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Essential 3D Geometry Unraveling the Mathematical Wonders First Edition By Bernardo Camou, John Olive, Margarita Colucci and Graciela Garcia Included in this preview: • Table of Contents • Introduction • Chapter 1 For additional information on adopting this book for your class, please contact us at 800.200.3908 x501 or via e-mail at [email protected] Essential 3D Geometry unraveling the mathematical wonders Bernardo Camou John Olive Margarita Colucci Graciela Garcia Bassim Hamadeh, CEO and Publisher Christopher Foster, General Vice President Michael Simpson, Vice President of Acquisitions Jessica Knott, Managing Editor Kevin Fahey, Marketing Manager Jess Busch, Senior Graphic Designer Zina Craft, Acquisitions Editor Jamie Giganti, Senior Project Editor Brian Fahey, Licensing Associate Kate McKellar, Interior Designer Copyright © 2013 by Cognella, Inc. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of Cognella, Inc. First published in the United States of America in 2013 by University Readers, an imprint of Cognella, Inc. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Printed in the United States of America ISBN: 978-1-60927-389-7 (pbk) / 978-1-60927-390-3 (br) Contents Introduction & Goals 1 Chapter 1: The Polygonal Faces of Regular Polyhedrons 3 Chapter 2: Constructing and Exploring the Platonic Solids 11 Chapter 3: The Surface Area of the Platonic Solids 17 Chapter 4: All About Radicals 21 Chapter 5: The Dihedral Angles of the Platonic Solids 27 Chapter 6: The Pentagon’s Contributions to Trigonometry 35 Chapter 7: Thales’ Theorem 39 Chapter 8: The Square-Based Pyramid and the Cube 45 Chapter 9: Volumes of the Platonic Solids 51 Chapter 10: How to Construct a Tetrahedron Knowing the Length of Its Six Edges 63 Chapter 11: Circumcircles and Circumspheres 67 Chapter 12: The Volume of a Tetrahedron with Three Congruent Concurrent Edges and Radius of Its Circumscribed Sphere 73 Chapter 13: The Volume of a Semi-Orthocentric Tetrahedron and Radius of the Circumscribed Sphere 79 Chapter 14: Spherical Angles, Lunes, and Angular Defects 87 Chapter 15: A World of Symmetry Within the Regular Polyhedrons 95 Chapter 16: Meeting the Archimedean Bodies 99 Appendix A: Trigonometric Table For The New Millennium (TTNM) 107 Introduction & Goals A ccording to astronomical data collected during the past ten years, the universe may be finite and possibly shaped like a dodecahedron—a 3D object with 12 twelve pentagonal faces.* The study of 3D mathematical objects has been and continues to be a tremendous challenge. Mathematics, as with all sciences, needs to experiment and the starting point of experimentation is representation, which has been always particularly complex when we are dealing with objects in space. So the goal of this course is to enter this marvelous, infinite world of 3D geometry, travelling just a few but worthwhile steps on the way to developing a systematic and fruitful approach to this fascinating but poorly studied branch of Mathematics. Our approach entails multi-type representations of the 3D objects using three types of technology: 1) craft technology, 2) computer technology, and 3) paper-pencil technology. Each technology provides more than one type of representation: 1) solids or lattice models, 2) Cabri 3D or GSP files, and 3) constructions using straight edge and compass or free-hand drawings. While we anticipate it will be fun to construct all these representations, the true goal of this activity is that with each new representation of the same mathematical object, we learn a new feature about * it. And though it could sound paradoxical, the more concrete representations we can have of an object the more abstract the object becomes in our mind. A second characteristic of our approach is the integrated method we will use throughout the course. We are not just studying geometry: it is geometry integrated with algebra and trigonometry. The three branches work together with the same goal of learning about these 3D objects. A third characteristic (maybe underestimated in mathematics) is the use and appreciation of approximations. Approximations are a natural path to exactness; they lead us eventually to exact results and they endow those results with sense and value. We will start with very simple activities such as constructing concrete models, and measuring and approximating properties of those models; then move towards more abstract representations, reasoning, and exact calculations. A major goal of this course is that all participating students construct a profound knowledge about the material of the course. And the ultimate goal is that the course would be inspirational for each of you in a way that whenever the opportunity arises, you would like to teach and further develop a 3D Geometry course for your own students. http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html Introduction & Goals 1 Chapter 1 The Polygonal Faces of Regular Polyhedrons A regular polyhedron is a three-dimensional body in which every face is the same regular polygon and every vertex of the polyhedron contains the same number of faces. There are five (and only five) regular polyhedrons: • The Regular Tetrahedron (the faces are four equilateral triangles) 1. an 8-cm side equilateral triangle Fig 1.1 2. a 7-cm side square • The Cube (the faces are six squares) • The Octahedron (the faces are eight equilateral triangles) • The Dodecahedron (the faces are 12 regular pentagons) Fig 1.2 3. a 6-cm side regular pentagon • The Icosahedron (the faces are 20 equilateral triangles) Four of these polyhedrons are shown on the cover of this book. Fig 1.3 Ac tivity 1.1 We shall begin our study of the 3D Regular Polyhedrons by first constructing and analyzing the 2D polygons that form the faces of these Polyhedrons (plus the regular hexagon). Your task is to construct the following regular polygons using any geometrical instruments: 4. a 5-cm side regular hexagon Fig 1.4 The Polygonal Faces of Regular Polyhedrons 3 4 Essential 3-D Geometry D The different constructions will be discussed in class. How to C o n str u c t a R eg u l a r Pentag o n w ith R u ler an d C o m pass E C 1. Draw a segment [AB] in the middle of a page. 2. Construct a perpendicular ray to [AB] at point B in the lower half-plane. 3. Construct a point P on this ray so that [BP] = 1/2 [AB]. 4. Draw the hypotenuse [AP] of the triangle ABP and extend it further from P. A A B P B Q Fig 1.6 P Q 9. You now have the 5 vertices of your regular pentagon; join them to produce the regular pentagon ABCDE. D Fig 1.5 5. Mark point Q on the extension of [AP] so that [PQ] = [BP]. E C 6. Set your compass with radius [AQ] and mark the intersection of arcs from points A and B using your compass to determine point D (D in the upper half-plane of border AB). A 7. Set your compass radius to length mark arcs from points A and D to point E. 8. Set your compass radius to length mark arcs from points B and D to point C. [AB] and determine [AB] and determine B Fig 1.7. Compass construction of a regular Pentagon 10. If instead of drawing the sides we draw the diagonals, we have the regular 5-point star! The mythic star!!! The Polygonal Faces of Regular Polyhedrons 5 Fig 1.8 This property (for Euclidean triangles) is well known so we consider its demonstration unnecessary (for this course). You could find or derive a proof of this property for yourself. Let’s go to the interior angles of the pentagon. Fig 1.9 At least two possible approaches can be considered: 1. First, we consider the center of the regular pentagon (the center of its circumscribed circle). The segments from this center to each vertex of the pentagon are all radii of the circumcircle and are therefore of equal length. These radii also partition the center angle into 5 equal angles. 72 Fig 1.10 We have seen here an effective and accurate procedure to construct both the regular pentagon and the five-point star. We lack, at this point, a step-by-step justification for why this procedure is valid. After the following activity, you will have an opportunity to figure out for yourselves the reasons that this procedure is certain and exact. Activity 1.2. In the first activity you constructed the first 4 regular polygons. You will now work on the following questions: • What are the measures of the interior angles of each regular polygon? • What is the area of each of the recently constructed polygons? The interior angles of equilateral triangles and squares are easy. But what about the others? Let’s state Property 1: “The sum of the interior angles of any triangle is always 180°.” Fig 1.11 If we divide the complete angle of 360° at the center into 5 equal angles we find the vertex angle of each of the 5 triangles at the center to be 72°. But each of the 5 triangles in the figure is isosceles (equal radii) and hence each has two equal base angles. Property 2: “If a triangle is isosceles, it has two equal angles and conversely, if a triangle has two equal angles, it is isosceles.” 6 Essential 3-D Geometry 54 54 540 = 108c 5 54 72 72 72 72 72 36 54 108 36 36 108 36 Fig 1.12 72 Each of the triangles is isosceles and has a 72° angle at the center of the pentagon. Thus, the other two angles (using Property 2) are calculated by 180 - 72 = 54 2 Hence the interior angles of the pentagon are calculated by: 54 x 2 = 108°. 2. For the second approach, we draw two diagonals of the pentagon to obtain three triangles as in Figure 1.13. 72 36 Fig 1.14 We have two isosceles triangles having a 108° angle. Using Prop 2 we can calculate the other two angles: 180 - 108 = 36° 2 The angles of the central triangle can then be calculated: 108 – 36 = 72° and 108 – 36 – 36 = 36°. 36 36 180 180 36 180 36 36 Fig 1.13 Fig 1.15 The sum of the 5 interior angles of the pentagon is equal to the sum of the angles of the three triangles, i.e.: 180 x 3 = 540 But since the 5 angles of the pentagon are equal, then each angle measures: Besides finding the angle of the regular pentagon, we have also found the angles of the star. This star is present on the flags of many countries, is used in the logos of numerous brands, and we step across hundreds of stars when Christmas is approaching. The Polygonal Faces of Regular Polyhedrons We know now that the angle of its points is exactly 36°! Activity 1.3 : A R atio n al e for the Pr o c edu re ( Algo rith m) for C ons tr u c tin g a R eg u lar Pe n t a g on Using what we have just found out about the angles of the interior triangle of a regular pentagon in Fig 1.14, we can now find a rationale for why the compass and straight edge construction of the regular pentagon works. 36 108 7 We can easily observe that we have three isosceles triangles: AEO, AOJ, and EJA. EJA is similar to AEO because they have the same angles. Hence, by Thales’ Theorem, their sides are proportional. Suppose that AE = 1 (the side of the pentagon) and the unknown AO = x (the pentagon’s diagonal). In terms of x and 1, what are the lengths of AJ and JE? Since EJA is isosceles, AJ=AE =1 36 36 Since AOJ is isosceles, OJ=AJ=1 108 36 Since AO= OE = x and OJ=1 then JE = x-1 72 72 36 Since triangle AEO is similar to triangle EJA, we can write the following equation: x = 1 1 x-1 Fig 1.16 Let’s work with the green central triangle which is equivalent to: O x2 - x - 1 = 0 36° O 1 J x J 36° 36° 1 72° A E Fig 1.17 We draw the angle bisector of A that intersects OE in point J. A 1 Fig 1.18 x–1 E 8 Essential 3-D Geometry This quadratic equation has two roots: BP = 1 then, using Pythagoras, 2 2 AP2 = 12 + ` 1 j = 1 + 1 = 5 2 4 4 5 and since PQ = 1 , then Hence: AP = 2 2 AQ = 1 + 5 . 2 If AB=1 and x1 = 1 + 5 and x2 = 1 - 5 2 2 The second root is a negative number and thus it should be discarded, as the diagonal’s length cannot be negative, but the first root, which is positive, is the solution to our problem. Therefore the diagonal’s length of a regular pentagon with side 1 is: 1+ 5 2 If AB = c instead of being 1, we will get from this construction that If the side’s length of the pentagon is c instead of 1, then the diagonal’s length would be applying similarity: AQ = c 1 + 5 m c . 2 But, by taking the ratio of the diagonal of the regular pentagon over its side-length, we still obtain the Golden Number: 1 + 5 . 2 c 1 + 5 mc 2 This number is famous not only in mathematics but also in art, in architecture, and even in biology. H ome w or k 1 . 1 It is named with the Greek letter (phi) and is known as the golden number or divine proportion. Review the derivation and construction of the diagonal of the regular pentagon. Write a definition of the Golden Number in terms of the diagonal and side of the regular pentagon. z = 1 + 5 can be defined in several ways but 2 the most illuminating, scientific, and significant is that is the length of the regular pentagon’s diagonal in terms of its side, or simply the ratio of the diagonal and the side of any regular pentagon. Our auxiliary construction in Fig 1.5 and reproduced in Fig 1.19 is the construction of the Golden Number (by applying Pythagoras) and is therefore the length of the diagonal of our regular pentagon starting with side AB. A B P Q Fig 1.19 H ome w or k 1 . 2 Bring to your next class three different colored sheets of poster board at least 12 inches by 18 inches (30 cm by 45 cm). Also bring your compass, ruler, and triangular square (or set-square). Complete the following activity on your own: Calculate the area of the following 4 regular polygons: The Polygonal Faces of Regular Polyhedrons G m GH = 5.00 cm H m AB = 8.00 cm B D m CD = 7.00 cm C E m EF = 6.00 cm F 9