Projective Geometry: GSP Sam and it’s Unique Educational Tool

A Capston Paper

Submitted to the Department of MathematicsGeorgia College

requirements for the degree ofBachelor of Science

in

Mathematics

byJoel Rapkin

November 2014

Acknowledgments

This capstone project would not be possible without Dr. Cazacu. It is a pleasureto thank Dr. Rodica Cazaci for her guidance as my advisor

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction to Geometrical Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

The Origins of the Projective Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Extending the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Introducing the Real Projective Plane Using the “swap infinite andfinite point” Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

My Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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Abstract

Projective geometry originated in the context of art, but evolved into a much moremathematical research field as more and more mathematicians started questioningthe uniqueness of the very popular Euclidean geometry. In this talk we will intro-duce the real projective plane and explore it using the Dynamic Geometry SoftwareSam. With its “swap finite & infinite points” function, students will be able tosee connected geometrical figures and explore some theorems from the Euclideanplane in the projective plane.

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Introduction

When I was first introduced to the idea of the projective plane in Geometry atGeorgia College and State University, I was very confused by this idea, and Iwanted to explore a better way to introduce this subject, but also do it at thehigh school level. I saw the Dynamic Geometry Software Sam as a great way todo this, and explored a way to introduce the important benefits of the projectiveplane in a more hands on way that high school students could more easily grasp.Hopefully, one day, students will be taught the following concepts in an order tofurther geometrical thought at an early stage in a students education.

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Introduction to GeometricalPlanes

Geometry is a subject in which many students, of primary and secondary levels,are inadequately familiar, and this unfamiliarity exposes a lack of abstract thinkingthat can greatly benefit students of all ages. One of the most prolific mathemati-cians once wrote, “Then, my noble friend, geometry will draw the soul towardstruth, and create the spirit of philosophy, and raise up that which is now unhap-pily allowed to fall down.”(The Republic) The mathematician that wrote this wasPlato, and he was a firm believer that Geometry, and the required thinking whenstudying geometry, is as key to living as is breathing. Many educational systemsdo not necessarily neglect geometry in schools, but instead inhibit the ability forstudents to learn in a geometrical way of thinking. Seeing Plato as one of the mostimportant philosophers in history can you give a good idea of just how importanthe thought geometry was when his school had the following inscribed above theentrance, “Let no man ignorant of geometry enter here”. It is due to note thatPlato’s school was not based on teaching Geometry, but was more geared towardsPhilosophy. This is an important distinction because Plato is acknowledging theimportance he saw in his students being seasoned on the geometries, and the think-ing that is required to study geometry.

Even though we will be mainly discussing the Projective Plane in this paper,we should first become familiar with a more common geometrical plane in aneffort to further understand the similarities between these planes, and the inherentdifferences that present themselves. Euclid is widely known as the “Father ofGeometry” and is best known for his collection of thirteen books known as theElements. Euclid sought out to build upon his predecessors with the collection,and laid down what is arguably one of the most prolific mathematical works inhistory. The Elements covers many geometrical topics in plane geometry, solidgeometry, proportions and other topics. However, Euclid ran into a problem ofsorts when he first began trying to prove his theories. How do you begin provinga geometrical theory without first having some defined statements? Let’s beginby looking at an example of why this can be an issue. “The Concise OxfordDictionary devotes over a column to the word ‘point’... ‘that which has positionbut not magnitude’.” [Co]. This is a great definition, however, what are the

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definitions of magnitude and position? We can illustrate this problem with a gameof “Vish”.

“Point = that which has position but not magnitude.Position = place occupied by a thing.Place = part of space...Space = continuous extension...Extension = extent.Extent = space over which a thing extends.Space = continuous extension...” [Co].

We can see that we get a vicious cycle (Vish) of having to inherently use otherwords to define a particular word. Getting back to Euclids work, we see thatthe only way to avoid this complication of a vicious circle is single out certainstatements that are deemed acceptable without justification. These statementsare known as postulates, and they are fundamental to proving theories, not onlyin the Elements, but in all of geometry as a whole. The postulates in Euclideangeometry are as follows:

• Euclid’s Postulate I: For every point P and for every point Q not equal to Pthere exists a unique line that passes through P and Q.

• Euclid’s Postulate II: For every segment AB and for every segment CD

there exits a unique point E on line←→AB such that B is between A and E

and segment CD is congruent to segment BE. (Any line segment can beextended)

• Euclid’s Postulate III: For every point O and every point A not equal to O,there exists a circle with center O and radius OA.

• Euclid’s Postulate IV: All right angles are congruent to one another.

• Euclid’s Postulate V: For every line l and for every point P that does notlie on l, there exists a unique line m through P that is parallel to l.(Euclid’sParallel Postulate).

These postulates are what lays the groundwork for Euclid’s work in the the Ele-ments and are absolutely necessary to begin deriving more advanced proofs. Thelast postulate, Euclid’s Parallel Postulate, is one that was debated for centurieson its legitimacy, and is also what spawned non-Euclidean geometries such as theProjective Plane that we will be discussing extensively in this paper.

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The Origins of the ProjectivePlane

The beginnings, of whats is known today as the Projective Plane, has a veryinteresting beginning in the fact that it actually originated from the fine arts.When one wants to accurately portray a 3-Dimensional scene on a 2-Dimensionalpiece of paper, or canvas, it is very important to have an origin of projection.

Figure 1: The parallel lines appear tomeet in the distance

Let us think about drawing a rail-road track that extends off into thedistance. It is easy to realize thatthe railroad tracks should never meet,hence are parallel, but if you extendthese two distinct lines far enough intothe distance they do appear to meet(figure 1). “It was in 1425 that theItalian architect Brunelleschi began todiscuss the geometrical theory of per-spective”...“‘Plane geometry may bedescribed as the study of geometri-cal properties that are unchanged by‘central projection,’ which is essentiallywhat happens when an artist draws apicture of a tiled floor on a verticalcanvas.”[Co] It is very intriguing thatthese concepts of projectivity first arose, not from the sciences, but from the arts.However, the mathematical field truly saw the projective plane as much more use-ful alternative to the Euclidean space in the way figures can be connected to oneanother and how some theorems are much better suited to be constructed in theprojective space, which we will discuss in more depth when we introduce the GSPprogram Sam.

To begin discussing the Projective plane and it’s benefits over the more familiar Eu-clidean plane, it is obviously necessary to introduce a definition that distinguishesthe Projective plane.

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Definition 0.1. Projective Plane: A projective plane is a model of incidencegeometry having the elliptic parallel property (any two lines meet) and such thatevery line has at least three distinct points lying on it. [Gr]

This idea of the elliptic parallel property is key to understanding the key differ-ences in the projective plane.

Figure 2: This can be visualzed as greatcircles on the earth, such as the equator

This property comes from the ideaof visualizing the geometric plane aspoints on a sphere, and lines as thegreat circles of sphere. This leads usto the statement of the elliptic paral-lel property of, any two lines meet, be-cause it negates the existence of paral-lel lines. This is fairly abstract, but theidea of any two lines, such as great cir-cles on a sphere, always having at leasttwo points in common is the founda-tion of the projective plane. We canhowever build this plane by systemati-cally adding certain elements to the eu-clidean plane that most people are moreaccustomed to. This approach can be

a very useful one that can be gentler to the learner that is not yet comfortablewith the more abstract elements of the projective plane. This brings us to the keyaxioms(postulates) that we will assume to be true in order to deduce certain ideas.Here are the following axioms we will use to construct our projective plane:

Axiom 1. There exists a point and a line that are not incident.

Axiom 2. Every line is incident with atleast three distinct points.

Axiom 3. Any two distinct points are incident with just one line. [Co]

These obviously differ from the axioms used in the euclidean plane, but by farthe most interesting change in the axioms is Axiom 2. This is a very interestingstatement and one that deserves a little further exploration. If you recall fromEuclid’s Postulate V: For every line l and for every point P that does not lie onl, there exists a unique line m through P that is parallel to l.(Euclid’s ParallelPostulate)., we see that each line is composed of, at least, two points. However, inthe projective space we will be adding a new line, called the line at infinity. Thisline is made up of a points at infinity, and like our railroad track analogy above,we define the point at infinity as the point where all parallel line eventually meet.This is a very strange concept, but we can now, by our Axiom 2 see that everyline is incident with atleast three points because all lines are incident with a point

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at infinity lying on the infinite line that we have constructed. So next we need tolook at our defintion of the projective plane. It says the the projective plane is amodel of incidence geometry. Incidence geometry is another topic that high schoolstudents are not introduced to, and I think could benefit from. Lets look at thedefinition of incidence geometry.

Definition 0.2. Incidence Geometry An incidence geometry (P, l, I) consists ofthe set of all points P , the set of all lines l, and the Incidence relation, I, betweenthese sets. (i.e. a point P and a line l are said to be incident if P lies on l.)

This type of geometry is much simpler than the euclidean geometry that istaught in high schools, and is not concerned with measurements or angles. Inci-dence geometry is only concerned with what points and lines are incidence witheach other, and the inherent relations that can be learned from these relations.This is not something currently taught in schools, and I think it could be verybeneficial to students to begin learning this more abstract form of geometry.

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Extending the Euclidean Plane

So if this were to be taught at the high school, you would first introduce theEuclidean plane, and then would introduce points at infinity. The projective planeis what we get by adding the line at infinity (l∞) to the affine plane A2 (in ourcase A2 = R2). This would be another opportunity to introduce another concept,the affine plane. The affine plane is any set of points and lines that satisfies thefollowing axioms:

Axiom 4. Any two distinct points lie on a unique line

Axiom 5. Each line has at least two points

Axiom 6. Given a point and a line there is a unique line which containsthe point and is parallel to the given line.

Axiom 7. There exists three non collinear points

As we can see, our projective plane satisfies all the following axioms exceptAxiom 6. We can get rid of this axiom by adding the points at infinity with thefollowing definition:

Definition 0.3. Point at Infinity: Any pair of parallel lines is said to meet ata unique point at infinity.

So we are stating that every pencil of parallel lines meets at unique point atinfinity, and therefore we are negating the existence of parallel lines by introducingthis definition. This idea is thought of as extending the euclidean plane by addingthe line at infinity to our more familiar euclidean plane, thus making it where nolines can be parallel because every set of parallel lines meets at this point at infinity.By Axiom 4 we can construct this line at infinity by the following definition:

Definition 0.4. Line at Infinity: The line created by connecting these pointsat infinity.

So now that we have a firmer grasp on what the real projective plane is, whatare the benefits at the high school level? The first main benefit is that introducingthese abstract ideas at the high school level can be very beneficial to high school

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students because it requires an out-of-the-box thinking that can make studentsabstractly think about the world around them. This would also be beneficial forhigh school students because they could be introduced to the axiomatic methodof proving theorems only using a few postulates or axioms. The main benefit tousing the projective plane, is the fact that certain theorems that have restrictionson them to be true, such as two lines having to be incident with each other, cannow not have these restrictions in the projective plane. For instance, even if twolines are parallel, in the projective plane, they still meet at their point at infinity.This leads to much more elegant theorems that dont have as many restrictionsas they do in the Euclidean plane. All of these concepts are very interesting andbeneficial, but for my capstone I wanted to find a way that high school studentscould be introduced to these concepts, and explore its benefits in a more hands onway. This lead me to the Dynamic Geometry Software Sam, and its unique featureof being able to swap finite points and infinite points.

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Introducing the Real ProjectivePlane Using the “swap infiniteand finite point” Function

When I was in Geometry at Georgia College and State University, we got to thetopic of the real projective plane, it was really confusing and not a very intuitiveidea that I could easily grasp. These ideas of points at infinity, and lines at infinitywere quite confusing at the time, and during a homework assignment for that classI stumbled upon the DGS program Sam, which is mainly used as a way to explorecomplicated geometry theorems at a high level, but I saw an opportunity for itto be used at the high school level to introduce the real projective plane and itsabstract concepts. Its special function swaps any finite point with the infinite pointthat a line is constructed to. If you construct a line through a finite point and aan infinite point, then you have a ability to then swap that infinite point with afinite point of your choosing. This function has the ability to introduce importantconcepts to high school students in a much more concrete and hands on way. Forthe first example, I constructed the following to show how the function works, andhow it can relate this geometry back to its artistic origins.

The image on the left shows the pencil of parallel lines which in constructedto all meet at the infinite point U , and the image on the right shows the abilityto swap this infinite point with an finite point, such that all the lines now appearto meet at a point at infinity. This is exactly what artists wanted to do whenthe origins of the projective plane began. The ability for students to constructthis themselves is critical to them understanding the projective plane, and is much

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more intuitive then just showing them a picture.

The next big benefit of using this program to introduce the projective planeis the ability to easily show students connected geometrical figures. The exampleI will be showing you is how a parallelogram, trapezoid and quadrilateral are allconnected figures and can all be constructed from one original construction byswapping strategic finite and infinite points.

(a) Img 1 (b) Img 2

(a) Img 3

As you can see in Img 1 we can construct a parallelogram by constructing twosets of parallel lines that meet at the infinite point U and the infinite point V . InImg 2 we can then swap the infinite point V with the finite point V and we obtaina trapezoid. Students will be able to see how these figures are connected! In thelast image we can then swap the infinite point U and the finite point U to obtaina quadrilateral. This is a very effective way to have students explore connectedfigures.

By far the most beneficial advantage of this program is when exploring whycertain theorems are more natural in the real projective plane. This can be veryabstract concept for high school students, but by utilizing this program and its“swapping finite and infinite points” students can easily explore this concept. Forthe first example lets look at Pappus’ Theorem. This theorem states the following:

Theorem 1. If A, B and C are three distinct points on line m and A′, B, and C ′

are three other distinct points on line l, then the intersections of lines A′B ∩B′A,C ′A ∩ A′C and B′C ∩ C ′B are collinear.

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So the main problem with this theorem in the Euclidean plane is the fact thatthe lines are required to intersect for Pappus’ Theorem to hold true. However ifwe embed this theorem in the real projective plane then we can see that it doesn’tmatter if two lines do not intersect (i.e. are parallel) because in our extendedeuclidean plane the lines still meet at a point at infinity, thus, the theorem holdstrue whether or not certain lines are parallel. With the benefit of our DGS Samstudents can construct these theorems themselves ans truly see the benefits ofusing the projective plane.

(a) Img 1 (b) Img 2

As you can see in Img 1 we have constructed it such that line A′B is parallel toline B′A. In the normal euclidean plane these points would no longer be collinearsince these lines no longer meet. But since we have constructed this theorem inthe extended euclidean plane, we see that lines A′B, B′A and the Pappus Line areparallel, and therefore do meet at the infinite point U . Students can easily verifythis by swapping the infinite point U with a finite point U to obtain Img 2. I thinkthis would be a huge benefit to students by being able able to actually see thisinteraction. Another example of using the real projective plane to show why it isbetter suited for some theorems is Desargues’ Theorem. This theorem states thefollowing.

Theorem 2. Let two triangles 4ABC and 4A′B′C ′ be perspectively related suchthat AA′, BB′ and CC ′ are concurrent at point P . If we extend the respectivesides of each triangle such that AC ∩A′C ′, AB ∩A′B′ and BC ∩B′C ′ then thesepoints of incidence are collinear.

Again, what if two sides of our triangle do not meet, and again are parallel,then our theorem will not hold in the usual Euclidean plane. However using DGSSam we can again show how this theorem is better suited for the extended eu-clidean plane.

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(a) Img 1 (b) Img 2

As you can see in Img1, lines AB and A′B′ fail to meet, but since lines AB,A′B′ and Desargues’ Line are parallel then we know they meet at a point at infinityand the theorem still holds true. Similarly, students have the ability to swap theinfinite point with the finite point and actually experience this benefit themselves!

I believe that this hands on experience would allow students to more firmlygrasp these concepts, and would also allow students to independently explore theseideas themselves.

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My Vision

The purpose of this capstone project was to find a way for students at the highschool level to be easily introduced to the real projective plane by extending theusual euclidean plane using the line at infinity, and begin to explore this planeusing the Dynamic Geometry Software Program Sam. I believe that by introducingthese abstract ideas to students in high school, then they would better preparedfor collegiate level math classes, having already delved into the abstact ideas. Thefollowing ideas I think would benefit high school students:

• Ideas of Infinity

• Line at Infinity

• Points at Infinity

• Incidence Geometry

• Affine Plane

• Perspectivities

• Projectivities (Objects being projectively related)

• Why the project plane is better for certain theorems

• the axiomatic method

• abstract thinking etc.

Also, many different concepts have been proved using projective geometry. Al-though I didn’t research these, here are a few subjects that have utilized theprojective plane.

• Elliptic Curve Cryptography

• Number Theory

• Group Theory

• Photography

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• Computer Vision

• Conics being projectively related

As you can see the projective plane is a very important concept that should beintroduced at the high school level! Hopefully, one day, utilizing the DGS Sam,students will already be familiar with the projective plane prior to college.

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Bibliography

[1] [GC] Great Circles of a Sphere. Digital image. Annenberg Learner. N.p., n.d.Web. 4 Nov. 2014.

[Gr] M.J. Greenberg, Euclidean and Non-Euclidean Geometries, W.H. Freemanand Company, 2008

[Co] H.S.M. Coxeter, Projective Geometry, University of Toronto Press, 1974

[Ro] H.L. Royden, Real Analysis, The Macmillan Company, New York, 1965

[Na] Nachbin, L.,Topology and Order, Van Nostrand, 1965.

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