TheInfoList

A finite geometry is any
geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ... system that has only a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
number of
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
. The familiar
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the
pixel In digital imaging Digital imaging or digital image acquisition is the creation of a representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imp ... s 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 space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
s because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and
Laguerre plane In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the France, French mathematician Edmond Nicolas Laguerre. The classical Lag ...
s, which are examples of a general type called
Benz plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s, and their higher-dimensional analogs such as higher finite inversive geometries. Finite geometries may be constructed via
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, starting from
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s over a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
; the affine and
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite
projective space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of dimension three or greater is
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... 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 the
non-Desarguesian planeIn mathematics, a non-Desarguesian plane is a projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect in a si ...
s. Similar results hold for other kinds of finite geometries.

# Finite planes The following remarks apply only to finite ''planes''. There are two main kinds of finite plane geometry:
affine Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology *Affine cipher, a special case of the more general substitution cipher *Aff ...
and projective. In an
affine plane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of fi ...
, the normal sense of
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
lines applies. In a
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 simple
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ... s.

## Finite affine planes

An affine plane geometry is a nonempty set ''X'' (whose elements are called "points"), along with a nonempty collection ''L'' of subsets of ''X'' (whose elements are called "lines"), such that: # For every two distinct points, there is exactly one line that contains both points. #
Playfair's axiom In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
: Given a line $\ell$ and a point $p$ not on $\ell$, there exists exactly one line $\ell\text{'}$ containing $p$ such that $\ell \cap \ell\text{'} = \varnothing.$ # There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not ''trivial'' (either
empty 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 the ''affine 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". More generally, a finite affine plane of order ''n'' has ''n''2 points and lines; each line contains ''n'' points, and each point is on lines. The affine plane of order 3 is known as the
Hesse configuration In geometry, the Hesse configuration, introduced by Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish Scottish usually refers to something of, from, or related to Scotland, inclu ... .

## Finite projective planes

A projective plane geometry is a nonempty set ''X'' (whose elements are called "points"), along with a nonempty collection ''L'' of subsets of ''X'' (whose elements are called "lines"), such that: # For every two distinct points, there is exactly one line that contains both points. # The intersection of any two distinct lines contains exactly one point. # 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 of duality 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 the
Fano plane In finite geometry, the Fano plane (after Gino Fano) is the Projective plane#Finite projective planes, finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 li ... . 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 called the ''projective plane of order'' 2 because it is unique (up to isomorphism). In general, the projective plane of order ''n'' has ''n''2 + ''n'' + 1 points and the same number of lines; each line contains ''n'' + 1 points, and each point is on ''n'' + 1 lines. A permutation of the Fano plane's seven points that carries
collinear In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
points (points on the same line) to collinear points is called a
collineation In projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a ...
of the plane. The full
collineation group In projective geometry, a collineation is a injective function, one-to-one and surjection, onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the Image (mathematics), images of collinear poin ...
is of order 168 and is isomorphic to the group
PSL(2,7) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
≈ PSL(3,2), which in this special case is also isomorphic to the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
.

## Order of planes

A finite plane of order ''n'' is one such that each line has ''n'' points (for an affine plane), or such that each line has ''n'' + 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 order ''n'' exist whenever ''n'' is a
prime power In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
(a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
raised to a
positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, in optics * Plus sign, the sign "+" used to indicate a positive number * Positive (electricity), a po ...
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
exponent Exponentiation is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
), by using affine and projective planes over the finite field with elements. Planes not derived from finite fields also exist (e.g. for $n=9$), but all known examples have order a prime power. The best general result to date is the Bruck–Ryser theorem of 1949, which states: :If ''n'' is a
positive integer In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of the form or and ''n'' is not equal to the sum of two integer
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...
s, then ''n'' 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 form , but it is equal to the sum of squares . The non-existence of a finite plane of order 10 was proven in a
computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large Proof by exhaustion, proofs-by-exhaustion of a mathematical theorem. The ...
that finished in 1989 – see for details. The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.

## History

Individual examples can be found in the work of Thomas Penyngton Kirkman (1847) and the systematic development of finite projective geometry given by von Staudt (1856). The first axiomatic treatment of finite projective geometry was developed by the
Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, a Romance language *** Regional Italian, regional variants of the ...
mathematician
Gino Fano Gino Fano (5 January 18718 November 1952) was an Italians, Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy. Fano made various contri ... . In his work on proving the independence of the set of axioms for projective ''n''-space that he developed, 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. In 1906
Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quant ...
and W. H. Bussey described
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
using
homogeneous coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
with entries from the
Galois field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
GF(''q''). When ''n'' + 1 coordinates are used, the ''n''-dimensional finite geometry is denoted PG(''n, q''). It arises in
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
and has an associated transformation
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
.

# Finite spaces of 3 or more dimensions

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

## Axiomatic definition

A projective space ''S'' can be defined axiomatically as a set ''P'' (the set of points), together with a set ''L'' of subsets of ''P'' (the set of lines), satisfying these axioms : * Each two distinct points ''p'' and ''q'' are in exactly one line. * Veblen's axiom: If ''a'', ''b'', ''c'', ''d'' are distinct points and the lines through ''ab'' and ''cd'' meet, then so do the lines through ''ac'' and ''bd''. * 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 an
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 plane as the two types of objects and ignore al ...
consisting of a set ''P'' of points, a set ''L'' of lines, and an
incidence relationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
''I'' stating which points lie on which lines. Obtaining a ''finite'' projective space requires one more axiom: * The set of points ''P'' is a finite set. In any finite projective space, each line contains the same number of points and the ''order'' of the space is defined as one less than this common number. A subspace of the projective space is a subset ''X'', such that any line containing two points of ''X'' is a subset of ''X'' (that is, completely contained in ''X''). The full space and the empty space are always subspaces. The ''geometric dimension'' of the space is said to be ''n'' if that is the largest number for which there is a strictly ascending chain of subspaces of this form: : $\varnothing = X_ \subset X_\subset \cdots \subset X_ = P .$

## Algebraic construction

A standard algebraic construction of systems satisfies these axioms. For a
division ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
''D'' construct an -dimensional vector space over ''D'' (vector space dimension is the number of elements in a basis). Let ''P'' be the 1-dimensional (single generator) subspaces and ''L'' the 2-dimensional (two independent generators) subspaces (closed under vector addition) of this vector space. Incidence is containment. If ''D'' is finite then it must be a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
GF(''q''), since by
Wedderburn's little theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 has points on a line, so the two concepts of order coincide. Such a finite projective space is denoted by , where PG stands for projective geometry, ''n'' is the geometric dimension of the geometry and ''q'' is the size (order) of the finite field used to construct the geometry. In general, the number of ''k''-dimensional subspaces of is given by the product:, where the formula is given, in terms of vector space dimension, by . :$_q = \prod_^k \frac,$ which is a
Gaussian binomial coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, a ''q'' analogue of a
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
.

## Classification of finite projective spaces by geometric dimension

*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 a ''projective line''. *Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for is a
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. These are much harder to classify, as not all of them are isomorphic with a . The
Desarguesian plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, paral ...
s (those that are isomorphic with a ) satisfy
Desargues's theorem In projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has ...
and are projective planes over finite fields, but there are many
non-Desarguesian planeIn mathematics, a non-Desarguesian plane is a projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect in a si ...
s. *Dimension at least 3: Two non-intersecting lines exist. The Veblen–Young theorem states in the finite case that every projective space of geometric dimension is isomorphic with a , the ''n''-dimensional projective space over some finite field GF(''q'').

## The smallest projective three-space The smallest 3-dimensional projective space is over the field
GF(2) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with ...
and is denoted by
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 following properties: * Each point is contained in ...
. 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 are
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... to the
Fano plane In finite geometry, the Fano plane (after Gino Fano) is the Projective plane#Finite projective planes, finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 li ... . 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 Gino Fano (5 January 18718 November 1952) was an Italians, Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy. Fano made various contri ... was the first to consider such a finite geometry.

### Kirkman's schoolgirl problem

PG(3,2) arises as the background for a solution of
Kirkman's schoolgirl problem Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in ''The Lady's and Gentleman's Diary'' (pg.48). The problem states: Fifteen young ladies in a school walk out three abreast ...
, 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 as ''packings''. A ''spread'' of a projective space is a partition 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 In combinatorics, combinatorial mathematics, a block design is an incidence structure consisting of a set together with a Family of sets, family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain condition ...
– a generalization of a finite projective plane. *
Generalized polygon In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
*
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, continuity, betweenness, and incidence (geometry), inc ...
*
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. The points in a line are said to be incident ...
* Near polygon *
Partial geometry An incidence structure C=(P,L,I) consists of points P, lines L, and flags I \subseteq P \times L where a point p is said to be incident with a line l if (p,l) \in I. It is a (Wikt:finite, finite) partial geometry if there are integers s,t,\alpha\geq ...
* Polar space

# References

* * * * * * * * * * * * * .

*
Incidence Geometry by Eric MoorhouseAlgebraic Combinatorial Geometry
by
Terence Tao Essay on Finite Geometry by Michael GreenbergFinite geometry (Script)Finite Geometry ResourcesJ. W. P. Hirschfeld
researcher on finite geometries

Galois Geometry and Generalized Polygons
intensive course in 1998 * *
Projective Plane of Order 12
on MathOverflow. {{Authority control Combinatorics