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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
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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