonical geometric properties of small finite sets.{{citation}}: CS1 maint: postscript (link)
• “Problem 31: Kirkman's schoolgirl problem” at theWayback Machine (archived August 17, 2010)
• Projective Plane of Order 12 on MathOverflow.

• Finite geometry
• Combinatorics

• Articles with short description
• Short description is different from Wikidata
• Pages using sidebar with the child parameter
• Webarchive template wayback links
• CS1 maint: postscript