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

A Capston Paper Submitted to the Department of Mathematics Georgia College requirements for the degree of Bachelor of Science in Mathematics by Joel Rapkin November 2014 Acknowledgments

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

ii 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 and finite point” Function ...... 9

My Vision ...... 13

Bibliography ...... 15

iii Abstract

Projective geometry originated in the context of art, but evolved into a much more mathematical research field as more and more mathematicians started questioning the 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 Software Sam. With its “swap finite & infinite points” function, students will be able to see connected geometrical figures and explore some theorems from the Euclidean plane in the projective plane.

iv Introduction

When I was first introduced to the idea of the projective plane in Geometry at Georgia College and State University, I was very confused by this idea, and I wanted to explore a better way to introduce this subject, but also do it at the high school level. I saw the Dynamic Geometry Software Sam as a great way to do this, and explored a way to introduce the important benefits of the projective plane 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 to further geometrical thought at an early stage in a students education.

1 Introduction to Geometrical Planes

Geometry is a subject in which many students, of primary and secondary levels, are inadequately familiar, and this unfamiliarity exposes a lack of abstract thinking that 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 towards truth, 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 was Plato, and he was a firm believer that Geometry, and the required thinking when studying geometry, is as key to living as is breathing. Many educational systems do not necessarily neglect geometry in schools, but instead inhibit the ability for students to learn in a geometrical way of thinking. Seeing Plato as one of the most important philosophers in history can you give a good idea of just how important he thought geometry was when his school had the following inscribed above the entrance, “Let no man ignorant of geometry enter here”. It is due to note that Plato’s school was not based on teaching Geometry, but was more geared towards Philosophy. This is an important distinction because Plato is acknowledging the importance 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 an effort to further understand the similarities between these planes, and the inherent differences that present themselves. Euclid is widely known as the “Father of Geometry” and is best known for his collection of thirteen books known as the Elements. Euclid sought out to build upon his predecessors with the collection, and laid down what is arguably one of the most prolific mathematical works in history. The Elements covers many geometrical topics in plane geometry, solid geometry, proportions and other topics. However, Euclid ran into a problem of sorts when he first began trying to prove his theories. How do you begin proving a geometrical theory without first having some defined statements? Let’s begin by looking at an example of why this can be an issue. “The Concise Oxford Dictionary devotes over a column to the word ‘point’... ‘that which has position but not magnitude’.” [Co]. This is a great definition, however, what are the

2 definitions of magnitude and position? We can illustrate this problem with a game of “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 other words to define a particular word. Getting back to Euclids work, we see that the only way to avoid this complication of a vicious circle is single out certain statements that are deemed acceptable without justification. These statements are known as postulates, and they are fundamental to proving theories, not only in the Elements, but in all of geometry as a whole. The postulates in Euclidean geometry are as follows:

• Euclid’s Postulate I: For every point P and for every point Q not equal to P there 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 be extended)

• 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 not lie on l, there exists a unique line m through P that is parallel to l.(Euclid’s Parallel 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. The last postulate, Euclid’s Parallel Postulate, is one that was debated for centuries on its legitimacy, and is also what spawned non-Euclidean geometries such as the Projective Plane that we will be discussing extensively in this paper.

3 The Origins of the Projective Plane

The beginnings, of whats is known today as the Projective Plane, has a very interesting 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-Dimensional piece of paper, or canvas, it is very important to have an origin of projection. Let us think about drawing a rail- road track that extends off into the distance. It is easy to realize that the railroad tracks should never meet, hence are parallel, but if you extend these two distinct lines far enough into the distance they do appear to meet (figure 1). “It was in 1425 that the Italian architect Brunelleschi began to discuss the geometrical theory of per- spective”...“‘Plane geometry may be described as the study of geometri- cal properties that are unchanged by ‘central projection,’ which is essentially Figure 1: The parallel lines appear to what happens when an artist draws a meet in the distance picture of a tiled floor on a vertical canvas.”[Co] It is very intriguing that these 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 one another and how some theorems are much better suited to be constructed in the projective space, which we will discuss in more depth when we introduce the GSP program 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 distinguishes the Projective plane.

4 Definition 0.1. Projective Plane: A projective plane is a model of incidence geometry having the elliptic parallel property (any two lines meet) and such that every 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. This property comes from the idea of visualizing the geometric plane as points on a sphere, and lines as the great circles of sphere. This leads us to 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 the idea of any two lines, such as great cir- cles on a sphere, always having at least two points in common is the founda- tion of the projective plane. We can however build this plane by systemati- Figure 2: This can be visualzed as great cally adding certain elements to the eu- circles on the earth, such as the equator clidean plane that most people are more accustomed to. This approach can be a very useful one that can be gentler to the learner that is not yet comfortable with the more abstract elements of the projective plane. This brings us to the key axioms(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 far the most interesting change in the axioms is Axiom 2. This is a very interesting statement and one that deserves a little further exploration. If you recall from Euclid’s Postulate V: For every line l and for every point P that does not lie on l, there exists a unique line m through P that is parallel to l.(Euclid’s Parallel Postulate)., we see that each line is composed of, at least, two points. However, in the projective space we will be adding a new line, called the line at infinity. This line 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 every line is incident with atleast three points because all lines are incident with a point

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

Definition 0.2. Incidence Geometry An incidence geometry (P, l, I) consists of the set of all points P , the set of all lines l, and the Incidence relation, I, between these 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 is taught in high schools, and is not concerned with measurements or angles. Inci- dence geometry is only concerned with what points and lines are incidence with each 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 very beneficial to students to begin learning this more abstract form of geometry.

6 Extending the Euclidean Plane

So if this were to be taught at the high school, you would first introduce the Euclidean plane, and then would introduce points at infinity. The projective plane 2 is what we get by adding the line at infinity (l∞) to the affine plane A (in our case 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 the following 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 contains the 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 except Axiom 6. We can get rid of this axiom by adding the points at infinity with the following definition: Definition 0.3. Point at Infinity: Any pair of parallel lines is said to meet at a unique point at infinity. So we are stating that every pencil of parallel lines meets at unique point at infinity, and therefore we are negating the existence of parallel lines by introducing this definition. This idea is thought of as extending the euclidean plane by adding the line at infinity to our more familiar euclidean plane, thus making it where no lines 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 points at infinity. So now that we have a firmer grasp on what the real projective plane is, what are the benefits at the high school level? The first main benefit is that introducing these abstract ideas at the high school level can be very beneficial to high school

7 students because it requires an out-of-the-box thinking that can make students abstractly think about the world around them. This would also be beneficial for high school students because they could be introduced to the axiomatic method of proving theorems only using a few postulates or axioms. The main benefit to using the projective plane, is the fact that certain theorems that have restrictions on them to be true, such as two lines having to be incident with each other, can now not have these restrictions in the projective plane. For instance, even if two lines 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 restrictions as they do in the Euclidean plane. All of these concepts are very interesting and beneficial, but for my capstone I wanted to find a way that high school students could be introduced to these concepts, and explore its benefits in a more hands on way. This lead me to the Dynamic Geometry Software Sam, and its unique feature of being able to swap finite points and infinite points.

8 Introducing the Real Projective Plane Using the “swap infinite and finite point” Function

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

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

9 more intuitive then just showing them a picture.

The next big benefit of using this program to introduce the projective plane is the ability to easily show students connected geometrical figures. The example I will be showing you is how a parallelogram, trapezoid and quadrilateral are all connected figures and can all be constructed from one original construction by swapping 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 two sets of parallel lines that meet at the infinite point U and the infinite point V . In Img 2 we can then swap the infinite point V with the finite point V and we obtain a trapezoid. Students will be able to see how these figures are connected! In the last image we can then swap the infinite point U and the finite point U to obtain a quadrilateral. This is a very effective way to have students explore connected figures.

By far the most beneficial advantage of this program is when exploring why certain theorems are more natural in the real projective plane. This can be very abstract concept for high school students, but by utilizing this program and its “swapping finite and infinite points” students can easily explore this concept. For the 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 A0, B, and C0 are three other distinct points on line l, then the intersections of lines A0B ∩ B0A, C0A ∩ A0C and B0C ∩ C0B are collinear.

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

(a) Img 1 (b) Img 2

As you can see in Img 1 we have constructed it such that line A0B is parallel to line B0A. In the normal euclidean plane these points would no longer be collinear since these lines no longer meet. But since we have constructed this theorem in the extended euclidean plane, we see that lines A0B, B0A and the Pappus Line are parallel, and therefore do meet at the infinite point U. Students can easily verify this by swapping the infinite point U with a finite point U to obtain Img 2. I think this would be a huge benefit to students by being able able to actually see this interaction. Another example of using the real projective plane to show why it is better suited for some theorems is Desargues’ Theorem. This theorem states the following.

Theorem 2. Let two triangles 4ABC and 4A0B0C0 be perspectively related such that AA0, BB0 and CC0 are concurrent at point P . If we extend the respective sides of each triangle such that AC ∩ A0C0, AB ∩ A0B0 and BC ∩ B0C0 then these points 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 DGS Sam we can again show how this theorem is better suited for the extended eu- clidean plane.

11 (a) Img 1 (b) Img 2

As you can see in Img1, lines AB and A0B0 fail to meet, but since lines AB, A0B0 and Desargues’ Line are parallel then we know they meet at a point at infinity and the theorem still holds true. Similarly, students have the ability to swap the infinite point with the finite point and actually experience this benefit themselves!

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

12 My Vision

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

• Elliptic Curve Cryptography

• Number Theory

• Group Theory

• Photography

13 • Computer Vision

• Conics being projectively related

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

14 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. Freeman and 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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