Aristotle's Organon

Aristotle was a Greek philosopher and scientist, a pupil of Plato, and teacher of Alexander the Great. The Organon (Instrument) refers to a collection of six of Aristotle's work in logic: Categories, Prior Analytics, De Interpretatione, Posterior Analytics, Sophistical Refutations, and Topics. Andronicus of Rhodes (c. 428-348 B.C.) determined the ordering of the six works around 40 B.C. Although Plato (c. 428-348 B.C.) and Socrates (c. 470-399 B.C.) delved into logical themes, Aristotle actually systematized the study of logic, which dominated scientific reasoning in the Western world for 2,000 years.
The goal of Organon is not to tell readers what it is true, but rather to give approaches to how to investigate the truth and how to make sense of the world. The primary tool in Aristotle's tool kit is the syllogism, a three-step argument, such as "All women are mortal; Cleopatra is a woman; therefore, Cleopatra is mortal." If the two premises are true, we know that the conclusion must be true. Aristotle also made a distinction between particulars and universals (general categories). Cleopatra is a particular term. Woman and mortal are universal terms. When universal are used, they are preceded by "all", "some", or "no." Aristotle analyzed many possible kinds of syllogisms and showed which of them are valid.
Aristotle also extended his analysis to syllogisms that involved modal logic - that is, statements containing words "possibly" or "necessarily." Modern mathematical logic can depart from Aristotle's methodologies or extend his work into other kinds of sentences structures, including ones that express more complex relationships or ones that involve more than one quantifier, such as "No women like all women who dislike some women." Nevertheless, Aristotle's systematic attempt at developing logic is considered to be one of humankind's greatest achievements, providing an early impetus for fields of mathematics that are close in partnership with logic and even influencing theologians in their quest to understand reality.  

Platonic Solids

A Platonic Solid is a convex multifaceted 3-D object whose faces are all identical polygons, with sides of equal length and angles of equal degrees. A Platonic solid also has the same number of faces meeting at every vertex. The best-known example of a Platonic solid is the cube, whose faces are six identical squares.
The ancient Greeks recognized and proved that only five Platonic solids can be constructed: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For example, the icosahedron has 20 faces, all in a shape of equilateral triangles.
Plato described the five Platonic solids in Timaeus in around 350 B.C. He was not only awestruck by their beauty and symmetry, but he also believed that the shapes described the structures of four basic elements thought to compose the cosmos. In particular, the tetrahedron was the shape that represented the fire, perhaps because of the polyhedron's sharp edges. The octahedron was air. Water was made up of icosahedra, which are smoother than the other Platonic solids. Earth consisted of cubes, which look sturdy and solid. Plato decided that God used the dodecahedron for arranging the constellations in the heavens.
Pythagoras of Samos - the famous mathematician and mystic who lived in the time of Buddha and Confucius, around 550 B.C. - probably knew of three of the five Platonic solids (the cube, tetrahedron, and dodecahedron). Slightly rounded versions of the Platonic solids made of stone have been discovered in areas inhabited by the late Neolithic people of Scotland at least 1,000 years before Plato. The German astronomer Johannes Kepler (1571-1630) constructed models of Platonic solids nested within one another in an attempt to describe the orbits of planets about the sun. Although Kepler's theories were wrong, he was one of the first to insist on a geometrical explanation for celestial phenomena.

Quadrature of the Lune

Ancient Greek mathematicians were enchanted by the beauty, symmetry, and order of geometry.Succumbing to this passion, Greek mathematician Hippocrates of Chios demonstrated how to construct a square with an area equal to a particular lune. A lune is a crescent-shaped area, bounded by two concave circular arcs, and this Quadrature of the Lune is one of the earliest-known proofs in mathematics In other words, Hippocrates demonstrated that the areas of these lunes could be expressed exactly asa rectilinear area, or "quadrature."
For the ancient Greeks, finding the quadrature meant using a straightedge and compass to construct a square equal in area to a given shape. If such construction is possible, the shape is said to be "quadrable" ( or "squarable"). The Greeks had accomplished the quadrature of polygons, but curved forms were more difficult. In fact, it must have seemed unlikely, at first, that curved objects could be quadrable at all.
Hippocrates is also famous for compiling the first-known organized work on geometry, nearly a century before Euclid. Euclid may have used some of Hippocrates' ideas in his own work, Elements. Hippocrates' writings were significant because they provided a common framework upon which other mathematicians could build.
Hippocrates' lune quest was actually part of a research effort to achieve the "quadrature of the circle" - that is, to construct a square with the same area as the circle. Mathematicians have tried to solve the problem of "squaring the circle" for more than 2,000 years, until Ferdinand von Lindemman in 1882 proved that it is impossible. Today, we know that only five types of lunes are quadrable. Three of these were discovered by Hippocrates, and two more were found in the mid-1700s.

Zeno's Paradox

For more than a thousand years, philosophers and mathematician have tried to understand Zeno's paradoxes, a set of riddles that suggest that motion should be impossible or that it is an illusion. Zeno was a pre-Socratic Greek philosopher from southern Italy. His most famous paradox involves the Greek hero Alchilles and a slow tortoise that Achilles can never overtake during a race once the tortoise is given a head start. In fact, the paradox seems to imply that you can never leave the room you are in.  In order to reach the door, you must first travel half the distance there. You;ll also need to continue to half the remaining distance, and half again, and so on. You won't reach the door in a finite number of jumps! Mathematically one can represent this limit of an infinite sequence of actions as the sum of the series (1/2 + 1/4 + 1/8 + ...). One modern tendency is to attempt to resolve Zeno's paradox by insisting that this sum of infinite series 1/2 + 1/4 + 1/8 is equal to 1. If each step is done in half as much time, the actual time to complete the infinite series is no different than the real time required to leave the room.
However, this approach may not provide a satisfying resolution because it does not explain how one is able to finish going through an infinite number of points, one after another. Today, mathematicians make use of infinitesimal (unimaginably tiny quantities that are almost but not quite zero) to provide a microscopic analysis of the paradox. Coupled with a branch of mathematics called nonstandard analysis and, in particular, internal set theory, we may have resolved the paradox, but debates continues. Some have also argued that if space and time are discrete, the total number of jumps in going from one point to another must be finite.

Pythagoras Founds Mathematical Brotherhood

Around the year 530 B.C., the Greek mathematician Pythagoras moved to Croton, Italy, to teach mathematics, music, and reincarnation. Although many of Pythagoras's accomplishments may actually being due to his disciples, the ideas of his brotherhood  influenced both numerology and mathematics for centuries. Pythagoras is usually credited with discovering mathematical relationships relevant to music harmonies. For example, he observed that vibrating strings produce harmonious sounds when the ratios of the length of the strings are whole numbers. He also studied triangular numbers ( based on patterns of dots in a triangular shape) and perfect numbers (integers that are the sum of their proper positive divisors). Although the famous theorem that bears his name, a^2 + b^2 + c^2 for a right  triangle with legs a, b and hypothenuse c, may have been known to the Indians and Babylonians much earlier, some scholars have suggested that Pythagoras or his students were among the first Greeks to prove it.
To Pythagoras and his followers, numbers were like gods, pure and free from material change. Worship of the numbers 1 though 10 was kind of polytheism for the Pythagoreans. They believed that numbers were alive, with a telepathic form of consciousness. Humans could relinquish their three-dimensional lives and telepatize with those numbers by using various forms of meditation.
Some of these seemingly odd ideas are not foreign to modern mathematicians who often debate whether mathematics is a creation of the human mind or if it's simply a part of the universe, independent of human thought. To the Pythagoreans, mathematics was an ecstatic revelation. Mathematical and Theological blending flourished under the Pythagoreans and eventually affected much of the religious philosophy in Greece, played a role in religion of the Middle Ages, and extended to philosopher Immanuel Kant in modern times.  Bertrand Russell mused that if it were not for Pythagoras, theologians wouldn't not have so frequently sought logical proofs of God and immortality.

Go

Go is a two-player board game that originated in China around 2000 B.C. The earliest written references to the game are from the earliest Chinese work narrative history, Zuo Zhuan ( Chronicles of Zuo), which describes a man in 548 B.C. who played the game. The game spread to Japan, where it became popular in the thirteenth century. Two players alternately place black and white stones on intersections of a 19x19 playing board. A stone or a group of stones is captured and removed if it's tightly surrounded by stones of the opposing color. The objective is to control a larger territory than one's opponent.
Go is complex for many reasons, including its large game board, multifaceted strategies, and huge numbers variations in possible games. After taking symmetry into account here, there are 32,940 opening moves, of which 992 are considered to be strong ones. The number of possible board configurations is usually estimated to be one of the order 10^172, with about 10^768 possible games. Typical games between talented players consist of about 150 moves, with an average of about 250 choices per move. While powerful chess software is capable of defeating top chess players, the best Go program often lose to skillful children.
Go-playing computers find it difficult to "look ahead" in the game to judge outcomes because many more reasonable moves must be considered in Go than in chess. The process of evaluating favorability of a proposition is also quite difficult because a difference of a single unoccupied grid point can affect large group of stones.
In 2006, two Hungarians researchers reported that an algorithm called UCT (for Upper Confidence bounds applied to Trees) could compete with professional Go players, but only on 9x9 boards. UCT helps the computer focus its search on the most promising move.

Pythagorean Theorem and Triangles

Today, young children sometimes first hear of the famous Pythagorean theorem from the mouth of the Scarecrow, when he finally gets a brain in MGM's 1939 film version of The Wizard of Oz. Alas, the Scarecrow's recitation of the famous theorem is completely wrong!
The Pythagorean theorem states that for any triangle, the square of the hypothenuse c is equal to the sum of the squares on the two (shorter) "leg" lengths a and b - which is written a^2 + b^2 = c^2. The theorem has more published proofs than any other, and Elisha Scott Loomie's book Pythagorean proposition contained 367 proofs.
Pythagorean triangles (PTs) are right triangles with integer sides. The " 3-4-5" PT - with legs of length 3 and 4, and a hypothenuse of length 5 - is the only PT with three sides as consecutive numbers and the only triangle with integers sides, the sum of whose sides (12) is equal to double its area (6).  After the 3-4-5 PT, the next triangle with consecutive leg lengths 21-20-29. The tenth such triangle is much larger: 27304197-273004196-38613965.
In 1643, French mathematician Pierre de Fermat (1601-1665) asked for a PT, such that both the hypothenuse c and the sum (a + b) had values that were square numbers. it was startling to find that the smallest three numbers satisfying these conditions are 4,565,486,027,761, 1, 061, 652,293,520, and 4,687,298,610,289. It turns out that the second such triangle would be so "large" that if its numbers were represented as feet, the triangle's legs would project from Earth to beyond sun!
Although Pythagoras is often credited wit the formulation of the Pythagorean theorem, evidence suggests that the theorem was developed by Hindu mathematician Baudhayana centuries earlier around 800 B.C. in his book Baudhayana Sulba Sutra. Pythagorean triangles were probably known  even earlier to Babylonians.

Les des

Les dés

Imaginez un monde sans nombres aléatoires. Dans les années 1940, la génération de nombres statistiques  aléatoires était essentielle parce qu’il était important pour les physiciens de simuler des explosions thermonucléaires. Aujourd'hui, de nombreux systèmes informatiques utilisent les nombres aléatoires pour sélectionner de façon objective des échantillons d’électeurs potentiels.

A l’origine, les dés, fabriqués à partir d'astragale (un os de cheville) d'ongulés, constituèrent l'une des premières méthodes de production de nombres aléatoires. Dans les anciennes civilisations, les dieux étaient censés contrôler le résultat des lancers de dés. Ainsi, les dés étaient invoqués pour prendre des décisions cruciales, depuis la sélection des dirigeants jusqu’au partage de propriétés à l’occasion d’héritage. Aujourd'hui encore, la métaphore de Dieu contrôlant les dés est ordinaire, comme le prouve la citation de l’astrophysicien Stephen Hawking : «non seulement Dieu joue aux dés, mais Il nous embrouille parfois en les jetant là où ils ne peuvent pas être vus."

Les plus anciens dés connus ont été exhumés avec un jeu de backgammon vieux de 5.000 ans près de la ville légendaire de Burnt au sud de l'Iran. La ville qui présente quatre étapes successives de civilisations détruites par des incendies, a été abandonnée en 2100 avant JC. Sur ce même site, les archéologues ont également découvert le plus ancien œil artificiel connu qui donnait un regard hypnotique au visage d'une antique prêtresse ou devineresse.

Pendant des siècles, les lancers de dés ont été utilisés pour enseigner les probabilités. Pour un seul lancer d’un dé à n côtés avec un nombre différent sur chaque face, la probabilité d'obtenir une valeur est 1 / n. La probabilité d’obtenir une séquence particulière de i nombres est 1 / n ^ i. Par exemple, les chances d’obtenir un 1 suivi d’un 4 sur un dé traditionnel est 1/6 ^ 2 = 1/36. En utilisant deux dés traditionnels, la probabilité d’obtenir une somme donnée est le nombre de façons d’obtenir cette somme divisée par le nombre total de combinaisons, c’est à dire la somme divisée par le nombre total de combinaisons. C’est pourquoi une somme égale à 7 est beaucoup plus plus facile à obtenir qu’une somme égale à 2.

Tic Tac Toe

The game of Tic Tac Toe (TTT) is among humanity's best-known and most ancient games. Although the precise date of TTT with its modern rules may be relatively recent, archeologists can trace what appear to be "three-in-a-row games" to ancient Egypt around 1300 B.C., and I suspect that similar kinds of games originated at the very dawn of human societies. For TTT, two players, O and X, take turns making their symbols in the spaces of a 3x3 grid. The player who first places three of his own marks on a horizontal, diagonal, or vertical row wins. A draw can always be obtained for the 3x3 board.
In ancient Egypt, during the time of the great pharaohs, boar games played an important role in everyday life, and TTT-like games are known to have been played during these ancient days. TTT may be considered an "atom" upon which the molecules of more advanced games of position were built through the centuries. With the slightest variations and extensions, the simple game of TTT becomes a fantastic challenge requiring a significant time to master.
Mathematicians and puzzles aficionados have extended TTT to larger boards, higher dimensions, and strange playing surfaces such as rectangular or square boards that are connected to their edges to form a torus (doughnut shape) or Kline bottle (a surface with just one side).
Consider some TTT curiosities. Players can place their Xs and Os on a TTT board in 9! = 362,880 ways. There are 255,168 possible games in TTT when considering all possible games that end in 5, 6, 7, 8, and 9 moves. In the early 1980s, computer geniuses Danny Hillis, Brian Silverman, and friends built a Tinkertoy computer that played TTT. The device was made of 10,000 Tinkertoy parts. In 1998, researchers and students at the University of Toronto created a robot to play three-dimensional (4 x 4 x4) TTT with a human.