Finite Geometry in a Nutshell

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Finite Geometry in a Nutshell

• Jeffrey Iverson

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An Introduction

• Consider a line segment, a stick of thin
• spaghetti, if you will.

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An Introduction

• Consider a line segment, a stick of thin
• spaghetti, if you will.
• This segment can be split into smaller and
  smaller segments, suggesting an infinite amount
  of points.

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An Introduction

• Consider a line segment, a stick of thin
• spaghetti, if you will.
• This segment can be split into smaller and
  smaller segments, suggesting an infinite amount
  of points.
• In modern times, things are known to be made up
  of atoms and subatomic particles.

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An Introduction

• This segment can be split into smaller and
  smaller segments, suggesting an infinite amount
  of points.
• In modern times, things are known to be made up
  of atoms and subatomic particles.
• So one is to believe that there are only a finite
  number of atoms in the universe.

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An Introduction

• In modern times, things are known to be made up
  of atoms and subatomic particles.
• So one is to believe that there are only a finite
  number of atoms in the universe.
• So the question presents itself. Does it make
  sense to investigate a geometry where the axioms
  talked about the existence of finite points?

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An Introduction

• So one is to believe that there are only a finite
  number of atoms in the universe.
• So the question presents itself. Does it make
  sense to investigate a geometry where the axioms
  talked about the existence of finite points?
• And does it make sense to talk about finite
  geometries?

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An Introduction

• 
  

Sure Does!
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Definition

• A finite geometry is any geometric system that
  has only a finite number of points. Euclidean
  geometry, for example, is not finite, because a
  Euclidean line contains infinitely many points.
  A finite geometry can have any (finite) number of
  dimensions.

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Definition (cont)

• Today, we will define objects in our geometry as
  a non-empty set of points and a non-empty set of
  lines, where a line is a given subset of the set
  of points that contains at least two elements.

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Definition (cont)

• Today, we will define objects in our geometry as
  a non-empty set of points and a non-empty set of
  lines, where a line is a given subset of the set
  of points that contains at least two elements.
• Also, the objects in a given geometry are defined
  by a set of axioms. Any constructed examples of
  the geometry must satisfy every axiom.

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Definition (cont)

• Today, we will define objects in our geometry as
  a non-empty set of points and a non-empty set of
  lines, where a line is a given subset of the set
  of points that contains at least two elements.
• Also, the objects in a given geometry are defined
  by a set of axioms. Any constructed examples of
  the geometry must satisfy every axiom.
• Lets go over some classical examples of finite
  geometries.

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Classical examples.

• Projective Planes
• Affine Planes
• Near Linear Spaces
• Linear Spaces
• Designs
• Biplanes

For the Axioms of each of these geometries,
visit http//home.wlu.edu/7Emcraea/Finite_Geometr
y/Introduction/Prob2FiniteGeometries/Problem2.htm
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Projective Planes

• Projective Planes
• Two distinct points are contained in a unique
  line
• Two distinct lines interest at a unique point
• There exists four points of which no three are
  incident with the same line.

• Projective Planes
• Two distinct points are contained in a unique
  line
• Two distinct lines interest at a unique point
• There exists four points of which no three are
  incident with the same line.

The first two axioms are pretty much
self-explanatory.
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Projective Planes

• Projective Planes
• Two distinct points are contained in a unique
  line
• Two distinct lines interest at a unique point
• There exists four points of which no three are
  incident with the same line.

• Projective Planes
• Two distinct points are contained in a unique
  line
• Two distinct lines interest at a unique point
• There exists four points of which no three are
  incident with the same line.

The first two axioms are pretty much
self-explanatory. The third axiom says there are
four points however, with the first two axioms
there must be at least 7 points.
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Projective Planes

• So what exactly do we get with only 7 points?
• Definition In finite geometry, the Fano plane
  is the projective plane with the least number of
  points and lines 7 each.
• How would we construct this?

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Constructing the Fano Plane

• By the axioms for a projective plane, there are
  four points (p, q, r, s), no three of which are
  collinear. So there must be two distinct
  intersecting lines l and l.

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Constructing the Fano Plane

• Let q and r be the points that lie on l and l
  respectively, that are distinct from the point of
  intersection s, and let p be the fourth point
  that lies on neither l nor l.

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Constructing the Fano Plane

• Any two points determine a unique line, and any
  two lines intersect at a unique point. So create
  a line from l to l from q through p. Also,
  create another line from l to l from r through
  p. Label the intersection q and r.

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Constructing the Fano Plane

• By the axioms for a projective plane, all points
  must be connected, thus, we connect q and r,
  creating line l. Note that for any two lines
  (not through p), we must be able to go from a
  given point on any line, through p, to some point
  on another given line. Thus, we
  must create a line from s through p, to l.

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Constructing the Fano Plane

• We will call the new point t. Finally, we must
  have the points r, q and t connected by a line.
  The result is the Fano plane.

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Order of a Projective Plane

• Theorem
• A projective plane of order n, has n2n1 points
  as well as the same number of lines. Each line
  contains n1 points and each point lies on n1
  lines.

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Order of a Projective Plane

• Proof (partial)
• Let P be an arbitrary point.
• By Axiom 1, every point in the plane, X
  determines a line together with P, which clearly
  passes through P.
• Hence, all the points of X lie on the n1 lines
  incident with P. (not proved).

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Order of a Projective Plane

• Proof (partial)
• Let P be an arbitrary point.
• By Axiom 1, every point in the plane, X
  determines a line together with P, which clearly
  passes through P.
• Hence, all the points of X lie on the n1 lines
  incident with P. (not proved).

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Order of a Projective Plane

• Proof (partial)
• Let P be an arbitrary point.
• By Axiom 1, every point in the plane, X
  determines a line together with P, which clearly
  passes through P.
• Hence, all the points of X lie on the n1 lines
  incident with P. (not proved).

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Order of a Projective Plane

• Proof
• determines a line together with P, which clearly
  passes through P.
• Hence, all the points of X lie on the n1 lines
  incident with P. (not proved).
• Each of these lines are incident with n points
  other than P, so there are n(n1) of them and
  including P, we get n2n1 points in X.

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Order of a Projective Plane

• Proof
• Hence, all the points of X lie on the n1 lines
  incident with P. (not proved).
• Each of these lines are incident with n points
  other than P, so there are n(n1) of them and
  including P, we get n2n1 points in X.
• There is a similar type of proof for the number
  of lines in X.

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More on Fano Planes

• A Fano Plane is a projective plane of order 2.
• Every line is incident with 3 points
• Every point is incident with 3 lines
• There are 7 points and 7 lines.

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What about larger planes?

• Today, it is well established that projective
  planes of order n exist when n is a prime power.
  
• It is conjectured that no finite planes exist
  with orders that arent prime powers (not
  currently proven).
• The best result is the Bruck-Ryser Theorem.

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What about larger planes?

• It is conjectured that no finite planes exist
  with orders that arent prime powers (not
  currently proven).
• The best result is the Bruck-Ryser Theorem.
• This theorem states If n is a positive integer
  of the form 4k1 or 4k2 and n is not equal to
  the sum of two integer squares, then n is not the
  order of a finite plane.

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Future Studies

• Finite Geometry is a huge field.
• It can involve any finite number of dimensions
• Incidence matrices
• And many other different types of finite
  geometries as stated earlier.

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Homework

• An affine plane is derived from a projective
  plane by removing a line from the projective
  plane as well as its respective points.
• Construct an affine plane using a well a simple
  projective plane geometry.

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Sources

• http//home.wlu.edu/7Emcraea/Finite_Geometry/Tabl
  eOfContents/table.html
• http//www-math.cudenver.edu/7Ewcherowi/courses/m
  6406/m6406f.html
• http//finitegeometry.org/
• http//www.answers.com/topic/fano-plane