Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics

333 Pages·2012·5.12 MB·English

by  Robert B. Banks

ROBERT B. BANKS
Slicing Pizzas, Racing
Turtles,
and Further Adventures in Applied
Mathematics
Princeton University Press
Princeton and Oxford
2
Copyright © 1999 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
press.princeton.edu
Cover design by Kathleen Lynch/Black Kat Design
Cover illustration by Lorenzo Petrantoni/Marlena Agency
All Rights Reserved
First printing, 1999
Fifth printing, and first paperback printing, 2002
Paperback reissue, for the Princeton Puzzlers series, 2012
Library of Congress Control Number 2012932651
ISBN 978-0-691-15499-2
British Library Cataloging-in-Publication Data is available
This book has been composed in Times New Roman
Printed on acid-free paper. ∞
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
3
To my mother;
Georgia Corley Banks,
and my sister and brothers,
Joan, Dick, and Barney
4
Contents
Preface
Acknowledgments
Chapter 1 Broad Stripes and Bright Stars
Chapter 2 More Stars, Honeycombs, and Snowflakes
Chapter 3 Slicing Things Like Pizzas and Watermelons
Chapter 4 Raindrops Keep Falling on My Head and Other
Goodies
Chapter 5 Raindrops and Other Goodies Revisited
Chapter 6 Which Major Rivers Flow Uphill?
Chapter 7 A Brief Look at π, e, and Some Other Famous
Numbers
Chapter 8 Another Look at Some Famous Numbers
Chapter 9 Great Number Sequences: Prime, Fibonacci, and
Hailstone
Chapter 10 A Fast Way to Escape
Chapter 11 How to Get Anywhere in About Forty-Two Minutes
Chapter 12 How Fast Should You Run in the Rain?
5
Chapter 13 Great Turtle Races: Pursuit Curves
Chapter 14 More Great Turtle Races: Logarithmic Spirals
Chapter 15 How Many People Have Ever Lived?
Chapter 16 The Great Explosion of 2023
Chapter 17 How to Make Fairly Nice Valentines
Chapter 18 Somewhere Over the Rainbow
Chapter 19 Making Mathematical Mountains
Chapter 20 How to Make Mountains out of Molehills
Chapter 21 Moving Continents from Here to There
Chapter 22 Cartography: How to Flatten Spheres
Chapter 23 Growth and Spreading and Mathematical Analogies
Chapter 24 How Long Is the Seam on a Baseball?
Chapter 25 Baseball Seams, Pipe Connections, and World
Travels
Chapter 26 Lengths, Areas, and Volumes of All Kinds of Shapes
References
Index
6
Preface
In large measure, this book is a sequel to an earlier volume
entitled Towing Icebergs, Falling Dominoes, and Other
Adventures in Applied Mathematics. As in the previous work,
this book is a collection of topics characterized by two main
features: the topics are fairly easy to analyze using relatively
simple mathematics and for the most part, they deal with
phenomena, events, and things that we either run across in our
everyday lives or can comprehend or visualize without much
trouble.
Here are examples of a few of the topics we consider:
You need to go from here to there in a pouring rainstorm.
To get least wet, should you walk slowly through the
rain or run as fast as you possible can?
What blast-off velocity do you and your spacecraft need to
entirely escape the earth's gravitational pull?
The colors of America's flag are, of course, red, white, and
blue. Which of the three colors occupies the largest area
of the flag and which color the smallest?
What is the length of the seam on a baseball or the groove
on a tennis ball?
What is the surface area of the Washington Monument and
what is its volume? How many golf balls could you put
into an entirely empty Washington Monument?
7
Some of these questions sound trivial, perhaps even silly. Even
so, they do depict settings or situations that are easy to
visualize and understand. With the help of mathematics, it is
not difficult to obtain the answers. As we shall see, the level of
mathematics ranges from algebra and geometry to calculus.
Several problems involve spherical trigonometry.
Throughout the book, topics involving various fields of
knowledge are investigated. For example, quite a few problems
featuring geography and demography are examined. In other
chapters, a number of topics concerned with hydrology,
geomorphology, and cartography are analyzed. In addition,
where it is appropriate and feasible, features are described that
relate to the historical aspects of a particular topic.
For over four decades, I was engaged in teaching and
research at several universities and institutes in the United
States and abroad (England, Mexico, Thailand, the
Netherlands). I collected most of the topics presented in the
book during that period. My primary interests, as a professor of
engineering, were in the fields of fluid mechanics and solid
mechanics (statics, dynamics, mechanics of materials).
Although this book deals with mathematics, it is certainly
not intended to be a textbook. It might, however, be a
worthwhile supplement to a text at the high school and
university undergraduate levels.
As was the case in my earlier book, my strong hope is that
this collection of mathematical stories will be interesting and
helpful to people who long ago completed their formal studies.
I truly believe that many of these “postgraduate students”
sincerely want to strengthen their levels of understanding of
mathematics. I think that this is especially true as all of us enter
a new century that assuredly will place heavy emphasis on
mathematics, science, and technology.
Here are some more examples of topics we examine in the
book.
A prime number is a number that can be divided only by
one and by itself. Some examples are 1, 2, 3, 5, 7, 11, 13, 17,
8
and so on. As you might want to confirm, there are 168
prime numbers less than 1,000. Can you guess the
magnitude of the largest prime number known at the
present time (1999)?
A beautiful Nautilus sea shell has a shape called a
logarithmic spiral. This attractive curve is
mathematically related to the well-known Fibonacci
sequence and the ubiquitous golden number,
The most famous “numbers” in all of mathematics are π (the
ratio of the circumference of a circle and its diameter)
and e (the base of natural logarithms). It is interesting
that these two important numbers are related by the
equation eiπ = –1 where The numerical values
of π and e (to five decimal places) are π = 3.14159 and e =
2.71828. Would you believe that at the present time
(1999), π has been calculated to more than 51 billion
decimal places?
The number of people in the world is approximately 6.0
billion as we begin the new century. Is this a large
percentage or a small percentage of the number of
people who have ever lived on earth?
Most of the ice in the world is in Antarctica and Greenland.
If all this ice melted, the oceans would rise by 75 meters
(246 feet). What would this increase in water level do to
Florida, Washington, D.C., the Mississippi River, and
Niagara Falls?
Finally, I hope you will enjoy going through the book. In
attempts to make things a bit easier, I have tried to be
somewhat light-hearted here and there. We all know that
mathematics is a serious subject. However, this does not mean
we cannot be a little frivolous now and then.
9
Acknowledgments
Numerous people gave me considerable assistance during the
period of preparation of this book. I am grateful to the
following persons for their willingness to read various chapters
of my manuscript: Philip J. Davis, J. Donald Fernie, Paulo
Ribenboim, O. J. Sikes, Whitney Smith, and John P. Snyder.
Others aided in a substantial way by providing information
dealing with all kinds of things. For this help, I would like to
express appreciation to Teresita Barsana, Brian Bilbray, Roman
Dannug, Bill Dillon, and Susan Harris.
As before, I express much gratitude to my editor, Trevor
Lipscombe, and to many others at Princeton University Press,
for their greatly appreciated efforts to turn my manuscript into
a book.
It is impossible to list the many contributions my wife,
Gunta, made to this endeavor. I thank her for all she has done.
In my earlier book, her wise advice and cheerful help certainly
improved the likelihood of a successful towing of that literary
iceberg. This time, once again, her wise counsel and cheerful
assistance surely strengthen the chances of a creditable race by
her husband, a literary turtle.
10