The Circle: A Mathematical Exploration Beyond the Line

345 Pages·2016·9.72 MB·English

by  Alfred S. Posamentier

Published 2016 by Prometheus Books The Circle: A Mathematical Exploration beyond the Line. Copyright
© 2016 by Alfred S. Posamentier and Robert Geretschläger. All rights reserved. No part of this publication
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Cover design by Jacqueline Nasso Cooke
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Library of Congress Cataloging-in-Publication Data Names: Posamentier, Alfred S. | Geretschläger, Robert.
Title: The circle : a mathematical exploration beyond the line / by Alfred S. Posamentier and Robert
Geretschläger.
Description: Amherst, New York : Prometheus Books, 2016. | Includes bibliographical references and
index.
Identifiers: LCCN 2016007783 (print) | LCCN 2016011144 (ebook) | ISBN 9781633881679 (hardcover) |
ISBN 9781633881686 (ebook) Subjects: LCSH: Circle. | Circle packing. | Geometry, Plane. |
Geometry.
Classification: LCC QA484 .P66 2016 (print) | LCC QA484 (ebook) | DDC 516/.18—dc23
LC record available at http://lccn.loc.gov/2016007783
Printed in the United States of America

Acknowledgments
Introduction
1. Circle Relationships in Basic Plane Geometry—and an Extension
2. The Circle's Special Role in Geometry
3. Famous Theorems about Circles
4. Circle-Packing Problems
5. Investigation of Equicircles
6. Circle Constructions: The Problem of Apollonius
7. Inversion: Circle Symmetry
8. Mascheroni Constructions: Using Only Compasses
9. Circles in Art and Architecture
10. Rolling Circles: Hypocycloids and Epicycloids
by Christian Spreitzer
11. Spherical Geometry—Circles on the Sphere
Afterword: A Cultural Introduction to the Circle
by Erwin Rauscher
Appendixes
Notes
Bibliography
Index
In the preparation of this book we would like to thank Dr. Hidetoshi Fukagawa
for his photos of the Sangaku in chapter 7 and Dr. Robert J. Lang for his
Treemaker graphic in chapter 4. Both have been a consistent source of
mathematical inspiration to coauthor Dr. Robert Geretschläger for many years.
We also wish to thank Dr. Bernd Thaller for his graphic of stacked spheres in
chapter 4. In addition to providing chapter 10, Dr. Christian Spreitzer was also
helpful with a number of geometric drawings in chapters 10 and 11, and for this
he deserves our thanks. Special thanks are also due to Dr. Erwin Rauscher, for
providing the afterword for the book.
A book of this complexity requires very capable editorial management and
copyediting. We wish to thank Catherine Roberts-Abel for very capably
managing the production of this book, and give special thanks to Jade Zora
Scibilia for the truly outstanding editing throughout the various phases of
production. The editor in chief of Prometheus Books, Steven L. Mitchell,
deserves praise for enabling us to approach the general readership to expose the
gems that lie among one of the most commonly seen geometric figures, the
circle. Special thanks also go to Hanna Etu, Mark Hall, Bruce Carle, Jacqueline
Nasso Cooke, Laura Shelley, and Cheryl Quimba for their hard work.
In elementary plane geometry, there are basically two kinds of lines that are
drawn: linear lines (straight lines) and circular arcs (complete circles). The basic
element of linear geometric shapes is the triangle.1 That is to say, most
geometric figures composed of straight lines are often broken down into
triangular components in order to investigate their properties. In the world of
linear geometry, this makes the triangle the key element for inspecting and
evaluating geometric relationships. However, the circle is as much a significant
portion of plane geometry as is any other component; furthermore, it is the only
kind of “line” that can be drawn on a sphere. This makes the circle perhaps truly
more ubiquitous in the world of geometry, in contrast to the straight line, which
is not present in spherical geometry. It is with this idea in mind that we embark
on a journey to inspect the most common aspects of the role of the circle in
geometry.
In the history of mathematics, the circle has fascinated mathematicians
perhaps more than any other shape. One aspect of how to trace the history of the
circle is through the evolution of the value of pi, π.2 The focus on a precise—or
almost precise—value of π has enchanted mathematicians for centuries. Circles
have been part of our culture both from a philosophical and from a theological
perspective. We will briefly address some of this after we have thoroughly
examined the many geometrical wonders that the circle presents.
Before delving into many of the exciting and truly amazing relationships
involving circles, we provide a brief review of basic circle relationships—many
of which are presented in high-school geometry courses.
There are many interesting geometric problems that, because of their
simplicity and cleverness, make for a very entertaining introduction to
appreciating the power of circles. At a more sophisticated level—or shall we say
more prominent level—there are many relationships in and about circles that
have become named theorems because of their significance. In chapter 3, these
are presented in a very comfortable and intelligible fashion. Even some unnamed
theorems should cause awe.
Circles that can be inscribed in a triangle, circumscribed about a triangle, or
simply tangent to the three sides of a triangle but not containing the interior of
the triangle, are often referred to as equicircles, which are explored at length in
chapter 5. Equicircles provide some clever insights into plane geometry and can
be very entertaining due to their simplicity.
Constructing circles to meet specific criteria, such as being tangent to other
given circles or to given lines, has been one of the challenging problems dating
back to antiquity. This problem—consisting of several parts—is known as “The
Problem of Apollonius.” In chapter 6, it is presented and solved in such a fashion
that you will feel a true sense of completion, and, we hope, appreciation. It is
always interesting to see the kinds of problems that have perplexed
mathematicians in antiquity, and we here solve them in a reader-friendly way, so
that they can be seen, simply, as solving a puzzle or problem.
We are all taught in high school to do geometric constructions with an
unmarked straightedge and a pair of compasses. Some of these constructions are
very simple, while others can be more challenging. However, for several
hundred years we know that all such constructions could also be done without a
straightedge and using just a pair of compasses. These are known as Mascheroni
constructions, which are explored in chapter 8. One may wonder how you can
construct a line using only a pair of compasses. Well, we will show that with
merely a pair of compasses one can place as many points on the line as is
necessary or desired—this is then tantamount to having constructed the line
itself. Once again, we will show how the circle can essentially replace the line.
Of course, this is more theoretical than practical!
Circles have a very important role in both art and architecture that merits
discussion and exposure. This is what we hope to provide in chapter 9, “Circles
in Art and Architecture,” which is devoted to this aspect of geometric
application. The manipulation of circles and spheres presents another aspect of
geometry that exhibits unusual properties of these important geometric shapes.
This is often seen in what is referred to as “packing spheres,” which we present
as a separate chapter in the book (chapter 4, “Circle-Packing Problems”).
Interesting questions also arise when we consider the surface of a sphere along
with the circles that lie on them. We often wonder why a flight from New York
to Vienna typically flies very near Greenland. When we look at a map, that path
looks like it would be a circumvention of sorts. Truth be told, the shortest
distance between two points on the sphere is the connection of points with a
great circle. We live on a sphere—the earth—and there is a lot of very
interesting geometry on the sphere. Recognizing this will give you a far greater
insight about the world in which we live.
Having now considered most aspects of the circle's manifestation on the plane,
we will devote a chapter of the book to present the role of circles on a sphere
(chapter 11). The circle plays a unique role on the sphere, namely, it is the only
form of “line” that can be drawn. We speak here of the “great circles,” which are
those drawn on a sphere, where the center of the circle is at the center of the
sphere. This chapter may open up a new vista for you to consider, where, for
example, the triangles drawn on the sphere no longer have an angle sum of 180°
—rather they have an angle sum greater than 180° and less than 540°.
In short, this book will open your mind to view circles with the importance
that they should have in our geometric thinking, both on the plane and on a
sphere. It will make geometry come alive in a fashion not typically anticipated.
That is the goal of this book!
As we mentioned earlier, the circle has for centuries represented a variety of
concepts beyond geometry, and these concepts can vary by culture. By tracing
the ancient knowledge of circles along with some of the early relationships and
representations that exist about the circle, we conclude our journey through the
world of the circle. We hope that the broad-spectrum concepts and applications
of circles and spheres we provide will entice you to delve even more deeply into
the many areas that we investigate.
Alfred S. Posamentier
Robert Geretschläger
As we begin our exploration of the circle, the one aspect that most people recall
is that the Greek letter π (pi) is somehow related to this important geometric
shape. Most will recall that π represents the ratio of the circumference of a circle
to its diameter. The two formulas that come instantly to mind are that the
circumference of the circle is equal to 2πr, and that the area of the circle is equal
to πr2. In each case, r is the length of the radius of the circle. However, before
we inspect the circle for its multitude of applications and appearances, we will
review some of the basic essentials that you may have forgotten from your high-
school geometry course.
First, we will review some basic terminology related to circles. We all know
that the radius is the line segment joining the center of the circle to any point on
the circle, while the diameter is the line segment joining two points on the circle
and that also contains the center of the circle. A line segment joining two points
on the circle is called a chord. A line is said to be tangent to the circle if it
touches a circle at exactly one point; and a line that intersects the circle in two
points is called a secant. Two more definitions to review: the area within a circle
bounded by two radii and the arc between them is called a sector, and the area
within a circle bounded by a chord and the arc of the circle is called a segment of
the circle.
Some relationships useful to our study of the circle are the following:
Any three non-collinear points determine a unique circle.
The non-collinear points A, B, and C determine the unique circle O.