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Into the Magical Realm of Mathematics

An exciting voyage through time and imagination in the land of numbers, where we meet the philosophers of  ancient Greece and find that nature is composed of mathematical forms

Dana Ashkenazi

Natural as They Come

The language of numbers is the only one almost all human beings can understand, even though they may not necessarily live in the same country. Math is as ancient as humankind, and it appears that even in very early periods, people knew how to use natural numbers (1, 2, 3 and so on). The ability to count represented a breakthrough in the development of human thinking. A key role in this development was played by the Greeks.

Ancient Greek Philosophers

The  ancient Greek scholar Pythagoras is justly considered one of the greatest mathematicians of all time. Pythagoras established a community of hundreds of students, whom he used to meet in secret to discuss mathematical riddles. The veil of secrecy surrounding Pythagoras' school spawned many legends related to its rituals. One of Pythagoras's students was the beautiful Theano, whom he was later to marry. Pythagoras and his learned students believed that understanding the relationships among numbers will help them decipher the secrets of the universe and bring them closer to the gods. These ancient Greeks adored math, made important mathematical discoveries and created the basis for the mathematics – and even the music – of today.

Fig. 1 - Pythagoras and his learned students, excerpt from Rafael's The School of Athens, 1510

The ancient Greeks discovered a ratio called the "golden ratio" or the "golden section". If we break a straight line into two sections, a and b, according to the golden ratio, the ratio between them will be identical to that between the original line and the longer section. In turn, a and b can form the sides of a golden rectangle, considered to be of ideal proportions.

The secret symbol of Pythagoras and his sect was the pentagram – a five-pointed star made up of five identical isosceles triangles. Like the golden rectangle, the pentagram was also considered of perfect proportions and beauty. Each triangle in the pentagram is golden sectioned: the ratio between each of its side and its base is 1.618.

Pentagrams may be naturally found in flowers, starfish and fruit. In Greek, pentagram simply means "five lines". Pentagrams were described already in the ancient writings of Mesopotamian scholars, as early as 3,000 BC. The ancient Babylonians and Greeks attributed supernatural and magical powers to the pentagram, and many used to wear jewels with the pentagram symbol as talismans against evil spirits. Thanks to the pentagram's unique inner proportions, this five-pointed star shape may be redrawn within itself ad infinitum.

Fig. 2 - The golden sectioned

Fig. 3 - The pentagram, or five-pointed star, was the secret symbol of the Pythagorean sect. This shape may also be found in nature. The hibiscus flower, the starfish and the carambola fruit section are all shaped this way. Thanks to the pentagram's unique inner proportions, this five-pointed star shape may be redrawn within itself an infinite number of times.

Pythagoras tried to capture the logic underlying numbers, and even aspired to reduce the whole of nature into numerical terms. He discovered, for instance, a distinct numerical relation between the length of musical strings and the sounds they produce. Thanks to Pythagoras' genius, numbers were no longer just a means to count and calculate, and began to attract special attention. Pythagoras acquired his mathematical knowledge in his voyages throughout the ancient world. Many of the mathematical tools he would later use for his discoveries were actually adopted from the Cretans, the Spartans, the Babylonians and the Egyptians.

Pythagoras gave his name to one of the best-known mathematical theorems. The Pythagorean Theorem maintains that the square of the hypotenuse (or longer side) of a right triangle is equal to the sum of the squares on the other two sides. In fact, it was the Babylonians who had discovered this theorem a thousand years before the Greeks, but Pythagoras was the first who managed to prove it theoretically. Almost every child in the world studies the Pythagorean Theorem in her early years in school.

Fig. 4 - Pythagorean Triangle

The ancient Greeks wrote many books about a variety of mathematical topics. They admired the perfect proportions of the golden rectangle, whose sides are proportioned identically to the sides of the isosceles triangles of which the pentagram is composed. The Greeks saw the golden rectangle as the epitome of beauty and a symbol of pure perfection. This beauty found its expression in classic Greek architecture, as in the Parthenon.

 For your information… The Parthenon is a Greek temple situated in ancient Athens' Acropolis (or "high  district") and is considered to be the most famous building of ancient Greece, and one of the most renowned in the world today.  The golden rectangle can still be identified in many places in our world today, as it is represented in many art pieces, flags and buildings.   Fig. 5 - Front of the Parthenon, and the golden section

 For your information…Geometry is an area of math dealing with the study of two- and three-dimensional forms. The word literally means "land measurement" (Geo means land, while metrio means to measure).

The Game of Kings
Mathematical thinking also exists in games. Chess, also called the Game of Kings, may be seen as a mathematical duel. It is one of the most favorite and challenging games ever invented by man. The battleground in this game is comprised of a checkered board with 64 alternately black and white squares. Each player has game pieces of a different color: a king, a queen, two bishops, two knights, two rooks, and a raw of eight pawns. Each piece moves differently across the chessboard.

German nobles playing chess, 1320

The Game of Kings – prints from 1474

The Game of Kings – prints from 1474

The name originates shah, the Persian word for king. The object of the game is to capture the adversary's king. The players play in turns, when they may move any piece according to the rules of the game. When the king is attacked by one of the adversary's pieces, this is called "check". When checked, the player under attack must do anything in his ability to protect the king against the threat, by moving the king away, capturing the attacking piece, or blocking the attack. When all attempts prove futile, the situation is called "mate" (from Persian for "dead"), and the game is lost.

Twelfth-century Scandinavian chess king

Excelling in chess requires concentration skills, creativity, imagination, logical thinking and sophistication. The game spawned a logical area called chess problems, which studies the way a desirable situation may be reached given a certain situation. The objective of chess problems is also to hone the skills of chess players. Apart for chess problems, there is also a branch of mathematics called chess riddles. These are mathematical riddles to all intents and purposes, presented and resolved using the chess board, pieces and rules.

 For your information…Lewis Carroll (1832-1898), a famous author and mathematician, used mathematical concepts in his books. In Alice's Adventures in Wonderland (1865) and the sequel Through the Looking Glass (1871), Carroll described an adventurous voyage by a girl named Alice, who reached strange worlds and met imaginary creatures. Among other things, she meets a white rabbit rushing along with a pocket watch and the Queen of Hearts. She drinks strange concoctions that make her shrink and expand and discovers mutual effects between her and her environment. Mathematicians are particularly fond of these two books since they discuss scientific concepts of expansion and shrinkage (physical dimensions) as well as the notion of time. In Through the Looking Glass, Carroll used chess as a setting for his story, as Alice met unfriendly game pieces. Cover of The Adventures of Alice in Wonderland (from 1899)   Illustration from Through the Looking Glass

 For your information…The 27-minute Disney film Donald in Mathmagic Land (1959), directed by Hamilton Luske, deals with the invention and rules of the language of math. It was nominated for the 1960 Academy Award. Through impressive animation and thanks to the Duck's unique sense of humor, children were invited to a wonderful adventure in the land of numbers.

Mathematics in Nature and Art
Many mathematical forms may be found in nature. The beehive is composed of thousands of hexagonal beeswax cells, in which the queen bee lays thousands of eggs, destined to become the next generation of bees.
The beautiful crystalline snowflakes are designed in myriad spectacular geometrical shapes, which may be seen through the microscope.

Beehive composed of hexagonal beeswax cells

Snowflake through the microscope: the crystals of which snowflakes are made create a spectacular collection of diverse and complex forms.

Mathematically, snowflakes may be described using equilateral triangles. When the mathematical snowflake is created, we replace the middle third of the triangle's side with two sections, the length of which is identical to the section removed. If we repeat this step almost an infinite number of times, we will have a snowflake whose circumference also approaches infinity. This manner of breaking up and repeating the same form on a different scale is called fractalization. Fractal is a geometrical form structured in a cyclic fashion, where on each level of detail, the shape obtain is identical to the original. The snowflake fractal was first presented in 1904 by Swedish mathematician Helga von Koch. Unlike the infinite theoretical snowflake fractal, natural snowflakes are finite, and the broad variety of their shapes is the result of material flaws.

Steps of creating a mathematical snowflake, called snowflake fractal, using a repetitively restructured geometric form.

God Save the Queen
Mathematics is called the Queen of Sciences, but it would actually be more accurate to call it the language of exact sciences. Math's unique status in the realm of science preoccupied many philosophers of science. The objective of science is to understand reality, and the language of mathematics enables this. Mathematical thinking begets new scientific discoveries, each potentially leading to others, just as in fractal evolution.
I wish to conclude with a saying by Italian physicist and philosopher Galileo Galilei, who lived four centuries ago: "Mathematics is the language with which God has written the universe".

Would you like to know more?
Bergamini, David, Mathematics, Life Science Library (1969).
Donald in Mathmagic Land, directed by Hamiltun Luske, Disney Studios (1959).

Dr. Dana Ashkenazi received her Ph.D in Mechanical Engineering from Tel-Aviv University and her first and second degrees in Materials Science and Engineering from Ben-Gurion University.She  conducts research and lectures  in the field of Materials Science and Engineering  and also has great interest in popular science, art and human culture.