The Central Idea

In ordinary Euclidean geometry, every line contains infinitely many points, and the plane itself contains infinitely many lines. Finite geometry asks a different question:

How much geometric structure can exist when the entire system has only finitely many points and lines?

The answer is: a great deal.

A finite geometry can preserve familiar geometric ideas such as

What changes is that the collection of objects is finite and highly organized.

Geometry as an Incidence System

In finite geometry, the most basic relationship is often incidence.

A point is incident with a line when the point lies on that line.

Instead of beginning with lengths and angles, we can begin with three pieces of information:

  1. a finite set of points;
  2. a finite set of lines;
  3. a rule describing which points lie on which lines.

This is enough to create a meaningful geometry.

A Tiny Example

Suppose the points are

\[ P_1,\quad P_2,\quad P_3. \]

Let the lines be

\[ L_{12}=\{P_1,P_2\}, \] \[ L_{13}=\{P_1,P_3\}, \] \[ L_{23}=\{P_2,P_3\}. \]

Every pair of points lies on exactly one line.

This small system already exhibits a geometric incidence rule even though it contains only three points.

Finite Does Not Mean Approximate

A finite geometry is not merely a rough pixelated version of Euclidean geometry.

It is an exact mathematical structure with its own axioms and consequences.

A finite projective plane, for example, is not obtained by simply taking a few sample points from the ordinary projective plane. Its finite point and line sets are arranged so that the projective incidence axioms hold exactly.

Incidence Geometry

An incidence geometry focuses on which geometric objects are incident with one another.

Typical questions include:

These questions are structural rather than metric. Distance and angle need not be defined at all.

The Two Major Models in This Lesson

Two especially important families of finite geometries are

They mirror the distinction already encountered between affine and projective geometry.

Affine planes contain parallel lines. Projective planes eliminate parallelism by arranging the system so that every two distinct lines meet in exactly one point.

Finite Affine Planes

A finite affine plane is a finite point-line system satisfying affine-style incidence and parallel rules.

One standard axiom system requires:

  1. Any two distinct points lie on exactly one line.
  2. Given a line \(L\) and a point \(P\) not on \(L\), exactly one line through \(P\) is parallel to \(L\).
  3. There exist three noncollinear points.

The second condition is the finite analogue of the Euclidean parallel postulate.

Order of a Finite Affine Plane

If every line in a finite affine plane contains exactly \(q\) points, the plane is said to have order \(q\).

From the affine-plane axioms, several counting facts follow.

A finite affine plane of order \(q\) has

\[ \boxed{q^2\text{ points}}. \]

It also has

\[ \boxed{q(q+1)\text{ lines}}. \]

Each line contains \(q\) points, and each point lies on \(q+1\) lines.

Parallel Classes

In an affine plane, lines can be grouped according to parallel direction.

A parallel class is a collection of mutually parallel lines that together cover every point of the plane.

In an affine plane of order \(q\):

Therefore,

\[ q(q+1) \]

lines occur altogether.

The Affine Plane of Order 2

The smallest finite affine plane has order

\[ q=2. \]

It contains

\[ q^2=4 \]

points and

\[ q(q+1)=2\cdot3=6 \]

lines.

Each line contains two points, and each point lies on three lines.

There are three parallel classes, each containing two lines.

A Coordinate Model for the Affine Plane of Order 2

Use coordinates whose entries come from the two-element field

\[ \mathbb{F}_2=\{0,1\}. \]

The four points are

\[ (0,0),\quad(0,1),\quad(1,0),\quad(1,1). \]

Arithmetic is performed modulo \(2\).

Thus,

\[ 1+1=0 \]

in \(\mathbb{F}_2\).

Lines in the Order-2 Affine Plane

The six lines can be represented by

\[ x=0,\qquad x=1, \] \[ y=0,\qquad y=1, \] \[ y=x,\qquad y=x+1, \]

with arithmetic in \(\mathbb{F}_2\).

These form three parallel classes:

Finite Fields

Coordinate constructions of finite planes rely on finite fields.

A finite field is a finite number system in which addition, subtraction, multiplication, and division by nonzero elements behave according to the field laws.

The notation

\[ \mathbb{F}_q \]

commonly denotes a finite field with \(q\) elements.

Prime Fields

If \(p\) is prime, the integers modulo \(p\) form a field:

\[ \mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}. \]

For example,

\[ \mathbb{F}_3=\{0,1,2\}. \]

Arithmetic is performed modulo \(3\):

\[ 2+2=1, \] \[ 2\cdot2=1. \]

Because every nonzero element has a multiplicative inverse, coordinate geometry can be carried out in this finite setting.

Prime Powers

Finite fields exist not only for primes but for every prime power

\[ q=p^n, \]

where \(p\) is prime and \(n\) is a positive integer.

Examples include

\[ 2,\ 3,\ 4,\ 5,\ 7,\ 8,\ 9,\ 11,\ 13,\ 16,\ldots \]

Each such finite field can be used to construct finite affine and projective planes of order \(q\).

The Affine Coordinate Plane over \(\mathbb{F}_q\)

The finite affine plane

\[ AG(2,q) \]

has points

\[ (x,y) \]

with

\[ x,y\in\mathbb{F}_q. \]

Because there are \(q\) choices for \(x\) and \(q\) choices for \(y\), the total number of points is

\[ \boxed{q^2}. \]

Affine Lines over a Finite Field

Nonvertical affine lines can be written

\[ y=mx+b, \]

where

\[ m,b\in\mathbb{F}_q. \]

There are \(q\) possible slopes and \(q\) possible intercepts, producing

\[ q^2 \]

nonvertical lines.

There are also \(q\) vertical lines

\[ x=c. \]

Hence the total number of affine lines is

\[ q^2+q=q(q+1). \]

Example: The Affine Plane of Order 3

Let

\[ q=3. \]

Then the plane has

\[ 3^2=9 \]

points.

It has

\[ 3(3+1)=12 \]

lines.

Every line contains \(3\) points, and every point lies on \(4\) lines.

Example: A Line in \(AG(2,3)\)

Consider

\[ y=2x+1 \]

over \(\mathbb{F}_3\).

Evaluate for \(x=0,1,2\):

\[ x=0\Rightarrow y=1, \] \[ x=1\Rightarrow y=0, \] \[ x=2\Rightarrow y=2. \]

Thus the line contains

\[ \boxed{(0,1),\ (1,0),\ (2,2)}. \]

Parallelism in Finite Coordinates

Lines with the same slope but different intercepts are parallel.

For example, in \(AG(2,3)\),

\[ y=x,\qquad y=x+1,\qquad y=x+2 \]

form one parallel class.

Together, the three lines contain all nine points exactly once.

Finite Projective Planes

A finite projective plane removes the exceptional role of parallel lines.

One standard axiom system requires:

  1. Any two distinct points lie on exactly one line.
  2. Any two distinct lines meet in exactly one point.
  3. There exist four points with no three on the same line.

The second axiom is the projective replacement for affine parallelism.

No Parallel Lines in a Projective Plane

In an affine plane, two lines in the same parallel class do not intersect.

In a projective plane, every pair of distinct lines intersects.

Informally, the projective plane adds a point at infinity for each affine direction and then places all of those new points on a line at infinity.

This converts each affine parallel class into a family of projective lines meeting at one common point.

Order of a Finite Projective Plane

A finite projective plane has order \(q\) when each line contains

\[ q+1 \]

points.

In a projective plane of order \(q\):

The Projective Counting Formula

A projective plane of order \(q\) therefore has

\[ \boxed{q^2+q+1} \]

points and exactly the same number of lines.

This equality between the point count and the line count reflects the strong point-line symmetry of projective geometry.

Why \(q^2+q+1\)?

Choose a point \(P\).

Exactly \(q+1\) lines pass through \(P\).

Each such line contains \(q\) additional points besides \(P\).

Distinct lines through \(P\) share no other point. Therefore,

\[ 1+q(q+1) \]

points occur altogether.

Simplifying,

\[ \boxed{1+q(q+1)=q^2+q+1}. \]

The Fano Plane

The smallest finite projective plane has order

\[ q=2. \]

It is called the Fano plane.

The Fano plane has

\[ 2^2+2+1=7 \]

points and

\[ 7 \]

lines.

Each line contains \(3\) points, and each point lies on \(3\) lines.

The Famous Fano Diagram

The Fano plane is often drawn using seven marked points arranged in a triangular pattern.

Six of the lines can be drawn as ordinary straight segments, while the seventh is often drawn as a circle through three points.

The circular appearance does not mean that one projective line is intrinsically curved.

The drawing is only a visual encoding of incidence. In the abstract geometry, all seven lines have the same status.

Every Pair of Fano Points Determines a Line

The Fano plane has

\[ \binom{7}{2}=21 \]

unordered pairs of distinct points.

Each line contains three points, producing

\[ \binom{3}{2}=3 \]

point pairs per line.

With seven lines,

\[ 7\cdot3=21. \]

Thus the seven lines account for every pair of points exactly once.

Every Pair of Fano Lines Meets

Dually, there are

\[ \binom{7}{2}=21 \]

pairs of lines.

Each point lies on three lines and therefore determines

\[ \binom{3}{2}=3 \]

pairs of incident lines.

Across seven points,

\[ 7\cdot3=21. \]

Every pair of lines therefore meets in exactly one point.

Projective Coordinates over a Finite Field

The finite projective plane over \(\mathbb{F}_q\) is commonly written

\[ PG(2,q). \]

Its points can be represented by nonzero triples

\[ (x,y,z) \]

with entries in \(\mathbb{F}_q\), where nonzero scalar multiples represent the same projective point.

Homogeneous Coordinates

In projective coordinates,

\[ [x:y:z] \]

and

\[ [\lambda x:\lambda y:\lambda z] \]

represent the same point whenever

\[ \lambda\ne0. \]

This is the same homogeneous-coordinate idea introduced in the Projective Geometry lesson, now carried out over a finite field.

Counting Projective Points from Coordinates

There are

\[ q^3-1 \]

nonzero triples in \(\mathbb{F}_q^3\).

Each projective point is represented by

\[ q-1 \]

nonzero scalar multiples.

Therefore,

\[ \frac{q^3-1}{q-1} \]

projective points occur.

Since

\[ q^3-1=(q-1)(q^2+q+1), \]

we obtain

\[ \boxed{q^2+q+1}. \]

Projective Lines in Coordinates

A projective line can be represented by a nonzero coefficient triple

\[ [a:b:c] \]

corresponding to the homogeneous equation

\[ ax+by+cz=0. \]

Scalar multiples of the coefficient triple describe the same line.

This parallel between point coordinates and line coordinates is one reason projective point-line duality becomes so natural.

Affine Planes Inside Projective Planes

A projective plane can be converted into an affine plane by removing one projective line and all of its points.

If the projective plane has order \(q\), the removed line contains

\[ q+1 \]

points.

Starting from

\[ q^2+q+1 \]

points leaves

\[ q^2 \]

affine points.

Projective Completion of an Affine Plane

The reverse process is called projective completion.

Begin with an affine plane of order \(q\).

It has \(q+1\) parallel classes. Add one new point for each parallel class, so all lines of that class pass through the corresponding new point.

Then place all \(q+1\) new points on one additional line, called the line at infinity.

The resulting incidence structure is a projective plane of order \(q\).

From Affine Counts to Projective Counts

The affine plane has

\[ q^2 \]

points.

Projective completion adds

\[ q+1 \]

points at infinity.

Therefore,

\[ q^2+(q+1)=\boxed{q^2+q+1}. \]

Similarly, the affine plane has \(q(q+1)\) lines, and projective completion adds one line at infinity:

\[ q(q+1)+1=\boxed{q^2+q+1}. \]

Point-Line Duality

The projective-plane axioms are nearly unchanged if we exchange the words point and line.

Compare:

This symmetry is called duality.

It helps explain why the number of points equals the number of lines in every finite projective plane.

Regularity

Finite affine and projective planes are highly regular structures.

In a projective plane of order \(q\):

No point or line is structurally exceptional.

This regularity is one reason finite planes connect so strongly with symmetric combinatorial designs.

Finite Projective Planes as Designs

A projective plane of order \(q\) can be viewed as a balanced incidence design.

The parameters are

\[ v=q^2+q+1, \] \[ k=q+1, \] \[ \lambda=1. \]

Here:

Incidence Matrices

A finite geometry can be encoded using an incidence matrix.

Put one row for each point and one column for each line.

Define

\[ M_{ij}= \begin{cases} 1,&\text{if point }P_i\text{ lies on line }L_j,\\ 0,&\text{otherwise.} \end{cases} \]

The geometry is then represented by a finite \(0\)-\(1\) matrix.

What an Incidence Matrix Reveals

In a projective plane of order \(q\), every row contains

\[ q+1 \]

ones because each point lies on \(q+1\) lines.

Every column also contains

\[ q+1 \]

ones because each line contains \(q+1\) points.

Thus algebraic properties of the matrix reflect geometric regularity.

Counting Incidences Two Ways

An incidence is a point-line pair \((P,L)\) with \(P\in L\).

In a projective plane of order \(q\), count incidences by lines:

\[ (q^2+q+1)(q+1). \]

Count instead by points:

\[ (q^2+q+1)(q+1). \]

The agreement is a simple example of double counting, a major technique in combinatorics.

Finite Geometry and Combinatorics

Finite geometry naturally becomes combinatorial because every collection can be counted exactly.

Typical problems ask:

Collinearity

Three or more points are collinear if they lie on the same line.

In a finite projective plane of order \(q\), every line contains exactly

\[ q+1 \]

collinear points.

For the Fano plane, every line contains three collinear points.

Triangles in a Finite Projective Plane

A projective triangle consists of three noncollinear points and the three lines joining them pairwise.

The definition is incidence-based.

No lengths or angle measures are required.

This is an important reminder that a geometric object can be meaningful even when familiar Euclidean metric quantities are absent.

Finite Geometry Is Usually Non-Metric

Many finite geometries studied here do not begin with a distance function.

That distinguishes them from the previous lesson on alternative metrics.

A finite incidence plane asks primarily

Which points lie on which lines?

rather than

How far apart are the points?

Metrics can be added in some contexts, but they are not part of the basic affine- or projective-plane structure.

Finite Geometry Is Not Discrete Geometry in General

The phrases finite geometry and discrete geometry overlap but are not synonymous.

Finite geometry specifically studies geometric structures with finitely many objects, often using finite fields and incidence axioms.

Discrete geometry is broader and may study infinitely many objects arranged discretely, as well as lattices, tilings, packings, polytopes, and other combinatorial structures.

Finite Geometry and Graphs

A finite geometry can generate a graph.

For example, construct an incidence graph with:

This creates a bipartite graph because edges always connect a point-vertex to a line-vertex.

Incidence Graph of a Projective Plane

For a projective plane of order \(q\), the incidence graph has

\[ 2(q^2+q+1) \]

vertices.

Every vertex has degree

\[ q+1. \]

Thus the graph is highly regular and inherits much of the symmetry of the geometry.

Finite Geometry and Error-Correcting Codes

Incidence structures from finite geometry can be converted into error-correcting codes.

One basic approach uses rows or columns of an incidence matrix as binary vectors.

Geometric regularity then produces predictable algebraic relationships among the vectors.

More advanced coding theory makes extensive use of finite fields, projective spaces, and algebraic geometry over finite fields.

Finite Geometry and Experimental Design

Balanced incidence patterns are useful when designing experiments.

The goal may be to arrange treatments, blocks, or comparisons so that pairs occur together in a uniform way.

The same combinatorial regularity seen in finite projective planes can therefore become a design principle in statistics.

Finite Geometry and Computer Science

Finite geometries appear in several computational settings because the structures are exact, finite, algebraic, and highly symmetric.

Applications and connections include:

Existence Is a Nontrivial Question

The counting formulas tell us what a finite projective plane of order \(q\) would have to look like numerically.

They do not prove that such a plane exists for every positive integer \(q\).

Coordinate constructions over finite fields guarantee affine and projective planes whenever

\[ q=p^n \]

is a prime power.

The general existence problem is deeper than the formulas alone suggest.

Prime-Power Orders

For every prime power \(q\), the finite field \(\mathbb{F}_q\) exists.

Consequently, the coordinate constructions

\[ AG(2,q) \]

and

\[ PG(2,q) \]

provide finite affine and projective planes of order \(q\).

These are called Desarguesian planes.

Not Every Finite Plane Must Come from Coordinates

Finite projective planes need not all arise from the standard coordinate construction over a finite field.

Planes not isomorphic to the usual \(PG(2,q)\) construction are called non-Desarguesian.

This shows that a single order can sometimes support more than one genuinely different projective incidence structure.

Isomorphism

Two finite geometries are isomorphic when their points and lines can be relabeled so that incidence relationships are preserved.

Isomorphic geometries may have very different drawings.

What matters is not the visual arrangement on the page but the abstract structure of which points lie on which lines.

Automorphisms

An automorphism is an incidence-preserving symmetry of a geometry onto itself.

It permutes points and lines while preserving all incidence relations.

Finite geometries often have rich automorphism groups, making them important examples in the study of symmetry.

Why Drawings Can Be Misleading

A finite projective plane may be drawn on ordinary paper, but the picture is not itself the geometry.

In a Euclidean drawing:

The incidence specification is authoritative.

A Useful Mental Shift

In Euclidean geometry, a line is often imagined first as a continuous straight path.

In finite incidence geometry, it is often better to think:

A line is a distinguished subset of points satisfying the geometry's incidence rules.

This abstract viewpoint prepares us for increasingly general geometric structures.

Example: Counting an Affine Plane of Order 5

Suppose an affine plane has order

\[ q=5. \]

Number of points:

\[ q^2=25. \]

Number of lines:

\[ q(q+1)=5\cdot6=30. \]

Points per line:

\[ 5. \]

Lines through each point:

\[ 6. \]

Example: Counting a Projective Plane of Order 5

For

\[ q=5, \]

the number of points is

\[ 5^2+5+1=31. \]

The number of lines is also

\[ 31. \]

Each line contains

\[ 6 \]

points, and each point lies on \(6\) lines.

Example: Completing an Affine Plane

Begin with an affine plane of order \(4\).

It has

\[ 4^2=16 \]

points and

\[ 4(5)=20 \]

lines.

There are

\[ 5 \]

parallel classes.

Add one point at infinity for each class and one line at infinity.

The projective completion therefore has

\[ 16+5=21 \]

points and

\[ 20+1=21 \]

lines.

Example: Recovering an Affine Plane

Suppose a projective plane has order \(3\).

It contains

\[ 3^2+3+1=13 \]

points.

Remove one projective line. That line contains

\[ 3+1=4 \]

points.

The remaining affine plane has

\[ 13-4=9 \]

points, as expected from

\[ 3^2=9. \]

Example: Number of Incidences

Consider a projective plane of order \(3\).

It has \(13\) points and \(13\) lines.

Each line contains \(4\) points.

Therefore the number of point-line incidences is

\[ \boxed{13\cdot4=52}. \]

Counting by points gives the same result because each point lies on four lines.

Example: Pairs of Points in a Projective Plane

In a projective plane of order \(q\), every pair of distinct points lies on exactly one line.

The number of point pairs is

\[ \binom{q^2+q+1}{2}. \]

Each line contains

\[ \binom{q+1}{2} \]

point pairs, and there are \(q^2+q+1\) lines.

Therefore,

\[ \binom{q^2+q+1}{2} = (q^2+q+1)\binom{q+1}{2}. \]

This identity can be verified algebraically.

Checking the Pair-Count Identity

The left side is

\[ \frac{(q^2+q+1)(q^2+q)}{2}. \]

Since

\[ q^2+q=q(q+1), \]

this becomes

\[ \frac{(q^2+q+1)q(q+1)}{2}. \]

The right side is

\[ (q^2+q+1)\frac{q(q+1)}{2}, \]

so the counts agree exactly.

Common Error: Assuming Finite Means Small

Finite does not mean that the geometry must contain only a handful of points.

For order

\[ q=101, \]

a projective plane has

\[ 101^2+101+1=10303 \]

points.

The structure is finite, but it may still be large.

Common Error: Treating the Drawing as Euclidean

A diagram of the Fano plane may contain a drawn circle.

That does not mean the finite geometry distinguishes between straight lines and circles.

All seven designated three-point subsets are projective lines.

Common Error: Assuming Parallelism in a Projective Plane

Finite affine planes contain parallel classes.

Finite projective planes do not contain parallel lines.

By definition,

\[ \boxed{\text{every two distinct projective lines meet in exactly one point}}. \]

Common Error: Using the Affine Counting Formula for a Projective Plane

For order \(q\):

affine point count:

\[ q^2; \]

projective point count:

\[ q^2+q+1. \]

The extra \(q+1\) projective points correspond to affine directions.

Common Error: Forgetting the Extra Projective Line

Projective completion adds one point for each affine parallel class, but it also adds the line at infinity containing all those new points.

Thus the line count changes from

\[ q(q+1) \]

to

\[ q(q+1)+1=q^2+q+1. \]

Common Error: Assuming Every Integer Is an Order

The formulas

\[ q^2,\qquad q(q+1),\qquad q^2+q+1 \]

describe the counts that a plane of order \(q\) must have.

They do not guarantee existence for arbitrary \(q\).

Prime powers provide standard constructions because finite fields of those sizes exist.

Common Error: Confusing \(\mathbb{Z}_n\) with a Field for Every \(n\)

Modular arithmetic modulo \(n\) does not always form a field.

For example, modulo \(6\),

\[ 2\cdot3\equiv0\pmod6, \]

even though neither factor is congruent to zero.

This prevents \(\mathbb{Z}_6\) from being a field.

When \(p\) is prime, however, \(\mathbb{Z}_p\) is a field.

A Reliable Procedure for Affine-Plane Counting

  1. Identify the order \(q\).
  2. Use \(q^2\) for the number of points.
  3. Use \(q(q+1)\) for the number of lines.
  4. Remember that each line contains \(q\) points.
  5. Remember that each point lies on \(q+1\) lines.
  6. Use \(q+1\) parallel classes.
  7. Use \(q\) lines in each parallel class.

A Reliable Procedure for Projective-Plane Counting

  1. Identify the order \(q\).
  2. Use \(q^2+q+1\) for the number of points.
  3. Use the same \(q^2+q+1\) for the number of lines.
  4. Use \(q+1\) points on each line.
  5. Use \(q+1\) lines through each point.
  6. Remember that every two points determine one line.
  7. Remember that every two lines meet at one point.

A Reliable Procedure for Coordinate Questions

  1. Identify the finite field being used.
  2. Perform arithmetic in that field rather than in ordinary integers.
  3. For affine points, use ordered pairs \((x,y)\).
  4. For affine nonvertical lines, use \(y=mx+b\).
  5. Include vertical lines \(x=c\).
  6. For projective points, use nonzero homogeneous triples.
  7. Identify scalar multiples as the same projective point.
  8. Check incidence by substituting coordinates into the line equation.

Practice 1: Affine Plane Counts

A finite affine plane has order \(7\).

Find:

  1. the number of points;
  2. the number of lines;
  3. the number of points on each line;
  4. the number of lines through each point.

Practice 2: Projective Plane Counts

A finite projective plane has order \(4\).

Find:

  1. the number of points;
  2. the number of lines;
  3. the number of points on each line.

Practice 3: Parallel Classes

An affine plane has order \(5\).

How many parallel classes does it have, and how many lines lie in each class?

Practice 4: Projective Completion

Start with an affine plane of order \(3\).

How many points and lines must be added to obtain its projective completion?

Practice 5: Removing the Line at Infinity

A projective plane has order \(5\).

Remove one line and all points lying on it.

How many points remain?

Practice 6: A Finite-Field Line

Work in \(\mathbb{F}_3\).

List the three points on the affine line

\[ y=x+2. \]

Practice 7: Fano Plane Incidences

The Fano plane has seven lines, with three points on each line.

How many point-line incidences occur?

Practice 8: Identify the Geometry

A finite plane has \(25\) points, \(30\) lines, \(5\) points on each line, and \(6\) lines through each point.

Is this count pattern affine or projective, and what is its order?

Practice 9: Projective Point Count

A projective plane contains \(13\) points.

Determine its order.

Practice 10: Explain the Difference

Explain why the statements

can both be correct in finite geometry.

Practice Answers

Practice 1

For \(q=7\):

\[ q^2=49 \]

points,

\[ q(q+1)=7\cdot8=56 \]

lines, \(7\) points per line, and \(8\) lines through each point.

Practice 2

For \(q=4\):

\[ q^2+q+1=16+4+1=21. \]

Therefore there are

\[ \boxed{21\text{ points and }21\text{ lines}}, \]

with

\[ q+1=5 \]

points on each line.

Practice 3

An affine plane of order \(5\) has

\[ q+1=6 \]

parallel classes, each containing

\[ q=5 \]

lines.

Practice 4

An affine plane of order \(3\) has \(q+1=4\) parallel classes.

Projective completion adds one point for each class:

\[ \boxed{4\text{ points}}. \]

It also adds

\[ \boxed{1\text{ line at infinity}}. \]

Practice 5

A projective plane of order \(5\) has

\[ 31 \]

points.

A line contains

\[ 6 \]

points.

Removing the line and its points leaves

\[ \boxed{31-6=25}. \]

Practice 6

In \(\mathbb{F}_3\), substitute \(x=0,1,2\):

\[ x=0\Rightarrow y=2, \] \[ x=1\Rightarrow y=0, \] \[ x=2\Rightarrow y=1. \]

Therefore the line contains

\[ \boxed{(0,2),\ (1,0),\ (2,1)}. \]

Practice 7

Count incidences by lines:

\[ 7\cdot3=\boxed{21}. \]

Practice 8

Since

\[ 25=5^2 \]

and

\[ 30=5(6), \]

the pattern is that of an

\[ \boxed{\text{affine plane of order }5}. \]

Practice 9

Solve

\[ q^2+q+1=13. \]

Then

\[ q^2+q-12=0, \] \[ (q+4)(q-3)=0. \]

The positive order is

\[ \boxed{q=3}. \]

Practice 10

The first statement is an affine statement: an affine plane contains parallel classes whose distinct lines do not meet.

The second is a projective statement: projective completion adds points at infinity so that formerly parallel lines meet.

Thus the two statements refer to different geometric axiom systems.

What to Remember

Where We Go Next

Finite geometries showed that geometry can be built from abstract incidence rules without requiring a continuous Euclidean plane. We now move in a different direction and ask how spaces can behave locally like familiar Euclidean space while having much more complicated global structure.

In Introductory Manifolds, we will develop the idea of a space that can be described using local coordinate neighborhoods. Curves and surfaces will become the first examples, providing a bridge from the geometry of specific shapes to the modern study of geometric spaces.