Pi

From Math Images

Revision as of 13:56, 13 December 2012 by Liesposito (Talk | contribs)
Jump to: navigation, search

Contents

Introduction

Pi
Pi

Note: π discussed in this article is confined to flat (Euclidean) geometry and does not contain curved geometries.

The symbol π is the sixteenth letter of the Greek alphabet, yet it has gained fame because of its designation in mathematics. π has some intriguing properties. For one, π is an irrational number[1], meaning that it cannot be written as the ratio of two integers and it has an infinite number of digits in its decimal representation. The first attempt to prove that π was irrational was made by Johaan Heinrich Lambert in 1761. Lambert used a continued fraction expansion of tan (x) to prove π's irrationality. Lambert's proof produced the theorem that if x is a non-zero rational number, then the number tan x is irrational, and if tan (x) is rational than x must be irrational. Lambert believed that since tan (π/4) = 1, a rational number, then π/4 must be irrational, hence π must be irrational. The results of Lambert's complex continued fraction of tan (x) proved his theory to be correct. [2] Lambert's proof seemed to be too simplified to be the answer to a complex problem. However, in 1794, Legendre proved the irrationality of π using a more rigorous procedure. In his book “Elements de Geometrie” Legendre provided a proof of π's irrationality, and also gave a proof that π2 is irrational. Legendre's work supported Lambert's proof, and put to rest the question of π's irrationality. Toward the end of his book, Legendre wrote : “It is probable that the number π is not even contained among the algebraic irrationalities, and that it cannot be the root of an algebraic equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly.” [3]

π is also a transcendental number[1], which means that it is not the solution of any non-constant polynomial with rational coefficients, such as \frac{x^3}{10}-x^2+1=0.The transcendence and irrationality of π has many important consequences. It is extremely difficult to prove that a number is transcendental. π was proven to be transcendental in 1882 by Ferdinand von Lindemann, based upon works of Charles Hermite and Euler. In 1873, Charles Hermite proved that the number e is transcendental. His proof showed that the finite equation aer + bes + cet + … = 0 cannot be satisfied if r, s, t, … are natural numbers and a, b, c … are rational numbers not all equal to zero. Lindemann extended Hermite's theorem to the case where r, s, t, … and a, b, c, … are algebraic numbers, not necessarily real. Lindemann's theorem states that if r, s, t, …, z, are distinct real or complex algebraic numbers, and a, b, c, …, n are real or complex algebraic numbers, at least one of which differs from zero, then the finite sum aer + bes + cet + … + nez cannot equal zero. Once Lindemann proved this, the transcendence of π quickly follows. Lindemann used Euler's theorem in the form e + 1 = 0, which gave an expression with a = b = 1 algebraic, and c and all further coefficients equal to zero; s = 0 is algebraic, leaving r = iπ as the only reason why the equation should vanish. Therefore, iπ must be transcendental, and since i is algebraic, π must be transcendental. [3]

However in this article, we will touch upon the more geometrical side of π. We will go back in time and retrace the steps of the ancient Egyptians, Archimedes, Euclid, and Cusanus in their arduous yet rewarding journey to estimate the value of the mysterious constant. We will learn each of their approximation methods, and actually calculate the π value they would have gotten with today's technology. (Calculators, Matlab, Mathematica, etc.) Then we shall explore many of π's applications, even in fields seemingly unrelated to geometry, like probability. We will explore the Reuleaux triangle, an analogous figure to the circle, later used as the prototype of the renowned Wankel engine. We shall find trace of π when the conventional wisdom led Cambridge geologist Hans-Henrik Stolum to calculate the ratio of the length of a river to that between its source and end point.

The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.[1]

Definition

The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter.

As stated in the first sentence of this article, we shall in the following sections deal with the calculation and usage of π in Euclidean geometry. We should also be aware that the previous definition of π is not universal, because it is not valid in curved (non-Euclidean) geometries.[4] For this reason, some mathematicians prefer definitions of π based on calculus or trigonometry that do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[4]

Calculating Pi

Throughout history, scholars have been trying to figure out the value of π. The polygon approximation era, infinite series, computer era and iterative algorithms have brought scholars closer and closer to the true value of the mysterious constant. We may wonder about their motivation for such research.

For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the universe with a precision of one atom.[4] Despite this, people have worked strenuously to compute π to thousands and millions of digits.[4] This effort may be partly ascribed to the human compulsion to break records. They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.

In the following sub-sections of this article, we shall explore approximations of π with a strong geometric blend to it. Such approximations include the method of the ancient Egyptians, Archimedes, Cosanus, and Euclid. Yet we have also added an algebraic sense by sometimes taking a detour along our ancestors' steps and using tools such as infinite series or Matlab to calculate results. Due to the orientation of this article, a large portion of π's approximations have been left out.

Ancient Egyptians

Figure 3-1: The Ancient Egyptian Method of Approximating π
Figure 3-1: The Ancient Egyptian Method
of Approximating π

An Egyptian scribe named Ahmes wrote the oldest known text to imply an approximate value for π. The Rhind Papyrus, written by Ahmes, said that if we construct a square with a side whose length is eight-ninths of the diameter of a circle, then their area will be equal. It was the effort to construct a square whose area is equal a circle that generated the early approximations of π.[5] If we inspected the Rhind Papyrus, we can replicate the work of the ancient Egyptians and find out how close they were to the true π value.

We will denote the diameter of the circle d, and the length of the side of the square to be a. Then we can calculate both of their areas with the corresponding formulas and equate them to get our (the ancient Egyptians') value of approximation.

We know a and d are related in the following way:

a=\frac{8}{9}d

The area of the square, A1 is:

A_1=a^2=\frac{64}{81}d^2

We know from today's knowledge the area of the circle, A2 is:

A_2=\pi(\frac{d}{2})^2=\frac{d^2\pi}{4}

We shall briefly stop here to comment on two things. First, one might argument the circle area equation was not available at the time of the ancient Egyptians. However, the purpose of our derivations is to merely find out how close they were to the true π value. Second, although the symbol "π" was not introduced to represent the ratio between the circumference and diameter of the circle till much later (three thousand years), we have used it for convenience and to avoid confusion.

We now set A1 equal to A2:


A_1 = A_2
\frac{64}{81}d^2 = \frac{d^2\pi}{4}
\pi = \frac{256}{81}

This is a reasonably close approximation of what we know the value of π to be by modern methods.

Euclid's Influence

Euclid also made a contribution to the history of π. Although Euclid himself never postulated an approximation for the value of π, his work hints at the possible awareness of such a constant. We shall further understand this by following Proposition 2, Book XII of Elements.

We first state Proposition 2, Book XII[6]:

Circles are to one another as the squares on the diameters.

Since the language may seem confusing after one read, let us instead appeal to symbols. Let Circle 1 have area A1 and diameter d1; let Circle 2 have area A2 and diameter d2. Proposition 2 is telling us the following relationship:

\frac{A_1}{A_2}=\frac{d_1^2}{d_2^2}

Some simple algebraic manipulation gives us the following:

\frac{A_1}{d_1^2}=\frac{A_2}{d_2^2}=C, where C is some constant value.

This says that the area of a circle is equal to the diameter squared times some constant C, eventually leading us to the formula for the area of the circle. Euclid's work is actually hinting to the existence of the constant which is known today as π.

Archimede's Method

Figure 3-3-1: A variety of inscribed regular polygons
Figure 3-3-1: A variety of inscribed regular polygons

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons. It was devised around 250 BC by the Greek mathematician Archimedes.[7] Since π is the ratio of any circle's circumference and its diameter, it will be greater than the circumference of any inscribed regular polygon and less than that of any circumscribed regular polygon. Shown in Figure 3-3-1, we see circles with inscribed regular polygons with 3, 4, 5, 6, 8, and 17 sides. It's obvious the more sides such a polygon has, the closer it will resemble an actual circle (notice how in Figure 3-3-1 when n=17 the circle is visually indiscernible from the parameter from the polygon), and thus the closer the length of its perimeter will be to that of the circle.

Figure 3-3-2: Inscribed hexagon inside circle
Figure 3-3-2: Inscribed hexagon inside circle

We will now retrace Archimedes' work. But instead of using both inscribed polygons and circumscribed polygons to "sandwich" the circle, we will only use inscribed polygons for our approximation process. With a regular hexagon as our visual guide (as shown in Figure 3-3-2), we wish to derive the general formula with which we can apply to any n-sided regular polygon to estimate the value of π.

Since we know the circumference of the circle to be C=πd, where d is the diameter of the circle, by setting the diameter to 1 the parameter of the polygon (C) will be equivalent to the value of π. This is done without loss of generality:

\pi=C

Now we will start our derivation of a general formula which will allow us to calculate the perimeter (C) of any n-sided polygon.

Since the polygon has n sides,

\angle AOC=\frac{360^\circ}{n}

\angle AOB=\frac{180^\circ}{n}

Applying trigonometry,

AB=\frac{1}{2}\sin(\frac{180^\circ}{n})

Because the polygon has n sides and the length of AB is half that of one side,

C=2nAB=n\sin(\frac{180^\circ}{n})

We can then plug in varoius values of n and compute the parameter of the regular polygon whose circumscribed circle has a diameter of 1.

The process is easily done with a calculator (or MATLAB). We will point out that when n=10,000, the approximation for the value of π is:

\pi=3.1415926 0191266 5692979

If we look at the known value of π for comparison:

\pi=3.1415926 5358979 3238462

We find that up to the seventh place, the approximation with a 10,000-sided regular polygon is perfectly accurate. Yet Archimedes himself obviously did not enjoy the luxury of calculating devices to assist him in the following:

\pi=96\sin(\frac{180}{96})=96\sin(\frac{15}{8})

Archimedes used the perimeters of many-sided polygons to approximate π and did an amazing job even viewed by today's standards. He managed his approximations through some geometric technique and for him it surely didn't come down to calculating 96sin(180/96). But in the following paragraphs I will take a detour while retracing Archimedes' route and actually tackle sin(180/96) through the double-angle and half-angle formulas:

Equa (1)         \cos^2\frac{\theta}{2}=\frac{1}{2}(1+\cos\theta)
Equa (2)         \sin^2\frac{\theta}{2}=\frac{1}{2}(1-\cos\theta)

When we try to take the square roots of Equa (1) and Equa (2), we have to consider the possibility that the left hand side might be negative. (It is obvious the right hand side of both equations will be positive.) Yet since as the number of sides of the regular polygon gets large, θ will become small and we're confident cos(θ/2) will stay positive. Thus after some simple manipulation, Equa (1) and Equa (2) can be rewritten in the following way:

Equa (3)         2\frac{\cos\theta}{2}=\sqrt{2+2\cos\theta}
Equa (4)         \frac{\sin\theta}{2}=\frac{1}{2}\sqrt{2-2\cos\theta}

We notice how we can generalize Equa (3):

Equa (5)         2\cos{2^k\theta}=\sqrt{2+2\cos{2^{k+1}\theta}}

We then realize if we start with sin(θ/2), the left hand side of Equa (4), we can keep substituting Equa (5) into the right hand side for however many iterations we like. The larger k gets, the more sides the regular polygon will have, and the more accurate our approximation of π will be:

Equa (6)         \sin{\frac{\theta}{2}}=\frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2+2\cos{2^k\theta}}}}}}

In fact we are using an infinite series (see Series for more) to estimate π. The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries.[7] An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques.

Cusanus' Method

Archimedes had used inscribed and circumscribed regular polygons within and about a given circle, eventually increasing the number of sides of the polygon. He "sandwiched in" the circle to get an upper and lower bound for the value of π. An analogous method developed by Cusanus has us "sandwiching" in regular polygons with increasing numbers of sides (the number of sides double after each iteration, as can be easily seen in Figure 3-4-1) by inscribed and circumscribed circles.[8] Like with Archimedes' method, we will only work with circumscribed circles. Inscribed circles are similar. We will start with squares to derive a general formula that relates the parameter of an n-sided regular polygon to that of a 2n-sided polygon. That way, with any random initial regular polygon, we can increase the amount of iterations with the help of some programming to get infinitely closer to the actual value of π.

As shown in Figure 3-4-1, we start with a square inscribed in a unit circle whose diagonal is length 2. If we denote the length of the square's side as an, then the circumference of the polygon (square) would generate our first approximation of the value of π. Without loss of generality, I picked the midpoint of line segment AB, D, and extended OD to intersect the circle at point C. CB is the side of the second polygon used in our approximation, whose length we shall denote as a2n. Notice these steps could be done on a paragon, a hexagon, or any other inscribed regular polygon. We shall now derive a general formula to calculate a2n with an, which can then be iterated on Matlab (along with an initial value) for however many loops we wish for our approximation of π:

Figure 3-4-2: Click to enlarge
Figure 3-4-2: Click to enlarge

We know that,

OB=1 \qquad DB=\frac{a_n}{2}

According to the Pythagorean Theorem,

OD^2=OB^2-DB^2=1^2-(\frac{a_n}{2})^2

Thus,

CD=CO-OD=1-\sqrt{1^2-(\frac{a_n}{2})^2}

And since DB=\frac{a_n}{2},

a_{2n}=CB=\sqrt{DB^2+CD^2}=\sqrt{(\frac{a_n}{2})^2+(1-\sqrt{1^2-(\frac{a_n}{2})^2})^2}

After some manipulation,

a_{2n}=\sqrt{2-2\sqrt{1-\frac{(a_n)^2}{4}}}

Figure 3-4-2: Click to enlarge
Figure 3-4-2: Click to enlarge

With the progressive relationship between an and a2n and the initial value a4=1.414, we can write a Matlab script that calculates the value of π up to infinite n value. However, after experimenting I found that 10 iterations can get us close enough to the "actual" π value.

Figure 3-4-2 is a screenshot of the Matlab script and the plot it generated. While the programming language is trivial, what the plot reveals is that after just 5 approximations, we have got ourselves a decent π value. In other words, a regular polygon with 32 (2^5=32) sides is as good as a circle in terms of calculating the π value.

A Brief Timeline[1]

Up till this point we have described the works of the Ancient Egyptians, Euclid, Archimedes, and Cusanus all of which while calculating the value of π have taken on a geometric approach. We shall now provide a general outline of the history of π.

  • The Ancient Egyptians, as early as 1650 BCE, have been recorded to have equated (although approximately) the area of a circle and a square whose side is 8/9 that of the circle's diameter.
  • Taking a leap forward in time, we come to the Babylonians, which spans from 2000 BE to about 600BCE. One tablet unearthed at Susa (not far from Babylon) compares the parameter of a regular hexagon to the circumference of its circumscribed circle. This result proved to be a little bit closer to the Egyptians' approximation.
  • In the Bible (Old Testament), written about 550 BCE,the Talmud's books of Kings and Chronicles describes King Solomon's water basin and hints a possible value of π, 3.
  • Then we come across the work of Archimedes. By both inscribing and circumscribing regular polygons around circles, he produced an upper limit as well as a lower limit for the value of π, between which our known value of π is smoothly squeezed in. He also visioned the circle as the limit of the ever-increasing number of sides of a regular polygon of a fixed parameter.
  • Meanwhile in China, works of Liu Hui and Zu Chongzhi paralleled that of the West. Liu's approximation,

\pi=\frac{3927}{1250}=3.1416

was perhaps the most most accurate approximation of π until Zu came up with his approximation,

\pi=\frac{355}{113}

  • As we enter the Renaissance, we shall note the works of Fibonacci. In his Liber abaci, published in 1202, by making use of a regular polygon of ninety-six sides, he computed the value of π to be:

\pi=\frac{1440}{458\frac{1}{3}}

Although for his approximations are not as accurate as the some approximations we have previously introduced, his contributions to mathematics are legendary, especially when considering the fact they took place after the Dark Ages.

  • Starting in the 16th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians including Mādhava of Sañgamāgrama, Isaac Newton, Leonhard Euler, Srinivasa Ramanujan, and Carl Friedrich Gauss.
  • In the 20th century, mathematicians and computer scientists discovered new approaches that – when combined with increasing computer speeds – extended the decimal representation of π to over 10 trillion digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records; but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.

Applications of π

Because π is closely related to the circle, it is found in many formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Formulae from other branches of science also include π in some of their important formulae, including sciences such as statistics, fractals, thermodynamics, mechanics, cosmology, number theory, and electromagnetism.[1]

Alternative Circle Area Formulation

Figure 4-1-1: Click to enlarge
Figure 4-1-1: Click to enlarge
Figure 4-1-2: Click to enlarge
Figure 4-1-2: Click to enlarge

Let's consider a relatively simple "derivation" for the formula A=πr2[1]. We divide the circle into 16 arcs, each being 22.5°, as in Figure 4-1-1. Then, let's cut the circle apart into 16 pieces and regroup them the manner shown in Figure 4-1-2.

We see that Figure 4-1-2 roughly resembles a parallelogram. That is, if the circle were equally divided into more sections, then the figure would look more like a true parallelogram. Let us assume it is a parallelogram. Now, we have transformed the task of finding the area of the circle into finding the area of the parallelogram, which we know to be base × height. By observation, the height of the parallelogram is the radius of the circle, r. Also, since half of the circle's arcs are used for each of the two sides of the approximate parallelogram, the base would have a length half of the circumference of the original circle.

To make the following derivations more clear, let us denote the base of the parallelogram as b, the height h, the area of the circle Acirc, the circumference of the circle C, and the area of the parallelogram Ap.

Since half of the circle's arcs are used for each of the two sides of the approximate parallelogram:

b=\frac{C}{2}=\frac{2\pi r}{2}=\pi r

And by observation:

h=r

Thus the area of the circle is:

A_{circ}=A_p=bh=\pi r^2

π in Probability

Figure 4-2-1: Buffon's Needle
Figure 4-2-1: Buffon's Needle
Figure 4-2-2: Zooming into One Particular Square
Figure 4-2-2: Zooming into One Particular Square

π shows up in areas that seemingly have nothing to do with geometry, such as probability. The French naturalist Buffon is primarily remembered for his work to popularize the natural sciences in France. Yet in mathematics he is remembered for two things: his French translation of Newton's Method of Fluxions, the forerunner of today's calculus, and more so even for the "Buffon needle problem."[9] We are primarily interested in the latter.


The Buffon needle problem goes like this: Suppose you have a piece of paper with ruled parallel lines throughout, equally spaced (at a distance d between both horizontal and vertical lines), and a thin needle of length l (where l<d). You can then toss the needle onto the paper many times. Buffon claimed that the probability that the needle will touch one of the ruled lines is \frac{2l}{d\pi}.. Let's find out why.[9]


Without loss of generality, we can choose to experiment with needles that are shorter than the spacing between the lines. (l<d) We try to form a mathematical argument of the condition satisfying which the needle will cross a line. In Figure 4-2-2, we zoom into one particular box. l is the length of the needle. x is the distance between the midpoint of the stick (point A) to its closest line. θ is the angle formed by the needle and the horizontal side of the square. d is the distance between the endpoint of the needle to the line parallel to the horizontal sides that also passes through A.


By observation, the needle crosses the upper line when:


x \le d


By trigonometry:


d=\frac{l}{2}\sin(\theta)


Thus, we arrive at the mathematical argument when the needle crosses a line,


x \le \frac{l}{2}\sin(\theta)


In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated U(a,b).[10]

Figure 4-2-3: Geometric Representation of the Continuous Uniform Distribution
Figure 4-2-3: Geometric Representation of the Continuous Uniform Distribution

By this definition, θ and x are both continuous uniform distribution (or rectangular distribution) that satisfies:


\theta:U(0,\pi)
x:U(0,\frac{d}{2})


In Figure 4-2-3, Region D (the rectangle with width π and height d/2) represents every possibility for tossing a needle with θ ranging from 0 to 180°, and at the same time with x going from 0 to d/2. x can not be larger than d/2 since it is defined as the closest distance from the midpoint of the needle to the parallel lines. Region A is the area under the curve x=lsinθ/2, which is the geometric representation of the condition under which a needle will cross a line, as derived above.


The area of A, A1, is computed using integration:


A_1=\int_{0}^{\pi} {{L \over 2} \sin(\theta) d \theta} = \frac{l}{2}(-\cos(\pi)-(-\cos(0))= L


The area of D, A2 is:


A_2=\frac{d\pi}{2}


With these two values, now we can calculate the probability of a needle crossing the parallel lines,


P=\frac{A_1}{A_2}=\frac{l}{\frac{d\pi}{2}}=\frac{2l}{d\pi}


This is how the equation for Buffon's experiment, P=\frac{2l}{d\pi} came about. And so such for theoretical manipulation. Say we had some spare time and wanted to try out Buffon's experiment by actually tossing needles onto a grid of parallel lines repeatedly. Without loss of generality, we can set l equal to d. So that the probability, P, of the needle touching one of the lines is 2/π:


P=\frac{2l}{d\pi}=\frac{2}{\pi}


This equation actually gives us another method to approximate the value of π,


\pi=\frac{2}{P}


Theoretically, the more you toss, the more accurate the estimate of π should be. In 1901 the Italian mathematician Mario Lazzarini tossed the needle 3408 times and got π=3.1415929[9], which is quite impressive. It differs from π by no more than 3×10−7.


While this method is not the most "accurate" measurement of π by any means, it is novel. Just think for a moment how the probability of a tossed needle intersecting a line relates to the value of a transcendental number, π.

It is quite curious that π is related to probability. Another such example: the probability that a number chosen at random from the set of natural numbers has no repeated prime divisors is 6/π2. This value also represents the probability that two natural numbers selected at random will be relatively prime. This is quite astonishing since π is derived from a geometric setting.[9]

Using π to Measure River Lengths

Figure 4-3: Meandering into semi-circles
Figure 4-3: Meandering into semi-circles

By now you should be fully convinced that π is not confined within geometry or even mathematics. For example, π also appears as the average ratio of the actual length and the direct distance between source and mouth in a meandering river. Hans-Henrik Stolum, a geologist at Cambridge University, calculated the ratio between the total length of a river to the direct distance between its source and end point.[6] He found the average ratio to be a bit larger than 3. It is actually around 3.14, which we recognize as an approximation of π. Coincidence?

Rivers have a tendency toward a loopy path. A slight bend will lead to faster currents on the outside shores, and the river will begin to erode and create a curved path. The sharper the bend, the more strongly the water flows to the outside, and in the consequence the erosion is in turn the faster.

The meanders get increasingly more circular, and the river turns around and around in semi-circles. It then runs straight ahead again, and the meander becomes a bleak branch. Between the two reverse effects a balance adapts. The process is demonstrated in Figure 4-3.

We have created a model where a fictitious river is superimposed by semicircles to represent the eventual curve of the river flow. Now by labeling the source point of the river A, end point B, the midpoints of every semicircle Mi (i=1,2,3,4), and the radius of each semicircle ri, we can check if the ratio of the length of the path of the river (C) to that of line segment AB (l) is actually π.

C=r_1\pi+r_2\pi+r_3\pi+r_4\pi

l=r_1+r_2+r_3+r_4

Thus,

\frac{C}{l}=\pi

Reuleaux Triangle

Figure 4-4: Reuleaux Triangle
Figure 4-4: Reuleaux Triangle

We will introduce the Reuleaux triangle by constructing one. The process is as follows: start by drawing a circle of center A with a random radius. Then we pick a random point B on the parameter of circle A. With B as the center, we draw a second circle whose radius is equal to that of circle A. We label one of the two interception points of circles A and B "C". With C as the center, we draw our third circle with a radius equal to that of the first two. The shaded figure is called a Reuleaux triangle, named after the German engineer Franz Reuleaux (1829-1905), who taught at the Royal Technical University of Berlin.[7]

Figure 4-4-1: Breadth of the Reuleaux Triangle
Figure 4-4-1: Breadth of the Reuleaux Triangle

Breadth of the Reuleaux

The ratio of the perimeter Reuleaux triangle to its "distant across" (the line segment AB, BC, or AC) is equal to π:

\pi=\frac{perimeter}{breadth}

While the Reuleaux triangle has many interesting properties, we shall start by examining its breadth. We refer to the distance between two parallel lines tangent to the curve (see Fig 234-234234) as the breadth of the curve.[7] Notice that no matter where we place these parallel tangents, they will always be the same distance apart: the radius of the arcs comprising the triangle, r. With our definition it's easy to see in the case of a circle, the breadth is always its diameter. Therefore, both the Reuleaux triangle and the circle have constant breadth. This property alone makes them analogous to each other.

Notice in the original three circles that constructed our Reuleaux triangle (namely circle A, B, and C in Figure 4-4). Arc AB, AC, and BC all correspond to a central angle of 60ˆ. The lengths of these arcs (which we shall denote as l) all are:

l=\widehat{AB}=\widehat{AC}=\widehat{BC}=\frac{1}{6}2\pi r=\frac{r\pi}{3}

Thus the perimeter of our given Reuleaux triangle, C1, is:

C_1=3\times l=r\pi

Figure 4-4-2: Calculating the Area
Figure 4-4-2: Calculating the Area

Same Perimeter, Different Areas

Yet we notice the circumference of a circle of diameter r is also πr. Thus, the circle with a diameter of length r has the same circumference with the Reuleaux triangle of breadth r. Their having the same perimeter causes us to wonder how their areas are related. Let us start with the more difficult task of calculating the area of the Reuleaux triangle (We shall denote it as AR).

As shown in Figure 4-4-2, we have divided the Reuleaux triangle into 4 parts and labelled them 1 through 4, which we shall also denote as Ai (i=1,2,3,4). The area of the Reuleaux triangle, AR, is:

A_R=A_1+A_2+A_3+A_4

By symmetry,

A_2=A_3=A_4

By observation,

A_3=Sector ABC - A_1

Using an alternative equation for the area of triangle, Area=abSinC, where a and b are two sides of the triangle and form angle C, the area the equilateral triangle, A1 is:

A_1=r\times r \times \sin(60^\circ)=\frac{\sqrt{3}}{4}r^2

Thus,

A_3= \frac{60^\circ}{360\circ}\pi r^2-\frac{\sqrt{3}}{4}r^2

After some scratch work we can calculate the area of the Reuleaux triangle:

A_R=\frac{r^2}{2}(\pi-\sqrt{3})

Meanwhile the area of a circle with diameter r (which we shall denote as ACirc) is:

A_{circ} =\frac{r^2\pi}{4}

Let's now compare the areas of two figures with the same perimeter, a circle of diameter r and a Reuleaux triangle with a "distance across" of r:

A_2-A_1= \frac{r^2}{4}(2\sqrt{3}-\pi)>0

Thus, through our comparison we have found out that the area of the Reuleaux triangle is smaller than that of the circle. In fact, it can be proven that of regular polygons the circle has the largest area for a given diameter. The Austrian mathematician Wilhelm Blaschke (1885-1962) proved that given any number of such figures of equal breadth, the Reuleaux triangle will always possess the smallest area, and the circle will have the greatest area.[7]

The Reuleaux and the Harry Watt Drill

Figure 4-4-3-1 Harry Watt Drill
Figure 4-4-3-1 Harry Watt Drill[11]

Another astonishing property of the Reuleaux triangle is that a drill bit in the shape of a Reuleaux triangle could bore a square hole rather than the expected round hole.[12] To paraphrase, the Reuleaux triangle is always in contact with each side of a square of appropriate size. Figure 4-4-3-1 gives such an example. The center of a Reuleaux triangle rotating in the square almost describes a circle——more exactly, it consists of four elliptical arcs. (The circle is the only curve of constant breadth that has a balanced center of symmetry.) Harry James Watt, an English engineer, recognized this in 1914. He received a patent (n0. 1241175), enabling these drills to be produced.[7]

Felix Wankel (1902-1988), a German engineer, built an internal combustion engine for a car that was the shape of a Reuleaux triangle and rotated in a chamber.[13] It had fewer moving parts and gave out more horsepower for its size than the usual piston engines. The Wankel engine was first tried in 1957 and then put into production in the 1964 Mazda.[13] Again, the unusual properties of the Reuleaux triangle made this type of engine possible.

As we look at the Envelope of the Reuleaux triangle in Figure 4-4-3-1, we notice it is almost a square, except for the four corners. What is the shape of those four corners? While we may be tempted to think the path of the geometric center (indicated by the black dot) is a circle, it is not. So what is it? We give answers to these two questions through the Parametric Equations of the curves.

Figure 4-4-3-2Left:First DKM Wankel Engine DKM 54 (Drehkolbenmotor), at the Deutsches Museum in Bonn, GermanyRight:A Wankel engine in Deutsches Museum in Munich, Germany
Figure 4-4-3-2
Left:First DKM Wankel Engine DKM 54 (Drehkolbenmotor), at the Deutsches Museum in Bonn, Germany
Right:A Wankel engine in Deutsches Museum in Munich, Germany[13]
Figure 4-4-3-3: Corners of the Reuleaux Triangle
Figure 4-4-3-3: Corners of the Reuleaux Triangle[12]
Figure 4-4-3-4: The Center of the Reuleaux Triangle
Figure 4-4-3-4: The Center of the Reuleaux Triangle[12]

We start by defining the four corners of the envelope of the Reuleaux triangle in the Cartesian coordinate system at (±1, ±1). The length of its side is 2. At the corner (-1, -1), the envelope of the boundary is given by the segment of the ellipse with parametric equations[12]:

x=1-\cos\beta-\sqrt{3}\sin\beta

y=1-\sin\beta-\sqrt{3}\cos\beta

Also shown in Figure 4-4-3-3, the ellipse is centered at (1, 1). The lengths of its semimajor axis and semiminor axis are as follows[12]:

a=1+\sqrt{3}

b=\sqrt{3}-1

In the same coordinate system, the path of the Reuleaux triangle's geometric center consists of a curve composed of four arcs of an ellipse (Wagon 1991). For a bounding square of side length 2, the ellipse in the lower-left quadrant has the parametric equations[12]:

x=1+\cos\beta+\frac{\sqrt{3}}{3}\sin\beta

y=1+\sin\beta+\frac{\sqrt{3}}{3}\cos\beta

As in Figure 4-4-3-4, the ellipse is also centered at (1, 1). The lengths of its semimajor axis and semiminor axis are as follows[12]:

a=1+\frac{1}{\sqrt{3}}

b=1-\frac{1}{\sqrt{3}}

Recent History of π

Ramanujan's Formula
Ramanujan's Formula

The more recent history of π includes the first electronic computation of π, which occurred in 1949 on the original ENIAC. It took 70 hours for ENIAC to compute the first 2037 decimal places of π. As technology continued to improve, so did the number of digits we were able to compute of π. In 1965, it was found that the fast Fourier transform (FFT) could be used to perform high-precision multiplications much faster than conventional schemes. [14] This advance dramatically lowered the time which it took the computer to calculate the digits of π. Despite all of these advances, until the 1970s all of the computer computations of the value of π still involved classic formulas. Some new infinite-series formulas were discovered by Ramanujan around 1910, but they were not well known until recently, when his writings were widely published. [14] One of the series is the formula for 1/π, which can be seen in the image to the right. Each term of this series produces an additional eight correct digits for π. In 1985, Bill Gosper used Ramanujan's formula to calculate seventeen million digits of π.


Formula for computing the n-th digit of π.
Formula for computing the n-th digit of π.

Before 1996, all mathematicians believed that if you wanted to determine the n-th digit of π, one would have to calculate all the previous digits before the n-th one. However, this is not true. In 1996, Borwein, Plouffe, and Bailey, found an algorithm which computed individual digits of π. Their algorithm only works for the digits of π which are hexadecimal (base 16) or binary (base 2). The algorithm they produced directly generates a single digit without needing to compute any of the previous digits. It can also be easily implemented on any modern computer, requires very little memory, and does not require multiple precision arithmetic software. Although this method is faster for finding a specific digit of π, it is not quicker than the best known method for computing all the digits of π up to a certain position. [14] The algorithm is based on the formula in the image to the left. In 1997, this algorithm was put into action by Fabrice Bellard, who computed 152 binary digits of π starting at the trillionth position. Bellard's work took twelve days to complete. [14]


The digits of π have been studied more than any other constant. In December 2002, Yasumasa Kanada of the University of Tokyo, along with a team of several other men, completed the computation of π to over 1.24 trillion decimal digits. Kanada and his team evaluated π with the use of formulas involving arc tan. [14] Kanada also studied the frequency of the ten decimal digits 0 through 9 in the first trillion digits of π. His findings reveal that the most frequent occurring digit is 8. The following table displays the total results of his study. [14]

References

  1. 1.0 1.1 1.2 1.3 1.4 1.5 Wikipedia, http://en.wikipedia.org/wiki/Pi
  2. Eymard, Pierre, and J. P. Lafon. The Number [Pi]. Providence, RI: American Mathematical Society, 2004. Print.
  3. 3.0 3.1 Beckmann, Petr. A History of Pi. Boulder: Golem, 1971. Print.
  4. 4.0 4.1 4.2 4.3 Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4., English translation by Catriona and David Lischka
  5. Rossi, Corinna Architecture and Mathematics in Ancient Egypt, Cambridge University Press. 2007. ISBN 978-0-521-69053-9
  6. 6.0 6.1 Health, L, Thomas. (2002). Euclid's Elements. Place of publication: Green Lion Press. ISBN 978-1888009187.
  7. 7.0 7.1 7.2 7.3 7.4 7.5 Posamentier, Alfred; Lehmann, Ingmar (2004). Pi: A Biography of the World's Most Mysterious Number. Place of publication:USA Prometheus Books.ISBN 978-1591022008.
  8. Polster, Burkard. (2004). Q.E.D.: Beauty in Mathematical Proof. Place of publication:USA Walker & Company. ISBN 978-0802714312.
  9. 9.0 9.1 9.2 9.3 Wikipedia, http://en.wikipedia.org/wiki/Buffon's_needle
  10. Wikipedia, http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)
  11. Weisstein, Eric W. "Reuleaux Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ReuleauxTriangle.html
  12. 12.0 12.1 12.2 12.3 12.4 12.5 12.6 Wolfram, http://mathworld.wolfram.com/ReuleauxTriangle.html
  13. 13.0 13.1 13.2 Wikipedia, http://en.wikipedia.org/wiki/Wankel_engine
  14. 14.0 14.1 14.2 14.3 14.4 14.5 Berggren, Lennart, Jonathan M. Borwein, and Peter B. Borwein. Pi : A Source Book. New York: Springer, 2004. Print.
Personal tools