Module 2 Finite Geometries PDF

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Modern GeometryModule 2: Finite Geometries17MODULE 2 – FINITE GEOMETRIESINTRODUCTIONWhen you think of finite, you always associate it with countable things. In Euclidean Geometry, you deal with an infinitely many points, lines and planes as well as bits of aggregate theorems that continues to grow. ...

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Modern Geometry MODULE 2 – FINITE GEOMETRIES

INTRODUCTION

When you think of finite, you always associate it with countable things. In Euclidean Geometry, you deal with an infinitely many points, lines and planes as well as bits of aggregate theorems that continues to grow. In this module, you are dealing with geometry with few numbers of axioms, few theorems and a finite number of elements which is called finite geometry. This type of geometry provides a rich opportunity from which to study geometric structure, as well as serve as an excellent springboard into discussion of other geometries like transformational, and projective geometry. This module provides you with the concepts of the different finite geometries such as three-point geometry, four-point geometry, four-line geometry, and five-point geometry and other types of finite geometries like incidence geometry, Fano’s and Young’s Geometry, and Pappus’s and Desargues’s Geometry.

LEARNING OUTCOMES After completing this module, you should be able to: a. discuss the properties of finite geometries; b. identify the axioms for three-point, four-point, four-line and five-point geometry and prove their theorems; and c. prove the finite geometries of Fano and Young, Pappus and Desargues.

LEARNING CONTENT Lesson 1: FINITE GEOMETRIES AND THEIR TYPES Before one understands the properties of finite geometry, you need to understand first what finite geometry is. WHAT IS FINITE GEOMETRY? A finite geometry is a geometry based on a set of postulates, undefined terms, and undefined relations which limits the set of all points and lines to a finite number.

Module 2: Finite

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A finite geometry is a geometry with a finite number of points. When confined to a plane, all finite geometries are either projective plane geometries (with no parallel lines) or affine plane geometries (with parallel lines).

NOTE: All the finite geometries of this discussion have point and line as undefined terms, and on as the main relation. Remember that figures in geometry are sets, thus, in a finite geometry each figure is a “finite” set. Your perception of a line must also be changed. In the previous module, you studied on how to build a model of a given set of axioms. You are going to use “models” to understand the underlying structure of the so-called finite geometries. You will start with the simplest type of finite geometries, the Three-Point Geometry. THREE-POINT GEOMETRY Undefined Terms. Point, line, on. Axiom 1. There exist exactly three points. Axiom 2. Two distinct points are on exactly one line. Axiom 3. Not all points are on the same line. Axiom 4. Two distinct lines are on at least one common point. Let’s consider the following set of axioms for Three-Point Geometry. We will now interpret the undefined terms such as point and line of this geometry to determine its model. Interpretation: In this geometry, point is interpreted as letter while line is interpreted as pairs of letters Point (p) = {A, B, C} Line (L) = {{A, B}, {A, C}, {B, C}} Verification: Verify whether each axiom in the given set of axioms is satisfied. This means that every axiom is true based from its interpretation. If all axioms in the set are satisfiable, then the model for the set of axioms exist (see Figure 1).

Figure 1. Model for Three-Point Geometry

Let’s now proceed to its theorems. Theorem 1: Any two distinct lines are on exactly one point. Proof: Let g and h be two distinct lines. By axiom there exists at least one point P on them. Suppose there exists point Q on both g and h distinct from P. By axiom there is exactly one line on P and Q. This is a contradiction to g and h being distinct. Thus, there can be exactly one point on g and h. ∎ Theorem 2: The three-point geometry has exactly three lines. Proof: From the three given points, there are three lines when points are taken two at a 3 time, there are three lines when points are taken two at a time, ( )2 = 3. Suppose there is a fourth line. It must have a distinct point in common with each of the other three lines. Thus, the line must be on two of the given three points, and therefore must be one of the other three lines (otherwise there would be two lines on two distinct points). Therefore, there are exactly three lines. ∎ Theorem 3: Each line contains only two points. (Try to prove this.) Another type of finite geometry is Four-Line Geometry. As you go through this geometry, you will be able to determine its plane dual which is called Four-Point Geometry. FOUR-LINE GEOMETRY Axiom 1: There exists exactly four lines. Axiom 2: Any two distinct lines intersect in one point. Axiom 3: Each point is on exactly two lines. 4L Theorem 1: There exists exactly six points. 4L Theorem 2: Each line contains exactly three points. Let’s consider the set of axioms for Four-Line Geometry. Devise a model for the system. Interpretation: P = {A, B, C, D, E, F} L = {{A, D, C}, {A, E, F}, {B, D, E}, {B, C, F}}

Verification: (This serves as an exercise to verify whether each axiom of the geometry is satisfiable.)

Figure 2. Model for Four-Line Geometry

Theorem 1: There exists exactly six points. 4

) = 6 pairing of lines, thus, there are at 2 least six points. If the six points were not (that is suppose that two points are the same) then the point would be on at least three lines which would be a contradiction. Suppose there is a seventh point. Then there would exist at least one more line other than the four used to get the original six points. There would be at least five lines, which is a contradiction. Thus, there are exactly six points. ∎ Proof: By Axiom 1 there are four lines. There are (

Theorem 2: Each line contains exactly three points. Proof: Let g, h, j, and k be the four distinct lines. Consider g paired with the other three lines. There would be at least three points on g. Further, suppose that there is a fourth point on line g. This would mean that there is a line distinct from h, j, and k on the point; a fifth line which is a contradiction. Therefore, there are exactly three points on each line. DEFINITION: The plane dual of a statement is formed by exchanging the words point and line in the statement. By exchanging these words, you create the axioms for a four- point geometry. FOUR-POINT GEOMETRY Undefined Terms: Point, Line, On. Axiom 1: There exist exactly four distinct points. Axiom 2: Any two distinct points are on exactly one line. Axiom 3: Each line is on exactly two points.

Definition 1.3.1: Two lines on the same point are said to intersect. Definition 1.3.2: Two lines that do not intersect are called parallel.

Devise a model for the system. Interpretation: P = {A, B, C, D} L = {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}} Verification: (This serves as an exercise to verify whether each axiom of the geometry is satisfiable.)

Figure 3. Model for Four-Point Geometry Theorem 1: There exists exactly 6 lines. 4. 3 2!= 6 pairs of distinct points. By Axiom 2 the geometry hast at least 6 lines. Assume there is a 7th line. BY Axiom 3 this line has to have two points on it. If one of these points is not one of the original 4 points then Axiom 1 is contradicted, otherwise Axiom 2 is contradicted. ∎

Proof: By Axiom 1 we have 4C2 =

Theorem 2: Each line has exactly one line parallel to it. Proof: By Axiom 2 is connected to each of the other points by a line. Hence each point hast at least three lines on it. Assume a point has 4 lines on it. This 4 th line cannot be on any of the other three points without contradicting Axiom 2. By Axiom 3 this line has two points on it so the geometry has at least 5 points in it which contradicts Axiom 1. Hence each point has exactly 3 lines on it. Now given a line ℓ, it has two points on it, say P1 and P2. By Axiom 1 there is a point P3 not on ℓ. Now this point has three lines on it, and by Axiom

2 two of them intersect ℓ. Hence ℓ has at least one line parallel to it. Assume there is a second line parallel to ℓ. This line is not on P3 since it already has three lines on it. It also is not on P1 and P2, hence either this line has one point on it which contradicts Axiom 2 or has two points on it which contradicts Axiom 1. Hence each line in the geometry has exactly one line parallel to it. ∎ Another type of finite geometry that has interesting structure is the Five-Point Geometry. Five-Point Geometry. Consider the following axioms: Axiom 1: There exist exactly five points. Axiom 2: Any two distinct points have exactly one line on both of them. Axiom 3: Each line has exactly two points. Proceed to Activity 1 and 2 to further test your understanding of the concepts presented in Lesson 1. Lesson 2 Other Types of Finite Geometries Fano’s Finite Geometry Axiom 1: There exists at least one line. Axiom 2: There exist exactly three points on every line. Axiom 3: Not all points are on the same line. Axiom 4: There exists exactly one line on any two distinct points. Axiom 5: There exists at least one point on any two distinct lines.

Figure 4. Model of Fano’s Finite Geometry

Fano Theorem 1: Two distinct lines intersect in exactly one point. Proof: By Axiom two distinct lines (say g and h) intersect in at least one point P. Suppose there is a second point, Q, at which the lines intersect. (P, Q ∈ g ∩ h). By axiom P and Q uniquely determine a line, therefore, P, Q ∈ g ∩ h is a contradiction. Thus, g and h intersect in exactly one point. ∎ Fano Theorem 2: There exists exactly seven points and seven lines. Proof: Consider line g and point P not in g. There exists points Q, R, and S on g. There are three unique lines on P and Q, P, and R, and P and S. There are three distinct points T, U, V on �, �, and �, respectively. Thus, there are at least seven points in the geometry. Suppose there exists an eight-point W.

Figure 5: Model of Fano Theorem 2 Consider the model above. Let there exists line distinct point, which is a contradiction to three points on of its three points, then

��. By axiom

�� must intersect ��. � intersects If

� at a

�� at one

� is not unique which is also a contradiction. Therefore, there

exists exactly seven points. ∎

Young’s Finite Geometry Axiom 1: There exists at least one line. Axiom 2: There exist exactly three points on every line. Axiom 3: Not all points are on the same line. Axiom 4: There exists exactly one line on any two distinct points. Axiom 5: For each line ℓ and each point P not on ℓ, there exists exactly one line on P that does not contain any points on ℓ.

1 A B c

Table 1: Model for Young’s Finite Geometry 2 3 4 5 6 7 8 9 10 11 A A A B B B C C C D E D H E D F F E H E I G F H I G I G D F

12 G H I

Figure 6: Model of Young’s Finite Geometry

Young’s Theorem 1: If two lines are each parallel to a third line, then they are parallel to each other. Proof: Let line g be parallel to line h (g||h) and line j||h. Suppose g∦ j then g would intersect j (g ∩ j ≠ ∅.

Figure 7 Let P, Q and R be on g (P, Q, R ∈ g) and suppose g ∩ j = {R}. R ∉ h by definition of parallel, therefore, both g & j are on R and parallel to h which is a contradiction. Thus, g ∩ j = ∅ or g||j. ∎

Finite Geometries of Pappus & Desargues Theorem of Pappus (From Euclidean Geometry) Given points A, B, C as distinct points on line g, and A’, B’ C’ as distinct points on line g’, then

�′ ∩ ′�, �′ ∩

′�, and

�′ ∩

′� are collinear.

Figure 8 This Euclidean theorem leads to the finite geometry of Pappus.

Axioms for the Finite Geometry of Pappus Axiom 1: There exists at least one line. Axiom 2: Every line has exactly three points. Axiom 3: Not all points are on the same line. Axiom 4: Given point P and line g. If P ∉ g then there exists exactly one line h such that P ∈ h and h || g. Axiom 5: Given P and g such that P ∉ g, there exists P’ ∈ g such that no lines contain both P and P’. Axiom 6: With the exception for the previous axiom, two distinct points uniquely determine a line.

Theorem 1: Each point is on exactly three lines. Proof: Let P be a point, then there exists a line g such that P ∉ g.

Figure 9 There exist Q, R S ∈ g. Let Q be such a point that no line contains both P and Q. Thus, there exist � and � �. By axiom there exists h ∈ P such that h||g. Therefore, there exists at least three line on P. Suppose there exists j on P. j ∩ g ≠ ∅ implies that j ∩ g = Q which is a contradiction or j ∩ g = T which is also a contradiction. Therefore, there exists exactly three lines on P.∎ Theorem 2: There exist nine points and nine lines. Prove the theorem. Definitions: (From Euclidean Geometry) Concurrent means three or more lines that intersect on the same point. Example: Triangle ABC (∆ ��� ) and (∆ �′�′�′ are perspective from point P if �  �′ are concurrent at P.

� ′,

�′, and

Figure 10 Two triangles perspective from a point are also perspective from a line. Two triangles are perspective from a line if the intersection of their corresponding sides is collinear.

Figure 11

If a point is a point of perspectivity for two triangles and a line is the line of perspectivity for the same two triangles, then the point is the pole and the line is polar. Finite Geometry of Desargues Definitions: The line g is a polar of point P if no lines contain P and a point on g.

A point P is a pole of line g if no lines contain P and a point on g. Axioms for the Finite Geometry of Desargues Axiom 1: There exists at least one point. Axiom 2: Each point has at least one polar. Axiom 3: Each line has at most one pole. Axiom 4: Two distinct points are on at most one line. Axiom 5: Every line has exactly three distinct points. Axiom 6: If a line does not contain a certain point, then there is a point of intersection for the line and any polar of the point. (Let p be the polar of P, if P ∉ g then p ∩ g ≠ ∅. Theorem 1: If P is on a polar of point Q, then Q is on each polar of P. Theorem 1: Each line has exactly one pole. Theorem 2: Each point has exactly one polar. (see Activity 3 in the Teaching and Learning Activities Section) After doing Activity 3, let’s now proceed to another type of geometry called Incidence Geometry. This type of geometry was introduced by German Mathematician David Hilbert in 1898 in his book Grundlagen der Geometrie to attempt and rectify the issues on Euclid’s axiomatic system. Hilbert’s set of axioms was independent and complete. His formulation also split the axioms into different sets: incidence axioms, betweenness axioms and congruence axioms. Incidence (Connection) Geometry The undefined terms are point, line and on. The Axioms are: Axiom 1: For each two distinct points there exists a unique line on both of them. Axiom 2: Every line contains at least two points. Axiom 3: There are at least three points that do not lie on the same line.

A geometry that satisfies all three axioms is called an incidence geometry. An incidence geometry has one of the following parallel postulates: Parallel Postulates (PP): Given a line ℓ and a point P not on ℓ, then three possibilities exist for parallel axiom:

PP1: There exist no lines on P that are parallel to ℓ. PP2: There exists exactly one line on P that is parallel to ℓ, or PP3: There exists more than one line on P parallel to ℓ. PP2 leads to Euclidean geometry, and any geometry whose axioms imply some equivalent statement is said to have the Euclidean parallel property. If we chose either of the options (PP1 or PP3), we will have a non-Euclidean geometry. Definition: A projective geometry is an incidence geometry having no parallel lines (parallel postulate 1) and in which each line has at least three points. Example: Four-Line, Five-Line and Fano’s Geometry Definition: An affine geometry is an incidence geometry that exhibits the Euclidean parallel postulate (PP2). Since we are dealing with Euclidean Geometry as of the moment, we can ask the following question: “What theorems can be proven without using any parallel axiom at all?” A set of Euclidean axioms without a parallel axiom is called a neutral geometry.

Proceed to Activity 3 and 4 to further test your understanding of the concepts presented in Lesson 2.

TEACHING AND LEARNING ACTIVITIES The activities provided in this section help you develop further your understanding of the terms and concepts of Finite Geometries. Activity 1 For Items 1 – 5, Refer to Axioms of Three-Point Geometry. 1. 2. 3. 4. 5.

What kind of drawing can be made to illustrate the geometry? How many lines are in the geometry? What, if any, theorems can be proved? What other objects can be used besides points and lines to represent the geometry? Are there any properties or theorems from Euclidean geometry that apply to this geometry?

For Items 6-8, refer to axioms of Four-Line Geometry. 6. Do two distinct points determine a line? 7. How many triangles exists? (Definition: A triangle consists of three distinct lines that intersect in three distinct points, vertices) 8. Are there parallel lines? (Definition: Two lines are said to be parallel if they do not intersect.) Activity 2 Five-Point Geometry. Consider the following axioms: Axiom 1: There exist exactly five points. Axiom 2: Any two distinct points have exactly one line on both of them. Axiom 3: Each line has exactly two points. a) Devise a model for the system of the Five-Point Geometry. b) State and prove two theorems in the Five-Point Geometry. c) Devise a model for the plane dual of the Five-Point Geometry.

Activity 3: Models for Finite Geometry of Desargues Refer to the set of axioms for the Finite Geometry of Desargues in Lesson 2. Devise a model for the system. Note that you can fine this online with a google search. Please attempt this on your own without resorting to copying. You will learn a lot more if you spend at least an hour working on this before resorting to copying.

Activity 4: Models for Incidence Geometries Each model below is an interpretation of the undefined terms points and lines. For each one, (a) determine whether it is an incidence geometry and (b) which of the three parallel postulate alternative it would satisfy. Model 1: Points are points on a Eu clidean plane, and lines are non-degenerate circles in the Euclidean plane. Model 2: Points are points on a Euclidean plane, and lines are all those lines on the plane that pass through a given fixed point P. Model 3: Points are points om a Euclidean plane, and lines are concentric circles all having the same fixed center. Model 4: Points are Euclidean points in the interior of a fixed circle, and lines are the parts of Euclidean lines that intersect the interior of the circle. Model 5: Points are points on the surface of a Euclidean sphere, and lines are great circles on the surface of that sphere.

Modern Geometry

ASSESSMENT TASK

Task 1: Conceptual Test Column A contains a description and definition of different types of finite geometries. Column B contains concepts on finite geometries. Match the concepts in Column B with the most appropriate definitions or descriptions in Column A. Write the letter of the correct answer on the space next to the numbers in Column A. Column A

Column B

1. A set of Euclidean axioms without a parallel axiom. 2. An incidence geometry that has no parallel lines and in which each line has at least three points.

a. Fano’s Finite Geometry b. Young’s Finite Geometry c. Four-Line Geometry d. Four-Point Geometry e. Affine Geometry

3. An incidence geometry that exhibits the Euclidean parallel postulate.

f. Projective Geometry g. Incidence Geometry h. Five-Point Geometry i. Pappus’s Geometry j. Neutral Geometry k. Duality l. Concurrent m. Finite Geometry

4. Three or more lines that intersect on the same point. 5. It is a way of interchanging the u...