# Beautiful Geometry

ELI MAOR
EUGEN JOST
https://doi.org/10.2307/j.ctt4cgb6n
Pages: 168
https://www.jstor.org/stable/j.ctt4cgb6n

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. ### Prefaces

• ART THROUGH MATHEMATICAL EYES
(pp. ix-x)
ELI MAOR
• PLAYING WITH PATTERNS, NUMBERS, AND FORMS
(pp. xi-xiii)
EUGEN JOST
4. 1 Thales of Miletus
(pp. 1-2)

Thales (ca. 624–546 BCE) was the first of the long line of mathematicians of ancient Greece that would continue for nearly a thousand years. As with most of the early Greek sages, we know very little about his life; what we do know was written several centuries after he died, making it difficult to distinguish fact from fiction. He was born in the town of Miletus, on the west coast of Asia Minor (modern Turkey). At a young age he toured the countries of the Eastern Mediterranean, spending several years in Egypt and absorbing all that their priests could...

5. 2 Triangles of Equal Area
(pp. 3-5)

Around 300 BCE, Euclid of Alexandria wrote hisElements, a compilation of the state of mathematics as it was known at his time. Written in 13 parts (“books”) and arranged in strict logical order of definitions, postulates (today we call them axioms), and propositions (theorems), it established mathematics as adeductivediscipline, in which every theorem must be proved based on previously established theorems, until we fall back on a small set of postulates whose validity we assume to be true from the outset. Euclid’s 23 opening definitions, 10 axioms, and 465 theorems cover all of classical (“Euclidean”) geometry—the...

(pp. 6-8)

Here is a little-known jewel of a theorem that never fails to amaze me: take any quadrilateral (four-sided polygon), connect the midpoints of adjacent sides, and—surprise—you’ll get a parallelogram! The surprise lies in the wordany. No matter how skewed your quadrilateral is, the outcome will always be a parallelogram. The theorem holds true even for the dart-shaped quadrilateral shown in the top-left corner of plate 3. And that’s not all: the area of the parallelogram will always be one half the area of the quadrilateral from which it was generated.

The proof is rather short and is...

7. 4 Perfect Numbers and Triangular Numbers
(pp. 9-12)

The Pythagoreans—the school founded by Pythagoras in the fifth century BCE — had a special relationship with numbers (the term here meaning positive integers). In their mind, numbers were not just a measure of quantity but symbols possessing mythical significance. The number 1 was not considered a number at all, but rather the generator of all numbers, since every number can be obtained from it by repeated addition. Two symbolized the female character, 3 the male character, and 5 their union. Five was also the number of Platonic solids—convex polyhedra whose faces are all identical regular polygons (although the...

8. 5 The Pythagorean Theorem I
(pp. 13-15)

By any standard, the Pythagorean theorem is the most well-known theorem in all of mathematics. It shows up, directly or in disguise, in almost every branch of it, pure or applied. It is also a record breaker in terms of the number of proofs it has generated since Pythagoras allegedly proved it around 500 BCE. And it is the one theorem that almost everyone can remember from his or her high school geometry class.

Most of us remember the Pythagorean theorem by its famous equation,${a^2} + {b^2} + {c^2}$. The Greeks, however, thought of it in purely geometric terms, as a relationship between...

9. 6 The Pythagorean Theorem II
(pp. 16-19)

The Pythagorean theorem is listed as Proposition 47 in the first book of Euclid’sElements. But you will not find Pythagoras’s name heading it: true to his terse, matter-of-fact style, Euclid avoided any reference to persons in his work, instead letting the geometry speak for itself. So the most famous theorem in mathematics simply became known as Euclid I 47.

Euclid’s proof of I 47 is anything but simple, and it has tested the patience of generations of students. In the words of philosopher Arthur Schopenhauer, “lines are drawn, we know not why, and it afterwards appears they were traps...

10. 7 Pythagorean Triples
(pp. 20-22)

A triple of positive integers (a,b,c) such that${a^2} + {b^2} + {c^2}$

is called aPythagorean triple; it represents a right triangle with sidesaandband hypotenusec, all of integer lengths. Some examples are (3, 4, 5), (5, 12, 13), and (8, 15, 17); one can find such triples even among large numbers: (4,601, 4,800, 6,649). These four examples are ofprimitivetriples—triples whose members have no common factor other than 1. Of course, from any given triple we can generate infinitely many others by multiplying it by an arbitrary integer; for example, the triple (6, 8,...

11. 8 The Square Root of 2
(pp. 23-25)

One of the most momentous events in the history of mathematics was the discovery of a new kind of number that had never been known before—anirrational number.

To the Pythagoreans, “number” meant either a positive integer or a ratio of two positive integers, arational number. Examples of such numbers are 2⁄1(or simply 2), 3⁄2, and 5⁄3. The Pythagoreans believed that any quantity, whether an abstract number or a physical entity, is represented by a rational number. This belief, in all likelihood, came from music, a discipline that in ancient Greece ranked equal in importance to arithmetic, geometry,...

12. 9 A Repertoire of Means
(pp. 26-28)

Another subject of great interest to the Pythagoreans was how to find the average, ormean, of two positive numbers. At first thought this seems to be a trivial question. Let the numbers beaandb. Add them and divide by 2, getting (a + b )/2: you are done. Today, of course, we would compute this mean numerically; for example, the mean of 3 and 5 is (3 + 5)/2 = 8/2 = 4. The Greeks, however, thought of it in geometric terms: they regardedaandbas the lengths of two line segments, drawing them end...

13. 10 More about Means
(pp. 29-31)

Turning again to the aircraft making its round trip between two cities, we found that the harmonic mean of the two ground speeds, 495 mph, was less—though just barely—than their arithmetic mean, 500 mph. This is not a coincidence. A well-known theorem says that of the three means, the harmonic mean is always the smallest, the arithmetic mean the largest, and the geometric mean somewhere in between. In the case of the aircraft, the geometric mean of the two speeds is$\sqrt {450.550} = 497.49$mph, rounded to two places, so we have$H < G < A$. This double inequality can actually be made...

14. 11 Two Theorems from Euclid
(pp. 32-35)

Theorem 35 of the third book of theElementssays,If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

To make sense of this enigmatic statement, we must understand that the Greeks always used geometric language to describe operations that nowadays would be stated in algebraic terms. Thus, “the rectangle contained by” is code for “the product of” [the sides of the rectangle]—in other words, the area of the rectangle. Translated into modern language, the theorem says:...

15. 12 Different, yet the Same
(pp. 36-38)

The two theorems we just met—numbers 35 and 36 in book III of Euclid—sound strikingly similar: both are about a circle, a pointP, and a line throughPthat cuts the circle at pointsRandS. The two theorems state that the product$\overline {PR} \times \overline {PS}$remains constant for all possible lines throughP. Yet there is a difference: in theorem 35Pis inside the circle, while in theorem 36 it is outside. And theorem 36 has the additional result that$\overline {PR} \times \overline {PS} = {\overline {PT} ^2}$, where$\overline {PT}$is the length of the tangent line fromPto the circle. So...

16. 13 One Theorem, Three Proofs
(pp. 39-41)

Theorem 13 of Book VI of Euclid tells us how to find the geometric mean (themean proportion,as the Greeks called it) of two line segments. In essence, it says that in a right triangle, the altitudehdivides the hypotenuse into two segmentsmandnsuch thath/m=n/h. From this it follows thath² =mn, so thathis the geometric mean ofmandn. Plate 13 illustrates this form= 9,n= 4, andh= 6.

We offer here three quite different proofs, with the question in mind, which...

17. 14 The Prime Numbers
(pp. 42-44)

The prime numbers have always enjoyed a special status among the integers. A prime number, orprime, for short, is an integer greater than 1 that can be divided only by itself and 1. The first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The smallest—and the only even prime—is 2. The largest, as of this writing, is 257,885,161-1, a gargantuan 17,425,170-digit number that would fill some 2,500 pages if printed. 1 An integer greater than 1 that is not prime is calledcomposite. The number 1 is considered neither prime...

18. 15 Two Prime Mysteries
(pp. 45-48)

Among the many unsettled questions about the primes, two stand out for their deceptive simplicity. Even a cursory glance at a table of primes will reveal the abundance of pairs of primes of the formpandp+ 2: 3 and 5, 5 and 7, 11 and 13, . . . , 101 and 103, and so on. One can find thesetwin primeseven among very large numbers: 29,879 and 29,881, 140,737,488,353,699 and 140,737,488,353,701. At the time of writing, the largest known twin pair is 3,756,801,695,685 ·2666,669± 1, each having 200,700 digits.¹ How many twin primes are...

19. 16 0.999 . . . = ?
(pp. 49-52)

When I ask beginning mathematics students, “is 0.999 . . .exactlyequal to 1, or only approximately so?” their responses are usually split evenly, but occasionally the majority will vote for the second option. Well, let’s see:

x= 0.999...

10x=9.999....

Subtract the first equation from the second:

9x=9

x=1.

Surprising? . . . If this simple question can cause such disagreement today, how much more so in ancient times, when the idea of anything going on to infinity was so confusing that it was avoided entirely, shunned by the Greeks ashorror infiniti, the horror of the infinite....

20. 17 Eleven
(pp. 53-55)

There is a parody about a mathematician who tries to prove that all numbers (here meaning positive integers) are interesting. Assume not. The number 1 is certainly interesting, being the generator of all numbers. So is 2, the first even integer and the only even prime. Three, being the sum of 1 and 2, makes it interesting as well. What about 4? We have 4 = 2 + 2 = 2 × 2 = 2²: no doubt about it, 4 is definitely interesting. And so it goes, until we arrive at the first uninteresting number. But this, of course, makes...

21. 18 Euclidean Constructions
(pp. 56-58)

According to tradition, it was Plato (ca. 427–347 BCE) who decreed that all geometric constructions should be done with a straightedge (an unmarked ruler) and compass alone. Of course, there is nothing intrinsically special about these tools, except perhaps their simplicity (you can still get them for a dollar or two at any drugstore), but Plato made their use into an art. Hundreds of constructions can be done with them, from very basic drawings to highly complex designs. Indeed, straightedge and compass constructions became so fundamental to geometry that Euclid incorporated them in hisElementsfrom the very beginning....

22. 19 Hexagons
(pp. 59-61)

Aregular polygonis a convex polygon whose sides all have the same length and meet each other at the same angle. Next to the equilateral triangle, the simplest regular polygon to construct—using only the Euclidean tools—is the six-sided hexagon. Let the side$\overline {AB}$be given (figure 19.1). Draw a circle with center atAand radius$\overline {AB}$, place the point of your compass atB, and without changing the compass’s opening, swing an arc, cutting the circle atC. Now place your compass atCand, with the opening still the same, swing a second arc, cutting...

23. 20 Fibonacci Numbers
(pp. 62-65)

Almost anyone with the slightest interest in mathematics will be familiar with the name Fibonacci. Leonardo of Pisa—he later adopted the name Fibonacci (son of Bonacci)—was born in Pisa around 1170, the son of a wealthy merchant. Pisa at that time was an important commercial center, serving both Christian Europe and Moslem Middle East and North Africa. Fibonacci was thus acquainted with the newly invented Hindu-Arabic numeration system, with the numerals (or “ciphers”) 0 through 9 as its centerpiece. Convinced that this system was superior to the cumbersome Roman numerals, he wrote a book entitledLiber Abaci(“Book...

24. 21 The Golden Ratio
(pp. 66-69)

Suppose you are being asked to divide a line segment into two parts such that the whole segment is to the longer part as the longer part is to the shorter. The Greeks were greatly intrigued by this seemingly simple problem, but exactly why is not quite clear: perhaps it was posed by an anonymous scholar as an exercise to his students, or it may have arisen from the challenge of constructing a regular pentagon with straightedge and compass (see the next chapter). Whatever its origins, this particular division of a line segment into two parts became known as the...

25. 22 The Pentagon
(pp. 70-72)

The five-sided pentagon has fascinated mathematicians for generations, and it still fascinates them today. In contrast to the three-, four-, and six-sided regular polygons, it is not at all obvious how to construct a regular pentagon, a fact that presented a challenge to the Pythagoreans. The secret to its construction is the 72–72–36-degree triangleABCformed by any of the pentagon’s five sides and the vertex opposite to it (figure 22.1).¹ This triangle is known as thegolden trianglebecause its side-to-base ratio is exactly the golden ratio$\varphi = (1 + \sqrt 5 )/2$. Since this ratio can be constructed using a straightedge...

26. 23 The 17-Sided Regular Polygon
(pp. 73-76)

Once we have constructed a regular polygon ofnsides—ann-gon, for short—it is easy to construct a regular polygon with twice as many sides—a 2n-gon: inscribe then-gon in a circle with center atO(figure 23.1 shows this for the hexagon). LetAandBbe two adjacent vertices of thisn-gon. Bisect$\angle AOB$and extend the bisector until it meets the circle atP. ConnectPto eitherAorB, and you have one side of the 2n-gon. So from the 3-, 4-, and 5-sided gons we can get the 6-, 8-, 10-,...

27. 24 Fifty
(pp. 77-80)

Number aficionados will tell you that every number has its own personality, its special features and its unique meaning—although people might differ as to what exactly that meaning is. But from the perspective of a number theorist, what matters most is a number’s prime factors. This prime factorization is the key to most, if not all, the mathematical properties of a number.

Let us take a look at 50—a nice, round number, the halfway point of a century, the age at which many of us start to think about our mortality. As decreed in the Bible (Leviticus 25:10),...

28. 25 Doubling the Cube
(pp. 81-83)

According to legend, at one time the Greek town of Delos was afflicted by a devastating plague that nearly decimated its population. In desperation, the city elders consulted the oracles, who determined that the god Apollo was unhappy with the small size of the pedestal on which his statue was standing. To appease him, they recommended to double the volume of the cubical pedestal. The task was given to the town’s mathematicians, who soon realized that doubling thesideof the cube would not do it—it would increase the volumeeightfoldand would make the pedestal unreasonably large. What...

29. 26 Squaring the Circle
(pp. 84-87)

At first glance, the circle may seem to be the simplest of all geometric shapes and the easiest to draw: take a string, hold down one end on a sheet of paper, tie a pencil to the other end, and swing it around—a simplified version of the compass. But first impressions can be misleading: the circle has proved to be one of the most intriguing shapes in all of geometry, if not the most intriguing of them all.

How do you find the area of a circle, when its radius is given? You instantly think of the formula$A = \pi {r^2}$....

30. 27 Archimedes Measures the Circle
(pp. 88-90)

By any measure, Archimedes (ca. 285–212 BCE) is considered the greatest scientist of antiquity, the equal of Newton and Einstein. As with most of the ancient Greek sages, much of what we know about him was written by later historians, who often confused legend with fact; thus many of the stories about him must be taken with a grain of salt. He was born in the town of Syracuse, on the southeast coast of the island of Sicily, where he spent all his life. In popular accounts Archimedes is best remembered for his spectacular engineering feats, but he considered...

31. 28 The Digit Hunters
(pp. 91-93)

In the second century BCE, during Archimedes’s lifetime, the Hindu-Arabic numeration system was still more than a thousand years in the future. So Archimedes had to do all his calculations in a strange hybrid of the Babylonian sexagesimal (base 60) system and the Greek system, in which each letter of the alphabet had a numerical value (alpha = 1, beta = 2, and so on). Today, of course, we associate the value of$\pi$with its decimal expansion—a nonrepeating, seemingly random string of digits that goes on forever. Terminate this expansion after any number of digits, and you’ll get...

32. 29 Conics
(pp. 94-98)

Imagine slicing a cone—for illustration’s sake think of it as an ice cream cone—with a swift stroke of a knife. If you slice it in a plane parallel to the cone’s base, you get a circular cross section. Tilt the angle slightly, and you get an ellipse. Tilt the angle even more, and the ellipse becomes narrower, until it no longer closes on itself: it becomes a parabola. This happens when the cut is parallel to the side of the cone. Increase the angle yet again, and you get two disconnected curves, the two branches of a hyperbola...

33. 30 $\frac{3}{3} = \frac{4}{4}$
(pp. 99-101)

We encountered the geometric progression in chapter 16 in connection with the runner’s paradox. Many interesting results can be obtained using a geometric progression, some quite unexpected. Consider a square of unit side, divide it into four equal smaller squares, and shade the upperright square, as in figure 30.1. Now divide each of the remaining, unshaded squares into four equal parts, and shade the upper-right quarter of each (figure 30.2). Repeating the process again and again, will the shaded area approach a limit? If so, what is it? In the first stage there is one shaded square of side 1/2...

34. 31 The Harmonic Series
(pp. 102-104)

In the last chapter we saw that the series 1-½+⅓-¼+... converges to In 2.

It is tempting to ask what will happen if we take the terms of this series in absolute value, that is, all positive. We then get theharmonic series, the sum of the reciprocals of the positive integers:

$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} +$...

The name “harmonic” comes from the fact that a vibrating string produces not only one note but infinitely many higher notes, whose frequencies are 1, 2, 3, 4, 5, . . . times the fundamental, or lowest, frequency. It is one more example of the influence that...

35. 32 Ceva’s Theorem
(pp. 105-107)

Giovanni Ceva (1647–1734) was born in Milan and got his schooling in a Jesuit institute there. After completing his education in Pisa he was appointed professor of mathematics at the university of Mantua, where he stayed for the rest of his life. His work was in geometry and in hydraulics—not an unusual combination in seventeenth-century Europe, before academic specialization became the rule: Leonardo da Vinci, Galileo, and many other scientists were working on engineering problems as much as on art, mathematics, and physics.

The prevailing view at the time was that after Euclid, geometry became largely a closed...

36. 33 e
(pp. 108-111)

To the trio of special numbers we have met so far—$\sqrt 2$,$\varphi$, and$\pi$—we now add a fourth number,e. This number, the base of natural logarithms, is of a more modern vintage than its ancient companions, tracing its origin to the seventeenth century. And unlike the others, it has its roots not in geometry, but in the world of business.

Financial matters have been of concern to people since the dawn of recorded history. A Babylonian clay tablet, dating back to about 1700 BCE and now in the Louvre, asks how long it would take for a sum...

37. 34 Spira Mirabilis
(pp. 112-115)

Of the numerous curves we encounter in art, geometry, and nature, perhaps none can match the exquisite elegance of the logarithmic spiral, shown in plate 34.1. This famous curve appears, with remarkable precision, in the shape of a nautilus shell, in the horns of an antelope, and in the seed arrangements of a sunflower (see page 64, figure 20.1). It is also the ornamental motif of countless artistic designs, from antiquity to modern times. It was a favorite curve of the Dutch artist M. C. Escher (1898–1972), who used it in some of his most beautiful works.

The logarithmic...

38. 35 The Cycloid
(pp. 116-118)

Rivaling the logarithmic spiral in elegance is thecycloid—the curve traced by a point on the rim of a circle that rolls along a straight line without slipping (figure 35.1). The cycloid is characterized by its arcs and cusps, each cusp marking the instant when the point on the wheel’s rim reaches its lowest position and stays momentarily ar rest.

The cycloid has a rich history. In 1673 Christiaan Huygens, whom we’ve just met in connection with the catenary, solved one of the outstanding problems that had intrigued seventeenth-century scientists: to find the curve down which a particle, moving...

39. 36 Epicycloids and Hypocycloids
(pp. 119-122)

Whereas the cycloid is generated by a point on the rim of a wheel rolling along a straight line, we may also consider a wheel rolling on the outside of a second, fixed wheel; the resulting curve is anepicycloid(from the Greekepi, meaning “over” or “above”). Or, we can let the wheel roll along theinsideof a fixed wheel, generating ahypocycloid(hypo= “under”). The epicycloid and hypocycloid come in a great variety of shapes, depending on the ratio of the radii of the two wheels. Let the radii of the fixed and moving wheels be...

40. 37 The Euler Line
(pp. 123-125)

Leonhard Euler (1707–1783) was arguably the most prolific mathematician of all time. Born in Basel, Switzerland, to a Calvinist minister, he studied with Johann Bernoulli and received his master’s degree from the University of Basel at the age of 16. When two of Bernoulli’s sons, Nicolas and Daniel (the latter famous for a law named after him in hydrodynamics), moved to St. Petersburg, Euler followed them and spent the next 14 years there. He then moved to Berlin to head the Prussian Academy of Sciences but returned to Russia in 1766 and never left again. His last years were...

41. 38 Inversion
(pp. 126-129)

The circle can be a source of never-ending fascination. Consider a circle with center at$O$and radius 1. Two points$P$and$Q$on the same ray through$O$are calledinversesof each other with respect to the circle if$\overline {OP} \cdot \overline {OQ} = 1$, or, equivalently,$\overline {OQ} = 1/\overline {OP}$(figure 38.1). Either point can be thought of as the “image” of the other. If$P$is inside the circle (that is,$\overline {OP} < 1$), its image$Q$will lie outside ($\overline {OQ} > 1$), and vice versa. Therefore, the entire interior of the circle is “mapped” onto the exterior in a one-to-one correspondence. And, conversely, the interior is...

42. 39 Steiner’s Porism
(pp. 130-133)

The first half of the nineteenth century saw a revival of interest in classical Euclidean geometry, in which figures are constructed with straightedge and compass and theorems are proved from a given set of axioms. This “synthetic,” or “pure,” geometry had by and large been thrown by the wayside with the invention of analytic geometry by Fermat and Descartes in the first half of the seventeenth century. Analytic geometry is based on the idea that every geometric problem could, at least in principle, be translated into the language of algebra as a set of equations, whose solution (or solutions) could...

43. 40 Line Designs
(pp. 134-137)

Julius Plücker (1801–1868) is not a household name among present-day mathematicians, but in the nineteenth century he carved himself a niche in geometry where few others had ventured before. He realized that a curve need not be regarded as a set of points; it can just as well be described as a set oftangent lines. The idea was not entirely new. It had been known for more than a century that certain formal statements about points and lines remain valid when the wordspointandlineare everywhere interchanged. For example, just as two points determine a unique...

44. 41 The French Connection
(pp. 138-140)

There is a time-honored French tradition, going back at least to the sixteenth century, that calls for scientists to pursue a career in the military or civil service in parallel to their academic careers: François Viète, René Descartes, Jean Baptiste Joseph Fourier, and Victor Poncelet—to name but a few—all served as army officers, military engineers, or public administrators at various levels of government. This tradition is embodied by the lives of two early nineteenth-century French geometers, Charles Julien Brianchon and Jean Victor Poncelet.

Not much is known about the early life of Brianchon (1783?–1864); even his year...

45. 42 The Audible Made Visible
(pp. 141-142)

Ernst Florens Friedrich Chladni (1756–1827) is not among the giant names in the history of science, but his discoveries had a visual impact—quite literally. Born in Wittenberg, Germany, in the same year as Mozart, but outliving him by nearly forty years, Chladni was a rare combination of musician and physicist. In his book,Discoveries in the Theory of Sound(1787) he showed how the vibrations of sound-generating objects could be seen visually. He experimented with thin metal plates, over which a fine layer of sand was strewn. When the plate was excited by rubbing a violin bow against...

46. 43 Lissajous Figures
(pp. 143-145)

Another scientist who transformed sound into visual patterns was Jules Antoine Lissajous (1822–1880). Lissajous was professor of mathematics at the Lyceé Saint-Louis in Paris, where he studied all kinds of vibrations and waves. In 1855 he invented a simple optical device for analyzing compound vibrations. He attached small mirrors to the prongs of two tuning forks vibrating at right angles to each other. When a beam of light was aimed at one of the mirrors, it bounced off to the other mirror and from there to a screen, where it formed a two-dimensional pattern, the result of superimposing the...

47. 44 Symmetry I
(pp. 146-148)

Few subjects have bridged the divide between the humanities and science more successfully than the concept of symmetry. Symmetry is as central to mathematics and physics as it is to the visual arts, architecture, music, and aesthetics. To the Greeks, symmetry meant a balance between the different parts of an object. Most Greek temples have a perfectbilateral,orreflection,symmetry: if you were looking at a mirror image of the Parthenon, you would hardly be able to distinguish it from the actual shrine. To the aesthetically minded Greeks, symmetry was synonymous with beauty and perfection; the human body was...

48. 45 Symmetry II
(pp. 149-153)

As we just saw, an equilateral triangle is endowed with six symmetry elements—three 120° rotations and three mirror reflections. Let the triangle beABC, with vertexAat the top, followed clockwise by verticesBandC. Let us denote the six symmetry operations by letters:r₁,r₂,r₃ for the 120°, 240°, and 360° clockwise rotations andm₁,m₂,m₃ for the reflections in the altitudes through the top, lower-right, and lower-left vertices (henceforth we’ll refer to these vertices as nos. 1, 2, and 3 rather thanA,B, andC, because they keep changing their position as...

49. 46 The Reuleaux Triangle
(pp. 154-156)

Franz Reuleaux (1829–1905), notwithstanding his French-sounding name, was a German scientist and engineer who is regarded as the founder of modern kinematics and machine design. He was born to a family of mechanical engineers and machine builders, an environment that nourished his future interests. He got his formal education at the Karlsruhe Polytechnic School and held his first academic appointment with the Swiss Federal Technical Institute (ETH) in Zurich. In 1864 he became professor at the Royal Industrial Academy in Berlin (later the Royal Technical College), being active as an educator, industrial scientist, and consultant. His views on various...

50. 47 Pick’s Theorem
(pp. 157-159)

Georg Alexander Pick (1859–1942) was born to a Jewish family in Vienna but spent most of his academic life in Prague. Pick began his career as assistant to some of the great names in mathematics and physics at the late nineteenth century, among them Ernst Mach (after whom the Mach number in aerodynamics is named) and Felix Klein, whose reform program in mathematics education would greatly influence subsequent generations of mathematicians. In 1892 Pick became professor at the German University of Prague, where 20 years later he would play a key role in offering a professorship to a young...

51. 48 Morley’s Theorem
(pp. 160-163)

Frank Morley (1860–1937) was an English-born mathematician who moved to the United States in 1887. He was professor at Johns Hopkins University from 1900 to 1928 and editor of theAmerican Journal of Mathematicsfrom 1900 to 1921. He published several books, including two on the theory of functions of complex variables (1893 and 1898), in which he introduced that subject—one of the highlights of nineteenth-century European mathematics—to American readers. He was known as an outstanding teacher who had no fewer than 45 doctoral students, an incredible number for any professor. He was also a gifted chess...

52. 49 The Snowflake Curve
(pp. 164-166)

Niels Fabian Helge von Koch (1870–1924) was a Swedish mathematician who is remembered today chiefly, if not solely, for a curious curve he discovered in 1904. Take an equilateral triangle of unit side, divide each side into three equal parts, each of length 1/3, and delete the middle part (figure 49.1). Over the deleted part, construct the two sides of an equilateral triangle of side 1/3. This gives you a 12-sided Star of David shape, whose perimeter is 4/3 that of the original triangle. Now repeat the process with each of the 12 new sides, resulting in a 48-sided...

53. 50 Sierpinski’s Triangle
(pp. 167-169)

Waclaw Franciszek Sierpinski (1882–1969) was born in Warsaw. He entered the University of Warsaw in 1899, graduating in 1904; four years later he was appointed to the University of Lvov. He became interested in set theory after reading about a theorem that allows for a seemingly impossible situation. We learn in analytic geometry that any point in the plane can be uniquely specified by two numbers, itsx- andy-coordinates. This has been the rock foundation of analysis ever since René Descartes invented his analytic geometry in 1637. Not so, said the theorem that Sierpinski came across: one number...

54. 51 Beyond Infinity
(pp. 170-174)

Can anything be larger than infinity? No, says common sense. But who is to say that common sense is always right—especially when it comes to the infinite, a world beyond our physical reach? “Infinity is a place where things happen that don’t,” an anonymous school pupil once said. Railroad tracks, though perfectly parallel, seem to meet far away on the horizon—at infinity; indeed, in projective geometry theydomeet at infinity (see chapter 40). Yet when we try to reach that elusive point, it recedes from us just as fast as we approach it.

Again, when comparing two...

55. Appendix: PROOFS OF SELECTED THEOREMS MENTIONED IN THIS BOOK
(pp. 175-182)
56. Bibliography
(pp. 183-184)
57. Index
(pp. 185-192)