In This Chapter

^ Identifying a president famous for a mathematical proof ^ Naming a world conqueror who did a proof while in exile ^ Pointing out some contemporary figures

Athematics has inspired so many people over the centuries. Picture a future U. S. president sitting through a wonderfully exciting meeting of Congress. Was he paying attention? No, he was discovering a proof of the Pythagorean theorem, of course! Other mathematicians came from wealthy families or monasteries. Mathematicians are found in all walks of life. Anyone can be a mathematician with just a bit of curiosity, respect for the basic mathematical rules, and willingness to stick her neck out a bit.

Pythagoras

How can Pythagoras be an Unlikely Mathematician when practically everyone knows his name? His contributions to the world of music and mathematics are legendary, although he would have preferred to keep his discoveries a secret. Pythagoras was a prophet and mystic. He established a secret society that had a strict code of conduct. Members of the society were vegetarians who were forbidden from eating beans (lentils). The Pythagoreans, as the society members were called, believed that odd numbers had male attributes and even numbers had female attributes. They based their worship on numerology and let it influence their way of life. The Pythagoreans believed that all numbers were Rational (could be written as a fraction), even though the Pythagorean theorem belies this with its need for numbers under Radicals (irrational numbers).

Napoleon Bonaparte

Emperor Napoleon Bonaparte is known for many things, but not many people know how much he contributed to mathematics. Napoleon was always a great supporter of mathematical inquiry and promoted mathematical study whenever he could. One of his better-known contributions is his discovery that, if you construct an equilateral triangle on each side of any other triangle, the centers of those equilateral triangles are the vertices of another equilateral triangle.

Figure 22-1 shows you three different triangles — one scalene, one isosceles, and one right. According to Napoleon, you construct an equilateral triangle (one with all the sides the same measure) on each side of any kind of triangle — the equilateral triangle has its sides measuring the same as that particular side of the base triangle. When the centers of the equilateral triangles are connected by segments, you see that Napoleon’s discovered another equilateral triangle formed in the middle.

Figure 22-1:

Napoleon kept himself busy while in exile.

Rene Descartes

Descartes was born into a wealthy family and received a thorough, general education at a Jesuit college. He studied law for a while but wasn’t really all that interested in it. For some years he traveled around with various military campaigns. Descartes wasn’t really a professional soldier. He took time off from his accompaniment of military campaigns for some interesting travel and study. He is known as the father of modern philosophy. His most serious interest in mathematics may have coincided with wanting to stay warm. He was traveling with the Bavarian army during a cold, winter campaign, and chose to stay in bed until 10 a. m. thinking about mathematical problems. Doesn’t sound like any army I know. In any case, Descartes made huge contributions to mathematics.

President James A. Garfield

President Garfield was the first ambidextrous U. S. president and the second president to be assassinated. He earned money to attend college by driving canal boat teams. He was a classics professor, then college president. He tired of the academic life, so he studied law and became a politician. While a member of the House of Representatives, in 1876, he discovered a novel proof of the Pythagorean theorem. Figure 22-2 shows his construction of a trapezoid, starting with a right triangle. His proof involves the areas of the triangles and trapezoid.

Figure 22-2:

President Garfield’s proof.

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President Garfield was shot in 1881 and died two months later of an infection. Alexander Graham Bell tried to find the bullet with a metal detector that he had invented, but failed — probably due to the metal in the president’s bed frame.

Charles Dodgson (Lewis Carroll)

More famous for his Alice in Wonderland And Alice through the Looking-Glass, Charles Dodgson is also well-known in mathematics circles for his work in probability, theory of elections, and algebra. Charles Dodgson studied at and later taught at Christ Church College, Oxford. He spent most of his time teaching and tutoring in algebra and developed study materials for students who were struggling with the material. His two Alice Books have many references to mathematics and logic. Dodgson liked working with children and amused them with his stories and word games — including an early version of Scrabble.

M. C. Escher

M. C. Escher was a Dutch graphic artist. His works are easily recognizable because of the many impossible-looking constructions and titillating tessellations. Figure 22-3 shows an example of the type of tessellation that Escher might have produced. (A Tessellation Is a tiling or filling of the plane with figures that leave no gaps between them.)

Escher was born in 1903 in the Netherlands and lived in and traveled to many other countries as world and political events affected his life. Although he didn’t have any formal mathematical training, his work has strong mathematical components including order and symmetry. His journey to the Alhambra in Spain resulted in his trying to improve upon those artworks using geometric grids as a basis for his own work. His art took many forms from his earliest work in 1937 until his death in 1972.

Sir Isaac Newton

Isaac Newton is probably best known for his discovery of the Law of Gravity, supposedly due to an apple falling on his head. Whether the apple story is true or not, his mathematical discoveries are even more remarkable, because most of his work was done during a two-year period when he had retired to the countryside to think and wait out the bubonic plague that was sweeping Europe. Even more startling is the fact that this two-year period ended with his 25th birthday.

Newton is recognized as the co-inventor of calculus. Both Newton and Gottfried Leibniz discovered calculus at the same time, and independently,

But Newton waited about 20 years to publish his findings, while Leibniz published almost immediately. Newton also discovered three laws of motion, the Corpuscular Theory of Light, and the Law of Cooling. He built telescopes and used them to calculate orbits of planets, but Newton’s main interest was really alchemy. After his early discoveries in mathematics and physics, he really did little more to add to the knowledge in these fields.

Marilyn Vos SaVant

Marilyn vos Savant’s column Ask Marilyn Is found weekly in Parade Magazine. She started writing this column after being featured in a Parade Article for her high IQ and then responding to a selection of questions in a follow-up article. Marilyn solves mathematical and logical problems in her column and also answers questions on physics, philosophy, and human nature.

Perhaps you remember her response to what has been dubbed the "Monty Hall problem" (see the nearby sidebar). In a 1990 column, she responded that you’d have a better chance of winning if you switched doors. This lead to all sorts of responses from academics and other readers — much of it criticism of her answer. She was, of course, found to be correct.

Marilyn, in addition to writing her column, is Chief Financial Officer of Jarvik Heart and assists her husband, Robert Jarvik, with cardiovascular disease research.

The Monty Hall problem

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, and behind the other two doors are goats. You pick a door, say door #1, and the host, who knows what’s behind the doors, opens another door, say door #3, which has a goat. He says to you: "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

Marilyn’s answer was that you should switch, giving you a two-thirds chance of winning rather than one-third, if you stayed with door #1.

Why was Marilyn correct? Consider the situation where the car is behind door #1 and goats are

Behind doors #2 and #3. The game show host is always going to show you one of the goats.

If you pick door #1, the host will show one of the goats behind door #2 or door #3, and if you switch, you lose.

If you pick door #2, the host will show you door #3, so if you switch to door #1, you win.

If you pick door #3, the host will show you door #2, so if you switch to door #1, you win.

You win two-thirds of the time if you switch. This same chance will appear, no matter where you put the car and where you put the goats.

Leonardo da Vinci

When you think of Leonardo da Vinci, you probably think first of the artist. As an artist, Leonardo turned to science as a means of improving his artwork. His study of anatomy and nature led to his remarkably realistic paintings. He was recognized as an inventor, scientist, engineer, musician, mathematician, astronomer, and painter.

His interest in the mathematics of art and nature led him to show how the different parts of the human body are related by the golden rectangle. Leonardo believed that artists should know the laws of nature as well as the rules of perspective.

Martin Gardner

Martin Gardner first came to my attention with his Mathematical Games Column in Scientific American, Which he wrote for about 25 years. In addition to this column, he has published over 60 books.

Martin Gardner grew up in Oklahoma, served in the U. S. Navy during World War II, and later earned his bachelor’s degree in philosophy from the University of Chicago. He decided to try for a life as a freelance writer after selling a humorous short story to Esquire. His second sale was a story based on mathematical topology — a story in science fiction. He is considered to be almost single-handedly responsible for creating the interest in recreational mathematics in the later part of the 20th century. Some subjects that he has popularized are

Flexagon: A Flexagon Is a hexagon made up from a long strip of equilateral triangles (most easily constructed from adding-machine tape). Folding and refolding reveals the three different faces of a Trihexa-flexagon, The six faces of a Hexahexaflexagon, And so on.

Uu Polyomino: A Polyomino Is a grouping of squares — three, four, five, and so on — such that no grouping is the same shape as any other, even when the grouping is flipped or rotated.

IU Soma cubes: Soma cubes Are the three-dimensional versions of polyomi-noes. Cubes are stacked or otherwise connected, producing different shapes that can’t be duplicated by any rotations or flips of the grouping.

IU Hex: The game of Hex Is played on a game board consisting of hexagons. Players take turns choosing hexagons (usually with different-colored game pieces) trying to form a path from one side of the board to the other.

LU Tangram: A seven-piece Tangram Starts out as a square. The different pieces — triangles, squares, parallelogram, and so on — are rearranged to form other shapes and pictures.

I Penrose tiling: A Penrose tiling Consists of Rhombi (a rhombus has all four sides the same measure) that appear to have no pattern or symmetry but that, in fact, have repeated patterns within the tiling.

I Fractal: A Fractal Is a geometric shape that can be continuously broken down into parts that are reduced copies of the original shape. The book Jurassic Park Introduced the dragon fractal and referred to chaos theory. (I’ll leave you with those two topics to search out on your own!)