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Projective Geometry: GSP Sam and It’S Unique Educational Tool

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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.
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