ROBERT B. BANKS
Slicing Pizzas, Racing Turtles,
and Further Adventures in Applied Mathematics
Princeton University Press
Princeton and Oxford
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
To my mother;
Georgia Corley Banks,
and my sister and brothers,
Joan, Dick, and Barney
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?
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
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?
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, 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 at
tempts 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.
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.
1
Broad Stripes and Bright Stars
These days we see much more of the flag of the United States than we ever did in the past. Old Glory flies over many more office buildings and business establishments than it did before. It is now seen far more extensively in parks and along streets and indeed in a great many programs and commercials on television.
With this greatly increased presence and awareness of the flag, there is understandable growing interest in learning more about numerous aspects of the U.S. flag, including its history and physical features and the customs and protocol associated with it.
There are a number of books that examine various topics concerning the flag of the United States. Representative are the publications of Smith (1975a) and Furlong and McCandless (1981). On the world scene, many references are available dealing with the flags of all nations. For example, the books by Smith (1975b) and Crampton (1990) cover many subjects relating to flags of the world and to various other topics of vexillology: the art and science of flag study.
The Geometry of the Flag of the United States
With that brief introduction, we come directly to the point. The colors of the U.S. flag are red, white, and blue. Now, are you ready for the big question? What are the area percentages of red, white, and blue? That is, which of the three colors occupies the largest area of the flag and which color the smallest? It's a good question. Do you want to guess before we compute the answer?
FIG. 1.1
Flag of the United States.
The flag is shown in figure 1.1 and its more important proportions and features are listed in table 1.1. Arbitrarily selecting the foot as the unit of linear measurement, here are some preliminary observations:
The total area of the flag is 1.0 × 1.9 = 1.9 ft2
The area of the union is 7/13 × 0.76 = 0.4092 ft2
The length of the seven upper stripes is 1.14 ft
The length of the six lower stripes is 1.9 ft
TABLE 1.1
Proportions and features of the U.S. flag
Item Quantity
Width of flag 1.0
Length of flag 1.9
Number of red stripes 7
Number of white stripes 6
Width of union 7/13
Length of union 0.76
Number of stars 50
Radius of a star 0.0308
FIG. 1.2
Definition sketches for (a) pentagram and (b) pentagram kite.
The width of a stripe is 1/13 = 0.07692 ft
The problem of computing the red area is easy
The problem of computing the white and blue areas is not so easy because of the 50 white stars in an otherwise blue union
So, before we can obtain the final answer we have to look at stars
The Geometry of a Five-Pointed Star
A five-pointed star, commonly called a pentagram, is shown in figure 1.2(a). Its radius, R, is the radius of the circumscribing circle. The five-sided polygon within the star is called a pentagon; the radius of its circumscribing circle is r.
The section ABOC is removed from the pentagram of figure 1.2(a) and displayed as the pentagram kite of figure 1.2(b). Some geometry establishes that α = 36°, β = 72°, and γ = 126°. Without much difficulty we obtain the expression
where A is the area of the pentagram. Substituting the values of α and β into this expression gives
For comparison, remember that the area of the circumscribing circle is πR2 where, of course, π = 3.14159. Equation (1.2) provides us with a simple formula to compute the area of a five-pointed star.
In elementary mathematical analysis we frequently run across the numerical quantity It is a very famous number in mathematics. It is called the golden number or divine proportion. Our pentagram is full of golden numbers. According to Huntley (1970), the following ø relationships prevail in figure 1.2 based on unit length BC (i.e., one side of the pentagon):
Utilizing these relationships, it can be established that the area of the regular pentagram can be expressed in terms of ø:
As we would expect from observing equation (1.2), the quantity in the brackets of equation (1.4) is 1.12257.
The perimeter of a pentagram is not difficult to determine. Using some geometry and trigonometry we obtain
The bracketed quantity has the numerical value 7.2654. The length of the circumscribing circle is, of course, 2πR.
Numerous other relationships could be established. For example, can you demonstrate that the ratio of the area of the five points of the pentagram to the area of the base pentagon is
How Much Red, How Much White, How Much Blue?
We now have the information we need to answer the big question. From table 1.1 we note that the radius of a star is R = 0.0308 ft and so, from equation (1.2), the area of a single star is A = 0.0010649 ft2. The area of 50 stars is 0.05325 ft2.
Color: red
The red area, Ar, is
Color: white
The white area, Aw, is
Color: blue
The blue area, Ab, is
The total area of the flag is Aflag = 1.9 ft2. So it is easy to calculate that the color percentages are 41.54% red, 39.73% white, and 18.73% blue. This means that if you are going to paint a really big U.S. flag, you will need 42 gallons of red paint, 40 gallons of white, and 19 gallons of blue to come out about even.
Here are several other items of information about the flag that you might want to confirm:
Based on a flag width of 1.0 foot, there are 10.26 feet of red stripe and 9.12 feet of white stripe. The total is 19.38 feet.
The union is 13.01% white and 86.99% blue.
If the 50 stars were replaced by a single big star with the same total area, it would have a radius of 0.218 feet.
The perimeter of a single star is 0.2238 feet; the total perimeter of all 50 is 11.19 feet. This is nearly double the perimeter of the entire flag.
This is sufficient. We stop here because we now have the information we set out to determine. That is, the red area is 41.6%, the white 39.7%, and the blue 18.7%.
Main Dimensions of Flags
The official flag of the United States has a ratio of width to length of 10:19. Why the relative length of the flag is precisely 1.9—or indeed why the relative length of the union is exactly 0.76 (could it be 1776?)—is not known. It just is. If you are interested in the historical aspects of vexillology, you might want to contact the Flag Research Center in Winchester, Massachusetts or the Flag Institute in Chester, England.
In any event, an interesting question has been raised by Nicolls (1987): “What are the ideal proportions of a flag?” Again, this is the kind of question for which there is no “scientific” a
nswer. A brief history of the changes of flag dimension proportions is given by Nicolls. During the Middle Ages, for example, flag proportions ranged from an extremely short l(width):0.5(length) ratio to a square 1:1 ratio.
Over the years, the poor visibility and inadequate flapping characteristics of quite short flags resulted in their gradual lengthening. At present, the official flags of the United Kingdom and thirty other nations have proportions of 1:2 and those of twenty five other countries possess ratios of 3:5. It is pointed out by Nicolls that these ratios are numerically close to that given by the golden mean, We see that collectively the geometric proportions of the world's flags are not greatly different from the “divine proportion.”
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