TABLE 1.2
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:ø, 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, ø. Furthermore, as we shall learn later on, ø 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
where r is the radius of the circumscribing circle of the pentagon. In addition, the perimeter of a regular pentagon is
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 ø: Three Problems
In the preceding sections we have seen that the golden ratio, 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 ø shows up not only in the geometry of five-pointed stars but in many other places as well.
FIG. 1.3
Definition sketch for the golden earring problem.
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 = ø.
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 = ø.
This problem was originally posed by the noted English mathematician E. H. Neville in 1915; it is discussed by Huntley (1970).
FIG. 1.4
The problem of the pentagon circles.
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 = ø and its volume is
2
More Stars, Honeycombs, and Snowflakes
Skinny Pentagrams and Fat Pentagrams
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 R is the radius of the external circumscribing circle and r is the radius of the internal circumscribing circle. It is clear that the angle β = 72° regardless of the value of r/R (because 5 X 72° = 360°). For small values of r/R, we have skinny pentagrams and for large values we have fat ones. In between, if r/R = 1/ø2, where we have our familiar regular pentagram. These three cases are illustrated in figure 2.2.
It is logical to identify the regular pentagram as that pentagram for which α = 36° or, equivalently, r/R = 1/ø2. An array of adjectives is available to define the cases for which α < 36° (long and skinny) and α > 36° (short and fat). The terminology acute-regular-obtuse seems to be the simplest and most descriptive way to define the three categories of pentagrams.
FIG. 2.1
Definition sketches for (a) pentagram and (b) pentagram kite.
FIG. 2.2
Pentagram shapes for (a) r/R = 0.20, (b) r/R = 1/ø2 = 0.382, and (c) r/R = 0.70.
Some Geometrical Features of Pentagrams
Utilizing figure 2.1 and the law of sines, we obtain the equations
We want to express the various quantities in our pentagram problem in terms of the variable r/R. Letting p = r/R and employing the above equations, we determine that the area of the pentagram is
and the perimeter is
The results of computations based on these equations are listed in table 2.1. We make the following observations:
Clearly, if p = 1/ø2 = 0.38197, the pentagram becomes the familiar five-pointed star.
When p = 0.5, the pentagram takes on the shape of the star used in the Berghaus projection of the world.
When p = 1/ø = 0.61803, both α and β are 72° and the pentagram is composed of ten identical triangles.
When p = ø/2 = 0.80902, the pentagram becomes a regular pentagon. As we observe in the table, the perimeter is a minimum for this value of p.
When p = 1, the pentagram becomes a regular decagon (i.e., a ten-pointed polygon).
TABLE 2.1
Geometrical features of pentagrams
Berghaus Star Projection of the World
When r/R = 0.5, the radius of the internal polygon is one-half the radius of the pentagram. As mentioned earlier, this is the radius ratio of the Berghaus star projection of the world.
Later on, in chapter 22 on cartography, this projection is examined again. For now, we simply make the observation that in the Berghaus projection, the northern hemisphere of the world is displayed in the internal circumscribing circle of the star. The southern hemisphere appears in the five points of the star defined at 72° intervals along the equator. A recommended reference on this topic is Snyder (1993).
Six-Pointed Stars: Hexagrams and Hexagons
We conclude our study of the geometry of stars with a brief analysis of six-pointed stars or hexagrams. A definition sketch appears in figure 2.3. The radius of the hexagram is R. The six-sided polygon inside the hexagram is called a hexagon; its radius is r.
In comparison with the geometry of a five-pointed star—the pentagram—that of the six-pointed star is simple. This is because the hexagram and its associated hexagon are composed of noth
ing but equilateral triangles. In figure 2.3, ABC is a representative triangle. Since the length of each of its three sides is r and each of the internal angles is 60°, it is not difficult to establish that the area of each triangle is and the perimeter of each is 3r.
FIG. 2.3
Definition sketch for the hexagram and hexagon.
With this information, we easily determine that the area and perimeter of a hexagon are
Likewise, the area and perimeter of a hexagram are
Finally, we note that and hence the radius ratio is The area and perimeter of the hexagram can be expressed in terms of radius R instead of r. The answers are
PROBLEM In the previous chapter, we determined the area percentages by color of the flag of the United States. It is observed that the Star of David, which appears on the flag of Israel, is a regular hexagram. What are the area percentages by color of this flag?
Honeycombs and Hexagons
Leaving the subjects of flags and stars, we proceed to an entirely different topic. Suppose we want to store a certain commodity, for example, corn, in a structure whose shape will provide the maximum volume of storage for a specified length of storage boundary. What shape should be used? The answer: a circle. That's why circular silos and elevators are used for the storage of corn, wheat, and other grains. It is easy to establish that the ratio of the area of a circle to its perimeter is A/P =
However, the use of circles may be troublesome. If we intend to build quite a few circular storage structures and place them side by side, we will have empty zones between the circles, no matter how we arrange them. In other words, circles are not “space-filling” shapes.
Well, what shapes are space-filling? The answer: equilateral triangles, squares, and hexagons. It is easy to establish that pentagons, octagons, and other n-gons do not qualify; they don't fit.
It is interesting to note that ordinary honeybees evidently know all about the geometry of optimized storage space. For a very long time indeed, these lively little insects have been constructing honeycombs for the storage of their very well-known commodity: honey. And what shape of structures do they utilize? If you said hexagons, you are absolutely correct.
Here is a historical note on this topic. Many years ago, the German astronomer-mathematician, Johannes Kepler (1571–1630), wrote an interesting little book entitled The Six-Cornered Snowflake as a New Year's gift for the emperor who provided his financial support. In his essay, Kepler (1611) raised the question of why snowflakes always have six sides. To quote him directly, “There must be a cause why snow has the shape of a six-cornered starlet. It cannot be chance. Why always six?”
As part of his wonderment concerning snowflakes, Kepler was curious about the six-cornered plan on which honeycombs are built. Again, to quote him:
…what purpose had God in putting these canons of architecture into the bees? Three possibilities can be imagined. The hexagon is the roomiest of the three plane-filling figures (triangle, square, hexagon); the hexagon best suits the tender bodies of the bees; also labour is saved in making walls which are shared by two; labour would be wasted in making circular cells with gaps.
Kepler concluded his short volume with a plea to chemists to study the problem and provide some answers. In addition, he presented a challenge “…to those who followed him to discover the mathematics of the emergence of visible forms in crystals, plants and animals.”
It is noteworthy that currently mathematicians and scientists are devoting much attention to research on the topics advocated by Kepler. In this regard, an intriguing book by Neill (1993), entitled By Nature's Design, presents a collection of remarkable photographs illustrating patterns, form, and shape of things ranging from honeycombs and snowflakes to seashells and spider webs.
PROBLEM It was established in a preceding paragraph that the ratio of the area of a circle to its perimeter is A/P = 0.500r. Determine the value of this ratio for a hexagon, a square, and an equilateral triangle. On the basis of these results, list some reasons why you think honeybees wisely use hexagons as the basic shape for their honeycombs.
Snowflakes and Hexagons
The same questions asked by Kepler regarding the hexagonal shape of snowflakes have been raised by countless others over the years. Scientists working in the field of crystallography now understand the molecular structure of snowflakes and ice crystals and the reasons for their two-dimensional hexagonal shape. A suggested reference on this topic is Knight and Knight (1973). The interesting book by Bentley and Humphreys (1962) displays more than two thousand photographs of beautiful hexagonal snow crystals.
Another topic relating to hexagons and other polygons is the remarkable work of the Dutch artist, M. C. Escher (1898-1972). His spectacular paintings and drawings invariably feature intriguing and beautiful displays of symmetry.
In 1985, an international conference was held in Rome, attended by many scientists, mathematicians, artists, and historians, to commemorate Escher's work. The proceedings of the conference were published the following year with H.S.M. Coxeter serving as senior editor. Many of Escher's spectacular drawings appear in these proceedings, Coxeter et al. (1986). It also includes numerous articles about Escher and his work prepared by specialists in various fields of science, mathematics, and the humanities.
Although Escher had essentially no training in mathematics, he possessed an incredible understanding of geometrical principles and space perception. Some of his works, for example, are remarkably beautiful displays of triangular and hexagonal symmetry.
Recently, Coxeter, one of the leading geometers of the twentieth century, carried out a mathematical analysis of one of Escher's woodcuts. His paper, Coxeter (1996), is interesting reading; only elementary geometry and trigonometry are needed to understand it.
The Koch Snowflake and a Brief Introduction to Fractals
We conclude our chapter with a look at the so-called Koch snowflake. It has nothing to do with a real snowflake. It is called one simply because successive mathematical embellishments of an equilateral triangle create a shape that ends up looking like a snowflake. By the way, Helge von Koch was a Swedish mathematician. In 1904, he devised the problem we now consider.
The problem begins with the equilateral triangle shown in figure 2.4(a); the side length of the triangle is r. We shall call this stage 1. In figure 2.4(b), stage 2, we attach an equilateral triangle of side length r/3, at the midpoint of each of the sides of stage 1. In our next step, shown in figure 2.4(c), stage 3, triangles are again attached at the midpoints, this time of side length r/9. The process is continued in this fashion for as long as we please. Two comments. First, noting that the shape gets fancier at each successive stage, we can understand why they call it a snowflake. Second, if we take a magnified look at the edge of our prickly snowflake, after a number of stages, we observe that there is no basic change in the geometric pattern of the boundary. Consequently, we say that the Koch snowflake is a self-similar curve.
FIG. 2.4
The first three stages of a Koch snowflake.
PROBLEM With reference to figure 2.4, the area and perimeter of stage 1 are and P = 3r. Likewise, the area and perimeter of stage 2 are and P = 4r. Continuing through an infinite number of stages, show that the area of the snowflake is finite — and equal to 8/5 of the area of the original triangle — but that its perimeter is infinite.
These remarks serve as a brief introduction to an interesting subject: fractals. Some aspects of this subject are certainly not new. For example, the snowflake devised by von Koch in 1904 attracted little attention until the classic publication of Mandelbrot (1982). Since then, scores of books have been published on the subject of fractals and the related topic of chaos theory. A recommended reference for the study of these topics is the interesting book by (Çambel (1993).
A final word: A fascinating interface between art and mathematics is provided by these fractal structures. By combining the techniques of so-called dynamical systems with those of computer graphics, it is p
ossible to create color displays of fractal geometry and chaos phenomena that are incredibly beautiful. The fascinating book by Peitgen and Richter (1986) displays many of these lovely pictures.
3
Slicing Things Like Pizzas and Watermelons
We Start with Pizza
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?
Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Paperbacks) Page 2