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

* By ROBERT B. BANKS * * Princeton University Press *

* *

Copyright © 1999 * Princeton University Press *

All right reserved. * ISBN: 0691102848 *

#### Chapter One

*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?

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 x 1.9 = 1.9 ft²

The area of the union is 7/13 x 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

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 [alpha] = 36º, [beta] = 72°, and [gamma] = 126º. Without much difficulty we obtain the expression

(1.1) *A* = [5 sin([alpha]/2)sin([beta]/2)/sin([alpha] + [beta]/2] R², where *A* is the area of the pentagram. Substituting the values of [alpha] and [beta] into this expression gives

(1.2) *A* = 1.12257 R².

For comparison, remember that the area of the circumscribing circle is [pi] *R²* where, of course, [pi] = 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 [phi] = (1/2)(1 + [square root of 5]) = 1.61803. 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 [phi] relationships prevail in figure 1.2 based on unit length BC (i.e., one side of the pentagon):

(1.3) *AE* = [phi]³; *AD* = [phi]²; *AC* = [phi]; *R/r* = [phi]²; *R/b* = 2[phi]; *b/r* = [phi]/2; *L/r* = [square root of 3 - [phi]]. Utilizing these relationships, it can be established that the area of the regular pentagram can be expressed in terms of [phi]:

(1.4) *A* = [5/[phi]² sin([beta]/2)]*R²*.

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

(1.5) *P* = [10 sin([beta]/2)/sin([alpha] + [beta]/2]*R*.

The bracketed quantity has the numerical value 7.2654. The length of the circumscribing circle is, of course, 2[pi]*R*.

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 [square root of 5]?

*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, [A.sub.r], is

[A.sub.r] = 4/13 x (1.9 - 0.76) + 3/13 x 1.9 = 0.78923 ft².

*Color: white*

The white area, [A.sub.w], is

[A.sub.w] = 3/13 x (1.9 - 0.76) + 3/13 x 1.9 + 0.05325 = 0.75479 ft².

*Color: blue*

The blue area, [A.sub.b], is

[A.sub.b] = (0.76 x 7/13) - 0.05325 = 0.35598 ft².

The total area of the flag is [A.sub.flag] = 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" answer. 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 1(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, 1:[phi], where [phi] = (1/2)(1 + [square root of 5]) = 1.618. We see that collectively the geometric proportions of the world's flags are not greatly different from the "divine proportion."

A word about flapping or fluttering of flags. This is an interesting and not very easy problem in fluid mechanics. If you are challenged by the mathematics of the phenomenon you can study the section on "surface waves" in Lamb (1945). If you are interested in some experimental work, get yourself an electric fan, a few sheets of paper, a pair of scissors, and go to work.

*PROBLEM 1.* Now that you know all about the geometrical features of the flag of the United States, your assignment is to select and analyze the flag of some other country. As in the case of the U.S. flag, you are required to determine the percentage distribution of the colors of the flag you choose. A list of flags appears in table 1.2. Since some flags are geometrically simple and others are rather complicated, there is considerable spread in the number of points you will be awarded.

*PROBLEM 2.* If the width to length ratio of the U.S. flag were altered from 1:1.9 to 1:[phi], and it was otherwise unchanged, what would be the area percentages of red, white, and blue?

*Answer.* Red: 39.4%, white: 38.6%, blue: 22.0%.

*Revisiting the Stars*

The pentagram-the five-pointed star-has a very long history. Its geometrical properties were known to the ancient Babylonians and it was regarded as a symbol and badge of the Grecian school of geometers in the days of Pythagoras.

We have seen the many relationships between the pentagram and the golden number, [phi]. Furthermore, as we shall learn later on, 4, is the ratio of successive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...) when these numbers become very large. So we note the connection between pentagrams and this famous mathematical sequence.

There are many interesting and practical problems involving pentagrams and pentagons; some are easy and some are difficult. For example, the area of the pentagon portion of a pentagram is

(1.6) *A* = [5/4 [square root of 2 + [phi]]*r²* = 2.37762r²,

where *r* is the radius of the circumscribing circle of the pentagon. In addition, the perimeter of a regular pentagon is

(1.7) *P* = 5 [square root of 3 - [phi]] *r* = 5.8779*r*.

For example, the famous Pentagon building near Washington has a side length *L* = 921 ft. Hence, the total perimeter is 4,605 feet. From equation (1.7) the radius *r* = 783 ft and so, from equation (1.6), the total area covered by the Pentagon is *A* = 1,457,680 ft². This is approximately 33.5 acres. Some questions: If they ever add the five points to the Pentagon building (thereby making it the Pentagram building) what will be the radius *R*, the perimeter *P*, and the total area *A*? The answers are that *R* = 2,050 ft, *P* = 14,895 ft, and *A* = 108.3 acres.

*The Ubiquity of [phi]: Three Problems*

In the preceding sections we have seen that the golden ratio, [phi] = (1/2)(1 + [square root of 2]) = 1.61803, makes numerous appearances in the patterns of pentagrams and pentagons. In later chapters, we take closer looks at this very interesting number. For the present, here are three problems that illustrate that [phi] shows up not only in the geometry of five-pointed stars but in many other places as well.

*GOLDEN EARRINGS.* A circle of radius *r* is removed from a larger circle of radius *R*, as shown in figure 1.3. The center of gravity of the remaining area is at the edge of the removed circle, *P*. Confirm the result obtained by Glaister (1996) that this "golden earring" indeed balances at point *P*, if the ratio of the radii of the two circles is R/r= [phi].

*PENTAGON CIRCLES.* An interesting design or pattern for a flag might be the array of circles shown in figure 1.4. In this figure, five circles, each of radius *r*, are placed with their centers at the corners of a regular pentagon and their circumferences passing through the centroid of the pentagon, *O*.

Show that the radius *R* of the largest circle that can be covered by the five smaller circles is given by *R/r* = [phi].

This problem was originally posed by the noted English mathematician E. H. Neville in 1915; it is discussed by Huntley (1970).

*FOLDED PENTAGRAM.* in figure 1.2, triangle ABC is folded about BC and the other four triangles are folded in the same fashion. The five sloping triangular sides meet at a point to create a "folded pentagram" pyramid with a pentagon base of radius *r*.

Show that the height of the pyramid is *H/r* = [phi] and its volume is

(1.8) *V* = [5/12 [phi] [square root of 2 + [phi]]*r³*.

Excerpted from *Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics* by *ROBERT B. BANKS*

Copyright © 1999 by Princeton University Press

Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.