finite geometry
   HOME

TheInfoList



OR:

A finite geometry is any
geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
system that has only a finite number of
points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
. The familiar
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
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, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...
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 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 only the properties relat ...
s 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. 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 (mathemat ...
, starting from
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
; the affine and
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 at a single point, but there are some pairs of lines (namely, paral ...
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, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of dimension three or greater is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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 planes. 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 may describe any of various topics concerned with 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 substi ...
and projective. In an affine plane, the normal sense of parallel lines applies. In 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 at a single point, but there are some pairs of lines (namely, paral ...
, 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 true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
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, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidea ...
: Given a line \ell and a point p not on \ell, there exists exactly one line \ell' containing p such that \ell \cap \ell' = \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". The affine plane of order 3 is known as the Hesse configuration. 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.


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. 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, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
points (points on the same line) to collinear points is called a collineation of the plane. The full collineation group is of order 168 and is isomorphic to the group PSL(2,7) ≈ PSL(3,2), which in this special case is also isomorphic to the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
.


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, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...
(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
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
exponent In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
), 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, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
of the form or and ''n'' is not equal to the sum of two integer
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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 proof Automation describes a wide range of technologies that reduce human intervention in processes, mainly by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machine ...
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 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 contr ...
. 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, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was lo ...
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 (''p ...
using
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
with entries from the
Galois field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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 geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates ...
and has an associated transformation
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
.


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 Point (geometry), points and Line (geometry), lines of the Euclidean plane as t ...
consisting of a set ''P'' of points, a set ''L'' of lines, and an
incidence relation In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point ''lies on'' a line" or "a line is ''contained in'' a plane" are used. The most basic incidence relation is that betw ...
''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, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
''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, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
GF(''q''), since by
Wedderburn's little theorem In mathematics, Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Art ...
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, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or ''q''-binomial coefficients) are ''q''-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as \binom nk ...
, a ''q'' analogue of a
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
.


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, 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 at a single point, but there are some pairs of lines (namely, paral ...
. 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. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that d ...
s (those that are isomorphic with a ) satisfy
Desargues's theorem In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...
and are projective planes over finite fields, but there are many non-Desarguesian planes. *Dimension at least 3: Two non-intersecting lines exist. The
Veblen–Young theorem In mathematics, the Veblen–Young theorem, proved by , states that a projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a division ring. Non-Desarguesian planes give examples of ...
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 with two elements. is the Field (mathematics), field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity ...
and is denoted by PG(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 are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the Fano 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 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 contr ...
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, 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.


See also

*
Block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that number of occurrences of each element satisfies certain conditions making the co ...
– a generalization of a finite projective plane. *
Discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
* Finite space *
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 generalized quadrangles (''n'' = 4). Ma ...
* Incidence geometry * Linear space (geometry) * Near polygon * Partial geometry * Polar space


Notes


References

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


External links

*
Incidence Geometry by Eric MoorhouseAlgebraic Combinatorial Geometry
by Terence Tao
Essay on Finite Geometry by Michael GreenbergFinite geometry (Script)Finite Geometry Resources

J. 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