Archive for October, 2008

Here is a link to a video of Benoit Mandelbrot, the father of fractals and chaos math, together with Nassim Nicholas Taleb, author of The Black Swan, discussing the economic crisis on PBS.

Click here for link to video

© Karthik Narayanaswami

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Some very interesting mathematical biology:

Scientists have known for nearly two centuries that larger animals have relatively slower metabolisms than small ones. A mouse must eat about half its body weight every day not to starve; a human gets by on only 2%. The first theories to explain this trend, developed in the late nineteenth century by the German nutritionist Max Rubner and the French physiologist Charles Richet, were based on the ratio between an animal’s surface area, which changes with the square of its length, and its volume, which is proportional to its length cubed. So large animals have proportionately less surface area, lose heat more slowly, and, pound for pound, need less food. The square-versus-cube relationship makes the area of a solid proportional to the two-third power of its mass, so metabolic rate should also be proportional to mass2/3. For many years, most biologists thought that it was.

But in 1932, Max Kleiber, an animal physiologist working at the University of California’s agricultural station in Davis, re-examined the question, and found that, for mammals and birds, metabolic rate was mass0.73—closer to three quarters than two thirds. Kleiber looked at animals ranging in size from a rat to a steer. By the mid-1930s, other workers had put together a “mouse to elephant” curve that supported the three-quarter-power law, and by the 1960s, the plot had been extended for everything from microbes to whales, still seeming to show the same relationship. Quarter-power scaling also began to stretch beyond metabolic rate. Biological times, such as lifespan and heart rate, were found to be proportional to mass1/4, and fractions related to one-quarter show up in other scaling relationships: the diameter of the aorta and tree trunks is proportional to mass3/8, for example.

It was, however, much harder to find a theoretical reason for why metabolic rate should be proportional to mass3/4—and more generally, why quarter-power scaling laws should be so prevalent in biology. The impasse meant that by the mid-1980s interest in scaling had waned. But it sparked back into life in 1997, when two ecologists—James Brown of the University of New Mexico, Albuquerque, and his graduate student Brian Enquist, now at the University of Arizona, Tucson—and a physicist, Geoffrey West of the Santa Fe Institute, developed a new explanation of why metabolic rate should equal the three-quarter power of body mass.

West, Brown, and Enquist’s theory is based on the structure of biological distribution networks, such as blood vessels in vertebrates and xylem in plants. The trio assumed that metabolic rate equals the rate at which these networks deliver resources, and that evolution has minimized the time and energy needed to get materials from where they are taken up—the lungs or roots, for example—to the cells. They also assumed that, although organisms vary greatly in size, the terminal units in their distribution networks, such as blood capillaries or leaf stalks, do not.

Bigger plants and animals take longer to transport materials, and so use them more slowly. In West, Brown, and Enquist’s model, the maximally efficient network that serves every part of a body has a fractal structure, showing the same geometry at different scales. And the number of uniform terminal units in such a network—and so the rate at which resources are delivered to the cells—is proportional to the three-quarter power of body mass.

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Last week, I wrote about the Arab mathematician Al-Khwarizmi and the origin of the word algebra.  The European and American cultures received much more than just a word from Arab mathematicians.  In the 12th century, a man named Guilielmo Bonacci lived in Bejaia, Algeria, where he represented the interests of the traders of Pisa, Italy.  His son, Leonardo, was tutored in the Hindu-Arabic mathematics that Al-Khwarizmi had written about several hundred years before.  Leonardo, who became known as Fibonacci, wrote in his book Liber Abaci:

When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians’ nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.

It was this book, Liber Abaci, written in 1202, that led to the adoption of the Hindu-Arabic numeral system by the Europeans.  At the time, Europeans were still using Roman numerals, which are cumbersome for calculation.  The Hindu-Arabic numerals and the system of calculation that goes along with them are very useful in both business and science.

The second section of Liber abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.

In 1478, the portions of Liber Abaci that dealt with trade and currency conversion were translated into Italian, so that people who didn’t read Latin could understand the ideas as well.  This book, known as the Treviso Arithmetic, spread these ideas to a much wider audience than was possible through a book written in Latin.  This allowed the Italian merchants to keep much better accounts, enabling trade to flourish.

Previously, calculations had been made using an abacus, but with the Hindu-Arabic numbers and calculation system, people had a much easier time keeping track of their money.  Before the advent of the calculation system, merchants had kept a table in their business that was used as a makeshift abacus in order tally up sales totals.  This table was known as a counter.  Although the counters in today’s businesses are not directly used to tally things up, the machines that do this job are kept on the “counter.”

Fibonacci is probably best known for a series of numbers that bears his name – the “Fibonacci Series.”  This series is derived from a problem in Liber Abaci and is created by adding the two numbers in the series to obtain the next.  It starts with ones, which are added to get 2.  Then the 1+2=3, then 2+3=5, and 3+5=8 and so on.  The series begins 1,1,2,3,5,8,13, 21,…  There are many applications where this seemingly trivial series of numbers is useful.

One of the more surprising uses of the Fibonacci Series is in stock trading, where investors use the ratio of Fibonacci Numbers to analyze the behavior of stock, bond or commodity markets.

Here is a link from the Forbes Investopedia

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Abu Ja’far Muhammad ibn Musa Al-Khwarizmi was an Arab mathematician who lived and worked in Baghdad during the 8th and 9th centuries.  Not much is known of his life.  He is primarily recognized as the author of two books, one on arithmetical computation and one on algebra.  The word algebra comes from the title of his most famous book Hisab al-jabr w’al-muqabala or The Compendious Book on Calculation by Completion and Balancing.  In this book Al-Khwarizmi discussed methods of solving linear and quadratic equations, although he used prose writing to do this.  The notation that we use today in algebra class was developed by European mathematicians between 1400-1800.

Al-Khwarizmi’s other notable book was about the Hindu-Arabic place-value number system that we learn about in elementary school.  The Hindu used 10 symbols to represent the numbers 0-9 with positional or place values to represent larger numbers.  The other important subject of this book were the methods of calculation that we learn in elementary school.  These “algorithms” for calculation in the number system allow us to add, subtract, multiply and divide any collection of numbers.  The Arab title of the book is not known.  It was translated into Latin as Algoritmi de numero Indorum or, in English, Al-Khwarizmi on the Hindu Art of Reckoning.  The word “algorithm” comes from the Latin representation of Al-Khwarizmi’s name.

The usefulness of Al-Khwarizmi’s work is explained by the author himself

… what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

Here is a link to a biography of Al-Khwarizmi

Here is a link to a Wikipedia article on his book

Next week I’ll write about the importance of Al-Khwarizmi’s work to European mathematicians.

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Pythagoras of Samos

Many people have heard of the Pythagorean Theorem (that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse), and many think of it as a²+ b²= c²

It’s pretty clear that many cultures were aware of this relationship.  The concept appears in Egyptian, Babylonian, Indian and Chinese mathematics.  Pythagoras himself may have learned of this relationship during his time in Egypt or Mesopotamia.

Pythagoras eventually ended up in Crotona, in southern Italy and established a community based on certain principles.  The exact nature of the Pythagoreans community is not well known as they were very secretive about their practices.  It is known that the Pythagoreans studied four interrelated subjects very closely -Music, Geometry, Arithmetic and Trigonometry.

The Pythagoreans are sometimes called number mystics because numerology played an important part in their belief system.  Part of their system of belief involved the idea that all numbers or quantities were rational numbers.  That is, they could be represented as a ratio of whole numbers.  Two lengths like 1/2 and 3/4 that could be represented as the ratio of whole numbers were called commensurable because they could be layed off as distances against each other.  Two lengths of 3/4 would equal three lengths of 1/2.  The Pythagoreans believed that this was true for all numbers.

However, being familiar with Geometry and the Pythagorean Theorem, they were familiar with the diagonal of a square with a side of length 1.  So they knew that this distance was the square root of 2, but they couldn’t figure out how to represent this number as a ratio of whole numbers.  Finally, one of the Pythagoreans named Hippasus of Metapontum showed that square root of 2 is not commensurable and cannot be written as the ratio of whole numbers.  He also showed that this is true of the length of the side of a regular pentagon.

Here’s the general proof that square root 2 is irrational (from Ask Dr. Math)

To prove: The square root of 2 is irrational. In other words, there
is no rational number whose square is 2.

Proof by contradiction: Begin by assuming that the thesis is false,
that is, that there does exist a rational number whose square is 2.

By definition of a rational number, that number can be expressed in
the form c/d, where c and d are integers, and d is not zero.
Moreover, those integers, c and d, have a greatest common divisor,
and by dividing each by that GCD, we obtain an equivalent fraction
a/b that is in lowest terms: a and b are integers, b is not zero,
and a and b are relatively prime (their GCD is 1).

Now we have

[1] (a/b)^2 = 2

Multiplying both sides by b^2, we have

[2] a^2 = 2b^2

The right side is even (2 times an integer), therefore a^2 is even.
But in order for the square of a number to be even, the number
itself must be even. Therefore we can write

[3] a = 2f

Using this to replace a in [2], we obtain

[4] (2f)^2 = 2b^2
[5] 4f^2 = 2b^2
[6] 2f^2 = b^2

The left side is even, therefore b^2 must be even, and by the same
reasoning as before, b must be even. But now we have found that both
a and b are even, contradicting the assumption that a and b are
relatively prime. Therefore the assumption is incorrect, and there
must NOT be a rational number whose square is 2.

It is believed that Hippasus died by drowning shortly after his discovery of irrational numbers.  The exact sequence of events that led to his death are not well known:

They say that the man (Hipassus) who first divulged the nature of commensurability and incommensurability to men who were not worthy of being made part of this knowledge, became so much hated by the other Pythagoreans, that not only they cast him out of the community; they built a shrine for him as if he were dead, he who had once been their friend. Others add that even the god became angry with him who had divulged Pythagoras’ doctrine; that he who showed how the icosagon (that is the dodecahedron, one of the five solid figures) can be inscribed within a sphere, died at sea like an evil man. Others still say that the same misfortune happened on him who spoke to others of irrational numbers and incommensurability. Hyamblicus (or Iamblichus of Chalkis), De vita pythagorica 246-247

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Someone once said to me that higher mathematics was useless.  I replied that since it controlled every computer on the planet, it seemed pretty useful to me.  One of my favorite areas of math is abstract algebra.  Abstract algebra grew out of the work of French mathematician Evariste Galois around 1830.

In the 1500’s Italian mathematicians had worked to find formulas to solve the cubic (x³) and quartic (x^4) equations.  Then, for almost 300 years, mathematicians worked to find a solution for the quintic, or fifth degree equation.  Finally, in the early 1800’s, the work of Galois and Abel proved that there is no general solution for equations with powers of x higher than 4.

From this work grew the field of abstract algebra.  Prior to Galois and Abel, between 1750 and 1820, many mathematicians contributed significant ideas to what would become known as abstract algebra.  Euler, Gauss, Legendre and Lagrange (1750-1820 or so) all worked with what is known as modular arithmetic.  Modular systems and discrete mathematics became very important in the mid-20th century with the development of the digital computer.

Here’s a link to a paper I wrote about some of these ideas

Mathematics, Communication and Secrecy

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