7may11, 24may11, 31may13, 4jan13

Lab on the Exterior Angle Theorem (XAT) with Geometry Explorer (GEX2.0)

Part I: In the Euclidean Plane

\begin{document} \maketitle \section{Introduction} This lesson explores the differences between Euclidean and non-Euclidean geometry by means of experimentation. It thus serves the same purpose as the first Project 1 of Hvidsten's text, and replaces it. \subsection{Why we use GEX2.0} This lesson will help you get started with Michael Hvidsten's geometry construction package, GEX2.0. The reason we use GEX in MA402 instead of other, similar construction kits, is this. Prof. Hvidsten has kindly augmented his already superior GEX1.0 which is bundled with his textbook, to include additional features specifically suited to our course. \subsection{Who should do this lab.} This document is intended for all students in the course, but does not assume that they had the opportunity for a real-time, hands-on and supervised lab introducing GEX. The Lab Report section below addresses the problem of providing feedback to your instructor that you have completed this lab. The style of this and subsequent labs is best compared to a recipe in a cookbook. First a list of "ingredients" followed by elaboration which may or may not be useful. \subsection{Geometrical significance.} As Hvidsten points out, contemporary college students need an experiential referent for axiomatic geometry. This is the more true the fewer students have had a rigorous treatment of Euclid's Postulates and their consequences in high school. We may think of the GEX as a simulation of of an axiomatic system. There are \textit{ primitives}, sometimes also called \textit{ undefined terms}, such as \textit{ points, lines} and the relation of \textit{incidence}. There are elementary \textit{ constructions} accessed by button presses. These may be treated as \textit{ axioms} and subsequent constructions you make are the \textit{ theorems}. But you must never forget that experiment without verification is bad science. Thus GEX is useful for discovering geometrical facts, and testing geometrical conjectures. The proof, however, proceeds in a more mathematical manner. \section{Experiment A: Euclid's Postulates} In the first experiment we examine Eulid's five Postulates, rephrase a few, and examine their validity in several models of the Euclidean and non-Euclidean plane. Be sure you have a working copy of GEX2.0 on your computer. You might also consult his webpage for operating instructions. Have a pencil and pad of paper handy when you work through this lab.. \subsection{The Postulates} Here are Euclid's postulates. See also the Appendix A of Hvidsten, and David Joyce's webpage on Euclid's Elements. \begin{itemize} \item E1: To draw a straight line from any point to any point. \item E2: To produce a finite straight line continuously in a straight line. \item E3: To describe a circle with any center and distance. \item E4: That all right angles are equal to one another. \item E5: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side side on which are the angles less than two right angles. \end{itemize} Our translation: \begin{itemize} \item E'1: Two points determine a (unique) line (segment) between them. \item E'2: This segment lies on an infinite line. \item E'3: Draw a circle with given a radius (center and point on circle). \item E'4: (Perpendiculars) From any point to any line drop the perpendicular. \item E'5: (Playfair's Postulate). Given a line and a point not on the line there is a unique line through the point parallel to the given line. (Lines with no common points are called parallel.) \end{itemize} \subsection{Comment on our version of Euclid's Postulates} Scholars do not really know what Euclid meant by his 4th postulate. Here we substitute Euclid's Proposition 12 for Postulate E4. Euclid's celebrated \textit{ Fifth Postulate}, E5, is also known as \textit{ Euclid's Parallel Postulate}. It is a mouthful. We shall spend the first third of the course understanding what he had in mind here. It is the central theme of the course. We have substituted John Playfair's (*1748) logically equivalent formulation, E'5, for Euclid's. Later in the course we shall study this equivalence, and other equivalences, in greater detail. \textbf{Vocabulary Exercise: } Euclid uses a number of technical terms in his postlulates, like \textit{draw, produce continuously, describe, equal, falling}. You should think about what they meant to his contemporary Greeks 2300 years ago. Compare this to what they mean in the Geometry Explorer application. Write an essay on this subject into your Journal. \subsection{GEX Creation versus Construction Buttons} Note that there are construction buttons on GEX for both E'4 and E'5. Once you have chosen a line and a point (hold the Shift-Key to choose more than one object), pressing "perpendicular" or "parallel" constructs that line. Choosing two points permits you to construct a circle, E'3, or a line segment, E'1. Do not confuse the construction buttons with advanced creation buttons for circles, line segments, rays and full lines. The latter are very convenient shortcuts for creating pairs of points and the constructing the object. Here is snapshot of the chrome in GEX. Explore each feature and make a note of it in your Journal with a short review of what is behind each button.. \section{Construction in the Euclidean Plane} \subsection{Recipe} \begin{itemize} \item X1. Open GEX2.0 $\rightarrow$ EuclModel $\rightarrow$ CentralOrthoModel. \item X2. Create a $\triangle{ABC}$. \item X3. Construct the ray extending segment $BC$ beyond $C$. \item X4. Create a point $D$ on the ray beyond $C$. \item X5. Bisect side $AC$ at $E$. \item X6. Double the median $BE$ to $EF$ \item X7. Construct segment $FC$. \item X8a. Measure and show that $\angle{FCA}=\angle{BAE}$. \item X8b. Measure and show that $\angle{DCF}=\angle{DCF}$. \item X8c. Conclude that \textbf{ in the Euclidean plane, the exterior angle of a triangle is the sum of the opposite interior angles.} \item X9. Switch to GEX2.0 $\rightarrow$ EuclModel $\rightarrow$ QuadModel. \item X10. Record your observations on scratch paper, edit, and enter into your Journal. \end{itemize} \subsection{Comments on the GEX construction} Note that in the Create-Palette there are buttons to shortcut some constructions. For example, to draw a triangle (X2), you can use the rubber-band cursor tool (fourth button in the creation palette) to draw a closed polygon of three edges. Note that to draw a triangle, you really only need two tools, the point creator, and the line-segment constructor. First \textbf{create} 3 points. Then choose 2 points and \textbf{construct} a side of the triangle. Do this three times. Hold the shift-key to choose multiple objects. Choose an empty spot to reject unwanted choices. In this way, Hvidsten's buttons play the role of primitives and compound actions. The latter can be reduced to a sequence of the former. Compare this to Euclid's postulates and propositions, and the axioms and theorems of an axiomatic system. Note that we have used Euclid's first Postulate three time. Apply the second Postulate by using the ray tool in the Create palette to extend the base of the triangle (X3). Note how you can \textit{ snap} the second point of the ray exactly onto $C$. So check that this was successful, \textit{ wiggle } the figure by choosing (first Create tool) one of the constructors, for example $C$, and moving it around. If the ray remains an elongation of $BC$ then your construction was correct. Similarly, for (X4) snap the point $D$ to the ray. For (X5) you choose a segment to bisect, and apply the bisection tool. In Euclid's Elements, it is a theorem that you can do this, not a postulate.
Question 1.
Look in the back of Hvidsten, or google Euclid's first book, and find the Proposition saying that segments can be bisected. To double the median (X6) with a ruler and compass would be quite easy. You apply the ruler to produce the median, the compass to measure off the distance, and mark off the new point.
Question 2.
Simulate this ruler-and-compass costruction in GEX by using the Ray-creator, and the Circle-constructor. To extend a segment create the ray through the endpoints of the segment. Then use a circle of the right center and radius to mark off the new point. For this you use the Intersection-constructor. Write here that you have completed the construction. We shall use a different, initially less intuitive construction, but one that has many dividends later in the the course. Do this \begin{itemize} \item V1. Choose the points $B$ and $E$ in that order. Click on \textit{ Mark} in the \textit{ Transform} palette. Choose \textit{ Vector}, in \textit{rectangular} coordinates. Henceforth, this construction shall be referred to simply as \texit{mark vector}$\vec{BE}$. \item V2. Once a transformation has been \textit{marked}, it can be applied to any parts of the existing figure. Here, choose the median $BE$ and the midpoint $E$ and \textit{ translate it} (1st button in Transform palette). \end{itemize} Henceforth we shall refer to the foregoing construction as \textit{producing the segment} to one-side of itself or the other. To measure an angle, for example $\angle{ FCE}$, choose the points \textit{ in this order} because that is the positive (counterclockwise) orientation for a positive angle. Then pull down the $Measure$ tab in the GEX chrome. \subsection{Vector Notation} We have used the conventional notation $BE$ for a \textit{directed line segment} with initial poit $B$ and final point $E$. In the prerequisite calculus course you learned that the set of all such segments parallel to the given segment comprises the \textit{vector} $\vec{BE}$. And here you have observed how a vector determines a transformation called a \textit{translation.} The line through two point $B$ and $E$ for example, is sometimes written as $\ell_{BE}$. Here the order is not important, $\ell_{BE}=\ell_{EB}$. But both notations, the over-arrow and the curly-ell with a subscript is inconvenient to write in a text document not specifically designed for it, as is LaTeX. Besides it takes an effort to write this on paper, and can't be written into an email. So we will abandon both notations. The line determined by two points will be writte $(EB)$. And the vector determined by the two points is written $B-E$, the difference of two points. Note the reversal of the order required by vector algebra. The segment in a figure can continue to be written just $BE$, as it has been for two millennia. Here is a little comic-strip on what we have been talking about here. \subsection{Geometrical Commentary} GEX2.0 allows us to break the lifelong habit of seeing the Euclidean plane in only one way, the way you were taught since grade school. C-Students in this course frequently answer the question, "What is non-Euclidean geometry?" something like this: "In non-Euclidean there are points and straight lines, except that they are not straight but curved." Using some 3-D geometries we shall study later, we can have a model of the Euclidean plane in which the straight lines are also curved. And we shall see models of the non-Euclidean plane in which the straight lines look straight to our Euclidean eyes. Now, Euclid would not have accepted the above argument simulated by GEX because the Greeks had no numerical sense of measuring angles. They could compare angles, but not by associating numbers to them. Even we should not trust a mere machine to have measured the angles to the very last decimal place. That is why geometry is a deductive science. This is a very good thing, because as we shall see in the second exercise, things are different in non-Euclidean Geometry. \section{Vocabulary and Review} \begin{itemize} \item \textit{Wiggle:} Choose one of the initially \textit{ created} objects and move them around to see that the rest of the construction hold together. \item \textit{ Produce a segment} Choose endpoints of a segment, mark this vector, choose the segment, translate what's chosen. \end{itemize} Here is a more detailed discussion of what you have, or are about to actually do. Do not worry about how verbose this lesson is and its redundancy. As you learn how to work with GEX the lesson will become shorter, terser, and leave a lot more to you. \begin{enumerate} \item Be sure you have a working copy of GEX2.0 on your computer. You might also consult his webpage for operating instructions. Have a pencil and pad of paper handy. \item Take a another look at Euclid's five postulates (Appendix A of text, or Google them). You might enter them into your Journal for future reference. \item. Euclid's formulation shows that he considered his first 3 postulates to be constructions. Therefore, we want to see how to construct them in the various models of Euclidean and non-Euclidean geometry available on GEX. \item Postulate 4 has been a mystery for 2300 years. So we'll ignore Euclid's formulation for now. But Postulate 5, the Parallel Postulate, is central to geometry and this course. Euclid's formulation of it is very strange, and for a good reason. For now we use an equivalent formulation due to John Playfair (1795). We will also substitute a theorem for E4, namely, that for a line and a point off the line, there is a unique perpendicular from the point to the line. Now all five Postulates assert that a certain construction can be performed. \item The idea of these experiments is to see HOW these five axioms hold in the three models of Euclidean geometry in GEX2.0, and whether or not they hold in the two non-Euclidean geometries. \item You should familiarize yourself with GEX 2.0, for instance by working with the appendix in the textbook (which is for GEX 1.0) and the relevant new parts in the documentation available for GEX 2.0. At this point we ONLY need the following features: \item Create Buttons: Euclid specifies that there are points and lines, and a point can either be on the line or not. So you can Create points. After every creation step you should immediately pick the Picker a.k.a. Chooser or the Arrow button, or you'll continue to create new stuff you'll want to erase. \item Given two points (pick two points, holding the shift key down for the second choice), you can Construct exactly those items that light up in the Construct menu. For Euclid, the line meant a line segment. Note that when you have chosen (picked) two points (no more nor less. Shake off anything accidentally sticking to your picker by picking a blank place.) the circle construction is also lit up. So you know that E3 is true. \item Shortcut Creations: Hvidsten provides 4 shortcuts in the Creation menu. Each of them ends up with a construction based on two points the: circle, segment, ray, and a full line. The two points mark the beginning and end position of your tool. Note that a creating point snaps to a another object if you get it near enough. That is usually a good thing. If it isn't, undo your action in the Edit menu. \item Euclidean Models: The EuclModel pulldown menu lets you choose one of 3 models to work in. You can see all three models with the QuadModel choice. But note, you can continue working ONLY in the chosen one in top left panel. While experimenting, it is faster to just pick a new canvas than to try and correct the one you're working in. Don't bother saving for now. This is scratch paper. \item Near the end of the course we will discuss how GEX 2.0 does this work. For now you need to understand that in addition to your usual idea of the Euclidan plane, as interpreted on a sheet of paper since your earliest schooling, we have two BOUNDED models, which consist of the points inside, but NOT on the boundary of a disk. We show the boundary for reference, but in this interpretation, all the Points are inside the disk. \item When discussing models of geometrical axiom systems, it is \texbf{essential} to distinguish between the names of objects in our common idea of a the plane (the points, lines, parallels, perpendiculars, circles you've used since you were wee children) and those geometrical figures that interpret the primitives of our axiom systems. We will capitalize words referring to the latter. It might be nicer to use color, and even musical jingles to alert you to this use of the same English words to mean two different things, or even use a foreign language. But all that is impractical. You have good brains and soon will have no difficulty keeping this straight. \end{enumerate} \section{Some final advice} Do not admit to being \textit{confused} by this lesson . It's OK to not quite understand all of it in one sitting. Use a knife and fork, and maybe keep some left-overs in the fridge for later. Take your time. But do not allow yourself to be confused. Befuddlement is for the ignorant or the lazy. In fact, we'll spend the entire semester really understanding these concepts. Right now we're learning by using, like the mother tongue babies learn by experience. \section{Lab Report} Your report on this lab should be a 2-4 page write up in .pdf format (15pc). There should be to be at least two properly captioned figures (15pc). Be sure that all mathematical formulas and letters are in LaTeX format (dollar signs!). There should be at least two mathematical formulas (15pc). Geometrical accuracy and expository quality counts for half the grade. \begin{itemize} \item Compose your lab report as a draft first. \item Draw hand sketches of what you plan to illustrate. \item Practice the LaTeX formulas you'll use in texPad first. \item Complete your figures in GEX2.0. Save your .gex files on your computer where you can find it easily. \item Make screen print of the figures. Edit them Paint/iPaint or some other picture editor if needed. Then upload your figures into texWins. \item Compose your report on texWins. Save a copy of your workspace on your computer. Be sure to change its generic name "mywspace.tex" to something that shows it belongs to you. Do NOT use spaces in the name! \item Download the .pdf, change its generic name "texdoc.pdf" to something more recognizable and upload it into the Moodle. \end{itemize} The following is a sequel to this lab. At a later date specified in the syllabus you should resume this experiment here.

Part II: In the Non-Euclidean Plane

\section{Introduction} \subsection{In Lab Session} Choose group membership of 3 or 4. Exchange email/phone coordinates. Choose name of group: Thales, Pythagoras, Euclid, Archimedes, Pappus, Klein, Poincare, Hilbert, Birkhoff. Elect \textbf{Presenter, Secretary, Demonstrator.} Record your experiments in preparation to write up a report. \subsection{Online Session} Despite the technical difficulties it is still possible for your to form study groups over the internet. In particular, if the Virtual Office is enabled, you and your group members can meet in this virtual lab and perform this lab, albeit with greater difficulty, but perhaps greater lasting benefit. \subsection{Non-Euclidean Utility in GEX2.0 } In the custom edition (GEX2.0) you should now Chose: File $\rightarrow$ New $\rightarrow$ Hyperbolic $\rightarrow$ Model $\rightarrow$ Klein . and you will see the same palette as in the default Euclidean mode of GEX. The circle in the picture plane reassures you that you have the non-default mode. There is also the Elliptic plane, which we do not treat in this course. You can read up on this geometry in Hvidsten's text on your own. The default model for Hyperbolic Geometry is the Poincare model. We will most often work in this model. The reason to start with the Klein model is that incidence geometry and the negation of Playfai're axiom is easiest to see in the Klein model. Also, if you do not have access to GEX on your computer, and have to depend on ruler-and-compass construction, the Klein modle constructions are easier. The third model, the Upper Half Plane (UHP), is the model most commonly used in higher mathematics, especially in 3-D. However, it is rarely used in high school geometry, and is not very intuitive because of it's anisotropy. Moreover, the UHP shares the disadvantage with the usual Euclidean plane that you cannot see all point and lines at the same time in a finite figure. We will very rarely refer to the UHP in this course. The important thing to remember is that all three models of Non-Euclidean (a.k.a. Hyperbolic) Geometry are isomorphic. Recalling the notions you have learned in Chapter 1 of this course, it is a theorem (which we do not prove in this course) that Hyperbolic Geometry is categorical, i.e. any model can be used to observe theorems in this geometry, whichever is the most convenient for the purpose. \subsection{Check Euclid's Postulates.} As a warmup exercise, determine which of Euclid's Postulates (as described above) hold or do not hold in the hyperbolic plane. Recall our vocabulary: \begin{itemize} \item[Wiggle:] Choose a collection of objects and move it around around with your mouse. If the relations between sub-ojects persist, your construction is correct. \item[Produce:] Starting with a line-segment, extend it to one side by create a translation vector and translating the segment, thus: \item Choose: endpoints(in order) $\rightarrow$ Transfrom Mark $\rightarrow$ Vector $\rightarrow$ 2 Pts $\rightarrow$ \item Choose: segment $\rightarrow$ Translate $\rightarrow$ repeat if your want a longer production of the segment.. \section{Non-Euclidean Exterior Angle Experiment} The following experiment consists in choosing an exterior angle of a triangle and performing the above construction of the Absolute Exterior Angle Theorem two times, once from each opposite interior angles. \subsection{Absolute Exterior Angle Construction} \begin{itemize} \item Choose an arbitrary triangle $\triangle ABC$ with exterior angle $\angle{DBC}$ by producing base $AB$ one length beyond $B$ to $D$. \item Produce the median $AA\prime$ to side $BC$ one length, ending at $A''$ \item Connect $A'' B$ to form exterior triangle $\triangle A'' A' B$ and prove that it is congruent to $\triangle$ AA' C$by SAS. (Watch the order of vertices!) \item Conclude the$\angle ACB \cong \angle A'' BC$. \item Observe that the opposite interior angle at$C$has a congruent image inside the exterior angle you chose at$B$. So it is smaller. \end{itemize} \subsection{Hyperbolic Exterior Angle Construction} Repeat the construction by extending median$CC'$, where$C'$is the midpoint of$AB\$. Using the Vertical Angle Theorem, you can transfer the second copy of an opposite interior angle into the same exterior angle as the first. This way you can compare their sum to the exterior angle itself. What are your conclusions? \section{The Conclusion} The above construction can be performed in all three geometries, Euclidean, Hyperbolic, and Elliptic. And it will show the same result, proper to each geometry, and in each of the seven models on GEX2.0 Write up your conclusion of the Exterior Angle Theorem in all three geometries. \end{document}