Geometry Explorer: User Guide Michael Hvidsten Gustavus Adolphus College DRAFT: August 3, 2015 ii Contents Introduction ix 1 Getting Started 1 1.1 The Main Geometry Explorer Window . 1 1.2 Selecting Objects . 4 1.2.1 Selections Using the Mouse . 4 1.2.2 Selections Using the Select All Menu . 7 1.3 Active vs Inactive Tools . 7 1.4 Labels . 7 1.5 Object Coloring . 9 1.6 Changing Palette Colors . 9 1.7 On-Line Help . 10 1.8 Undo/Redo of Actions . 10 1.9 Clearing, Resizing the Canvas . 12 1.10 Saving, Opening, Printing of Files . 13 1.11 Saving Files as Images . 14 2 Tutorials 17 2.1 Tutorial 1 Working With Basic Geometric Figures . 17 2.2 Tutorial 2 Constructions . 20 2.2.1 Euclid's Equilateral Triangle . 21 2.3 Tutorial 3 Transforming Geometric Figures . 24 2.3.1 Rotation of a Figure { Using a Geometric Angle . 24 2.3.2 Dilation of a Figure { Using a Numerical Angle . 26 2.4 Tutorial 4 Measurement . 27 2.4.1 Triangle Area . 28 2.4.2 Triangle Angle Sum . 29 2.5 Tutorial 5 Analytic Geometry . 32 2.6 Tutorial 6 Hyperbolic Geometry . 35 iii iv CONTENTS 2.7 Tutorial 7 Elliptic Geometry . 41 2.8 Tutorial 8 Recording Geometric Macros . 45 2.8.1 Recorder Windows . 45 2.8.2 Custom Tools . 48 2.9 Tutorial 9 Turtle Geometry . 51 3 Constructions 57 3.1 Tools in the Construct Panel . 58 3.2 Using the Locus Tool . 62 3.2.1 A Simple Example Using the Locus Tool . 62 3.2.2 The Ellipse as a Locus of a Point . 63 3.3 Tangent to a Circle . 67 4 Measurements 69 4.1 Neutral Measurements . 70 4.1.1 Point Measurements . 71 4.1.2 Segment Measurements . 71 4.1.3 Circle Measurements . 72 4.1.4 Arc Measurements . 72 4.1.5 Filled Object Measurements . 72 4.2 Euclidean-only Measurements . 72 4.2.1 Point Measurements . 72 4.2.2 Linear Object Measurements . 73 4.3 Hyperbolic-only Measurements . 73 4.4 Elliptic-only Measurements . 74 4.5 Precision in Measurements . 75 4.6 Compound Measurements . 75 4.7 Using the Calculator . 76 4.7.1 Circle Area . 77 4.7.2 The Button Pad . 79 4.7.3 Evaluation of Expressions . 84 4.8 Sliders . 84 4.9 User Input Parameters . 87 4.10 Using Tables . 91 4.10.1 Quad Interior Angles . 91 5 Transformations 95 5.1 Quick Overview of Transformations . 96 5.2 Defining Transformations . 97 5.2.1 Setting Geometric Transformation Data . 97 CONTENTS v 5.2.2 Defining Custom Transformations . 98 5.3 Example: The Hyperbola . 99 5.4 Copying Figures in Transformations . 102 5.5 Compound Transformations - Fixed . 104 5.6 Compound Transformations - Random . 107 5.7 Compound Transformations - IFS . 112 5.8 Transformations Based on Measurements . 116 5.9 Affine Euclidean Transformations . 118 5.9.1 Affine Transformations on Circles . 120 5.10 Editing Custom Transformations . 120 6 Analytic Geometry 123 6.1 The Coordinate System Used in Geometry Explorer . 124 6.2 Plotting Points Based on Measurements . 125 6.3 Analysis of Functions . 128 6.3.1 Plotting y = f(x) . 128 6.3.2 Plotting Polar Functions r = f(θ) . 131 6.3.3 Plotting Parametric Functions . 132 6.3.4 Adding Input Boxes for Functions . 133 6.3.5 Attaching Points to the Graph of a Function . 134 6.3.6 Tangents to Functions . 136 6.3.7 Derivatives of Functions . 138 6.3.8 Iterated Functions and Dynamical Systems . 140 6.3.9 Controlling the Appearance of Plotted Functions . 147 7 Hyperbolic Geometry 151 7.1 Background and History . 151 7.2 The Poincar´eDisk Model . 153 7.3 The Klein Disk Model . 155 7.4 The Upper Half-Plane Model . 157 7.5 Working in the Hyperbolic Canvas . 158 7.6 Saccheri Quadrilateral . 161 7.7 Translation, Parallel Transport, and Holonomy . 164 7.8 M¨obiusTransformations . 168 8 Turtle Geometry 173 8.1 Basic Turtle Geometry in Geometry Explorer . 174 8.2 Turtles, Fractals, and Grammar Re-writing . 181 8.3 Plant Grammar . 186 8.4 Color Index Tables . 188 vi CONTENTS 8.5 Saving, Opening, Printing Grammars . 190 8.6 Turtle Geometry in Non-Euclidean Environments . 191 9 Tessellations 193 9.1 Regular Tessellations of the Plane . 194 9.2 A Tessellation Construction . 197 9.3 Hyperbolic Tessellations . 200 10 Recording Constructions 205 10.1 Using the Recorder Window . 205 10.1.1 Starting a Recording . 206 10.1.2 Playing a Recording . 208 10.1.3 Recursive Recordings . 210 10.1.4 Saving, Opening, Printing . 214 10.1.5 Playback on Sets of Basis Elements . 215 10.2 Custom Tools . 216 10.2.1 Managing Custom Tools . 219 11 Animation 223 11.1 User Interface for Animation . 223 11.2 Animation in the Euclidean Plane . 225 11.2.1 Animating Circles along Segments - The Cycloid . 225 11.2.2 Animating Circles along Circles - The Hypocycloid . 228 11.3 Animation in the Hyperbolic Plane . 231 11.4 Animation in Elliptic Geometry . 233 12 Geometry Explorer and the Internet 235 12.1 The Geometry Explorer Web Browser . 235 12.2 The Geometry Explorer Help System . 235 12.3 Using Web Links Directly from the Canvas . 236 12.4 Saving Constructions as HTML Files . 238 13 Other Features 241 13.1 The Edit Menu . 241 13.1.1 Undo/Redo . 241 13.1.2 Cut/Copy/Paste . 242 13.1.3 Clear, Select All . 242 13.1.4 Point Size, Pen and Fill Styles . 242 13.1.5 Properties . 248 13.1.6 Setting User Preferences . 249 CONTENTS vii 13.2 The View Menu . 249 13.2.1 Helper Windows . 249 13.2.2 Hiding and Showing . 250 13.2.3 Tracing Objects . 252 13.2.4 Animation . 252 13.2.5 Miscellaneous View Options . 252 13.2.6 Zooming and Panning the Canvas . 252 13.3 Saving The Canvas as an Image . 253 13.4 Control Buttons . 253 13.5 The Info Tool . 254 13.5.1 Parents and Children . 255 13.6 Editing Text Areas in Geometry Explorer . 255 Bibliography 259 Index 261 Introduction It may well be doubted whether, in all the range of science, there is any field so fascinating to the explorer-so rich in hidden treasures-so fruitful in delightful surprises-as Pure Mathematics. Lewis Carroll (Charles Dodgson) (1832-1898) An explorer is one who seeks out new worlds and ideas. As Lewis Carroll would probably agree, exploration is not always easy|the explorer can at times find the going tough. But the treasures and surprises that active exploration of ideas brings is worth the effort. Geometry Explorer is designed as a geometry laboratory where one can create geometric objects (like points, circles, polygons, areas, etc), carry out transformations on these objects (dilations, reflections, rotations, and trans- lations), and measure aspects of these objects (like length, area, radius, etc). As such, it is much like doing geometry on paper (or sand) with a ruler and compass. However, on paper such constructions are static|points placed on the paper can never be moved again. In Geometry Explorer, all constructions are dynamic. One can draw a segment and then grab one of the endpoints and move it around the canvas, with the segment moving accordingly. Thus, one can create a construction and test out hypotheses about the construc- tion with an infinite number of possible variations. Geometry Explorer is just what the name implies|an environment to explore geometry. Geometry Explorer can easily be used to access Web-based information. There is an Internet browser built in to the program that allows hyperlinks to Web pages to be inserted directly into a geometry construction. The Help system consists of a series of inter-linked Web pages that are accessed via the built-in browser. (You do not need to be connected to the Internet to use the Help system.) Non-Euclidean geometry can easily be explored using Geometry Ex- plorer. Constructions can be carried out in Euclidean, Hyperbolic, Elliptic, or Spherical geometry environments using the same user interface. Almost ix x INTRODUCTION all actions that apply in the Euclidean environment can be carried out in the non-Euclidean environments (with a few important exceptions that depend on the parallel postulate). Fractal geometry can be explored using turtle graphics and grammatical descriptions of fractals. In turtle graphics, one controls a \turtle" on the screen by telling it to move, draw, rotate, change color, etc. Grammar- based descriptions of fractals encapsulate a fractal's structure by sentences of symbols. These sentences can then be interpreted as a series of turtle actions. Geometry Explorer is designed to assist the classroom teacher. Text areas can be created on the screen so that additional information can be included with a construction. If a large amount of textual information must accompany a construction, this can be included in a separate Notebook with the construction. Web pages can be referenced directly from the.