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

ROBERT B. BANKS
Pages: 304
https://www.jstor.org/stable/j.ctt7sbwv

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
4. Acknowledgements
(pp. xiii-2)
5. 1 Broad Stripes and Bright Stars
(pp. 3-12)

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...

6. 2 More Stars, Honey combs, and Snowflakes
(pp. 13-22)

So far, in our study of pentagrams—that is, five-pointed stars—we have considered only the quite familiar pentagram that appears on the flag of the United States and in so many other places. We shall say that the five points that appear on this particular pentagram are normal or “regular” in shape.

In contrast, our next step is to examine the geometry of pentagrams with points that range in shape from “long and skinny” to “short and fat.” Let figure 2.1 serve as our definition sketch.

The main geometrical variable in our problem is the ratio$r/R$, in which...

7. 3 Slicing Things Like Pizzas and Watermelons
(pp. 23-33)

Our problem begins with the supposition that you have a large pizza in front of you and you want to obtain the maximum number of pieces with a certain number of straight line slices. With one slice you get two pieces of pizza, two slices give you four pieces, and three slices get you six, right?

Not necessarily. If your third slice avoids the intersection of the first two slices, you will have a total of seven pieces.

A short pause while you get a pad of paper, a pencil, and a ruler. All set? Draw a straight line and...

8. 4 Raindrops Keep Falling on My Head and Other Goodies
(pp. 34-43)

Which weighs more, all of the air in the world or all the water, and how much more? What would happen if all the ice in the world were to melt? What is the average rainfall in the world? What is the velocity and kinetic energy of a raindrop? How much power is in rainfalls? How many times has the world’s air been breathed by humans and the world’s water drunk by humans?

We now obtain answers to these awesome questions. Nothing difficult in the way of theoretical analysis is introduced. However, there will be quite a few numbers and...

9. 5 Raindrops and Other Goodies Revisited
(pp. 44-48)

Our saga now continues. Readers will recall that the incredible Amazon river in South America delivers 180,000 cubic meters of water per second to the Atlantic ocean. This is about thirty times more than the flow over Niagara Falls. The drainage area of the Amazon is about 7 million square kilometers—about ten times the size of Texas. Much of the Amazon basin receives heavy rainfall to sustain the well-known rain forests.

On a smaller scale, the same things can be said about the world’s second river: the Congo (or Zaire) in Africa. It collects water from a drainage area...

10. 6 Which Major Rivers Flow Uphill?
(pp. 49-56)

As all of us know, jellied biscuits fall to the rug (frequently upside down), baseballs fall to the center fielder (frequently with two outs and bases loaded), and car keys fall to the pavement (frequently through sewer gratings). And rivers fall—or as we usually say—flow downhill. So how can a river flow uphill? Well, be patient. That matter will be explained in a page or two.

We start our consideration of this topic with a short trip through some trigonometry and analytic geometry. As illustrated in figure 6.1, suppose that you have a circular cone made of cheese...

11. 7 A Brief Look at $\pi ,e$, and Some Other Famous Numbers
(pp. 57-68)

Many years ago, the writer received a magnificent $15 for submitting this poem, which, not long afterward, appeared in the mathematical nursery rhyme corner of a trade journal. Though the poem is pathetic enough, the real misfortune is that space did not allow acknowledgment to the famous Swiss mathematician, Leonhard Euler. In any event, the episode serves as a prologue to our next endeavor: an examination of several of the important numerical constants of mathematics. Without question, the most famous of these “numbers” is the one we call π; it has the approximate numerical value π = 3.14159. Although its... 12. 8 Another Look at Some Famous Numbers (pp. 69-77) In the preceding chapter we examined several of the most important numerical constants appearing in mathematics. The best known of these “famous numbers” are, of course,$\pi $(the ratio of the circumference and diameter of a circle) and$e$(the base of natural logarithms). We also looked briefly at three other important numbers:$\phi $(the golden ratio),$\gamma $(Euler’s constant), and δ (the Feigenbaum number). In this chapter, the subject of famous numbers is continued but aimed in a somewhat different direction. We begin with what are called real numbers, imaginary numbers, and complex numbers. First, some rules are given... 13. 9 Great Number Sequences: Prime, Fibonacci, and Hailstone (pp. 78-96) The answer to this question is that number sequences are simply lists of numbers appearing in a particular order. For example, the sequence 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is the list of the natural numbers or integers. The sequence 1, 3, 5, 7, 9, 11 is the list of the odd integers and the sequence 1, 4, 9, 16, 25, 36 is the list of the squares of the integers. Another example: The sequence 1, 2, 4, 8, 16, 32, 64 is the list of numbers generated by doubling successive numbers. Sometimes, as in this... 14. 10 A Fast Way to Escape (pp. 97-104) Take a tennis ball and toss it straight upward. It goes to a certain height and falls back to your hands. Keep tossing or throwing the ball, each time with greater velocity. Observe that each time the ball goes higher and the total flight time is longer. An amazing observation! Anything else new? Most likely; we shall get to that shortly. However, before going further we do magic: we remove all the air in the world. The resulting perfect vacuum greatly simplifies our mathematical analysis because now we can neglect the effect of air resistance. In this case, as we... 15. 11 How to Get Anywhere in About Forty-Two Minutes (pp. 105-113) Some rainy afternoon, when you have little to do, you might want to spend some time looking up the following rather interesting information. If you dug a hole straight downward from where you live, through the center of the earth, where would you come out? If you live quite close to the intersection of Montana, Alberta, and Saskatchewan you are quite fortunate. You will come out on dry land, even though it is bleak and dreary Kerguelen Island in the south Indian Ocean. If you live anywhere else in the continental United States, Alaska, or Canada, you’d better be ready... 16. 12 How Fast Should You Run in the Rain? (pp. 114-122) There is a steady downpour of rain, you have no raincoat and no umbrella, and you must get from here to there without delay. And instinctively, you want to get the least wet as you make the excursion of specified distance. If you walk slowly in the rain, only the top of your head and shoulders get wet; your front stays relatively dry as do your shoes and socks. However, with a slow walk you are out in the rain that much longer. If you jog at moderate speed, your front gets wet and your shoes get soggy from splashing... 17. 13 Great Turtle Races: Pursuit Curves (pp. 123-130) We have some small turtles that are very intelligent and well trained; they are what we call smart turtles. They can be ordered to stay put, to walk or trot along a straight line or some curved path, and to follow other specific instructions. They refuse to swim and they cannot fly; hence their movements are restricted to zero, one, or two dimensions, that is, to a point, a straight line, or a plane, respectively. With that as a prologue, we examine what is termed thecurve of pursuit problem. This is quite an ancient problem in mathematics. Evidently it... 18. 14 More Great Turtle Races: Logarithmic Spirals (pp. 131-137) More turtle racing: This time we shall use several or many turtles and instruct all of them to move at the same speed. So it is not really racing; as we shall see, it is more a matter of symmetrical pursuit. We start with a configuration involving two turtles. Turtle$A$is located at the west end of a straight line of length$a$, facing east. Turtle$B$is at the east end facing west. When the starting gun is fired, each turtle heads directly toward the other. Obviously, the two turtles will collide at the midpoint of the line... 19. 15 How Many People Have Ever Lived? (pp. 138-145) In 1990 the population of the world was approximately 5.32 billion people. This is an increase of 844 million over the 1980 population, which was an increase of 755 million over the 1970 population, which was an increase of 671 million over the 1960 population, . . . , and so on. Interesting, but perhaps we are going the wrong way in time. Who cares about the population of the past—the demography of yesterday? We want to know about the population of the future—the shape of things to come. Two comments: First, the best way to make forecasts,... 20. 16 The Great Explosion of 2023 (pp. 146-152) Our previous chapter concluded on a note depicting a very alarming situation: (1) a 1990 world population of over five billion people and a population doubling time of around 17 years and (2) a projection to a year-2000 population of over eight billion and a doubling time of around 12 years. Clearly, it is urgent that something be done about this serious crisis of rampant growth of world population. However, we need to look at the data more closely. Before we try to resolve this incredible crisis, perhaps it is necessary to examine how serious it really is. So we... 21. 17 How to Make Fairly Nice Valentines (pp. 153-162) Have you noticed how dramatically the cost of greeting cards has risen during recent decades? Do you realize that each year we pay trillions of dollars for birthday cards and holiday cards? Many people find this difficult to believe. Well, we can begin reducing our budgets for greeting cards by creating our own, at least some of them. The problem is that unless one has considerable artistic talent, it is impossible to trim expenses by making one’s own fantastically beautiful Christmas cards, Labor Day cards, and Groundhog Day cards. Likewise, homemade Easter cards and Thanksgiving cards would probably look pathetic.... 22. 18 Somewhere Over the Rainbow (pp. 163-176) Nature is very generous. Frequently, and always gratuitously, it provides all of us with spectacular exhibitions of magnificent beauty. And surely, at or near the top of everyone’s list of incredibly beautiful displays of nature would be the rainbow. For thousands of years, poets and artists, scientists and mathematicians have attempted to describe the rainbow with words and paintings, symbols and equations. According to historical records, the noted Greek philosopher Aristotle (384-322 B.C.) made extensive studies of rainbows, trying to understand what they are and how they are created. But for many hundreds of years, the magnificent arc of color... 23. 19 Making Mathematical Mountains (pp. 177-183) In this chapter and the one to follow, we are going to fabricate and analyze various kinds of mathematical mountains. To do this, we shall need some algebra and trigonometry and some analytic geometry and calculus. In addition, for problems with large numbers and diverse shapes of mountains, it will be necessary to utilize various topics of statistics and probability theory. These are the basic mathematical tools used in the science of geomorphology: the study of the characteristics, origins, and changes of land forms. We begin our studies of the subject with a very simple mountain: a circular cone. What... 24. 20 How to Make Mountains out of Molehills (pp. 184-195) In the preceding chapter, in an introduction to the subject of geomorphology, the so-called hypsometric curve was defined and several examples were computed. So we ask the question: Why do geomorphologists study things like hypsometric curves and related topics? The following paragraphs may provide some answers. Geomorphology deals with the characteristics, origins, and changes of land forms. The incredibly powerful forces of nature—aided and abetted by mankind these days—continually alter the topography and geography of our world. Rainfalls and snowfalls, floods, droughts, winds, tornadoes, hurricanes, typhoons, landslides, avalanches, earthquakes, volcano eruptions, forest fires, ocean waves, and tides—not... 25. 21 Moving Continents from Here to There (pp. 196-203) It turns out the bear was white, because it was a polar bear, because the hunter lived right at the North Pole, because only at the North Pole could one carry out the geometrical journey described in the question. The crux of the matter, as shown in figure 21.1, is that we are dealing with asphericaltriangle, not aplanetriangle. You might want to examine this geometry on your globe of the world or your basketball. When dealing with problems involving distances on the earth’s surface, our computations are simplified if we usenautical miles, instead ofstatute... 26. 22 Cartography: How to Flatten Spheres (pp. 204-218) It is said that Columbus must surely have been an economist, a stockbroker, or something like that to have successfully raised the funds for the journey of his three small ships across the Atlantic back in 1492. But even though Columbus did receive fairly strong financial support from the Spanish crown, he definitely was not an economist. In fact, he was a very competent seaman and, more importantly, he possessed considerable knowledge of and experience in the principles of navigation and geography. The fact that Columbus mistook the vast land mass we now call the Americas for south or east... 27. 23 Growth and Spreading and Mathematical Analogies (pp. 219-231) How fast does a plant or a person grow? What is the rate of increase of the population of a state or a nation? How quickly does a rumor or a disease spread through a certain community? How rapidly is a new technology adopted in a particular geographical setting? How fast does an innovation replace an established methodology? To help us obtain answers to these and similar kinds of questions, we need to construct an appropriate mathematical framework. Such a framework is provided by the following simple differential equation:$\frac{{{\rm{dN}}}}{{dt}} = aN\left( {1 - \frac{N}{{{N_*}}}} \right)$, (23.1) in which$N$is the magnitude of the growing... 28. 24 How Long Is the Seam on a Baseball? (pp. 232-246) Or, if you prefer, how long is the groove on a tennis ball? Now we all know that (a) this is not one of the great unsolved problems of mathematics and (b) you will enjoy the World Series or Wimbledon just as much never knowing. However, if you welcome an opportunity to deflate the egos of certain friends or relatives who, by their own admission, know practically everything there is to know about baseball or tennis, your moment has arrived. We have long been aware of the fact that the seam on a baseball, the groove on a tennis ball,... 29. 25 Baseball Seams, Pipe Connections, and World Travels (pp. 247-255) We have established that the length of the seam on a baseball is 10.99 times the radius of the ball and that the length of the groove on a tennis ball is 9.39 times its radius. Very interesting. What else can we get out of our lengthy analysis? This is a good question. We are now going to look at two problems that have nothing to do with a baseball but everything to do with the mathematics we developed in analyzing a baseball. The first problem is quite a practical one in civil engineering; the second problem deals with some... 30. 26 Lengths, Areas, and Volumes of All Kinds of Shapes (pp. 256-278) An intriguing book you might want to examine sometime isEngineering in History, by Kirby et al. (1990). One thing becomes quite apparent as you read this informative text: for thousands of years, humans have been making calculations about things they build. The Great Pyramid of Cheops in Egypt is an excellent example. Constructed over 4,500 years ago, this structure had a height$H$of 480 feet and an average side length$L$of 750 feet. So its volume$V = K{L^2}/3\$was approximately 90 million cubic feet. This required 2.1 million blocks of limestone, each block measuring 3.5 feet...

31. References
(pp. 279-284)
32. Index
(pp. 285-286)
33. Back Matter
(pp. 287-287)