# PYTHAGORAS (c.560 – c.480 BCE)

Sixth Century BCE – Greece

‘In a right-angled triangle, the square on the hypotenuse is the sum of the squares on the other two sides’

The Theorem may also be written as a general law:  a2 + b2 = c2  where c is the length of the hypotenuse of a right-angled triangle, and a and b the lengths of the other two sides. Pythagoras’ theorem is a starting point for trigonometry, which has many practical applications such as calculating the height of mountains and measuring distances.

c.525 BCE – Pythagoras taken prisoner by the Babylonians

c.518 BCE – establishes his own academy at Croton (now Crotone) in southern Italy

c.500 BCE – Pythagoras moves to Metapontum

Pythagoras was the first to prove the relationship between the sides of a right-angled triangle, but he did not discover it – it was known to Babylonians for nearly 1000 years before him.

His disciples, members of the semi-religious, philosophical school he founded, may have actually found many of the mathematical discoveries credited to Pythagoras. The inner circle of followers were known as mathematikoi and, unusually for the time, included women among its membership. An outer circle, the akousmatics, lived in their own homes and came in to the school by day.

Of the five key beliefs the Pythagoreans held, the idea that ‘all is number’ was dominant; the belief that reality at its fundamental level is mathematical and that all physical things like musical scales, or the spherical earth and its companions the stars and the universe, are mathematically related. Pythagoras was responsible for the widely held Greek belief that real knowledge had to be like mathematics – universal, permanent, obtained by pure thought and uncontaminated by the senses.

Because of the reverence with which the originator of the Pythagoreans was treated by his followers and biographers, it is difficult to discern legend from fact, such as the notion that he was the first to offer a three-part argument that the shape of the Earth is spherical:
The field of stars changes with the latitude of the observer; the mast of a ship comes into view before its hull as the ship approaches the shore from a distance; and the shadow of the Earth cast on the moon during a lunar eclipse is always round.

After Pythagoras, the idea of a ‘perfect’ mathematical interrelation between a globe moving in circles and the stars behaving similarly in a spherical universe inspired later Greek scholars, including ARISTOTLE, to seek and ultimately find physical and mathematical evidence to reinforce the theory of the world as an orb.

Attributed to the Pythagoreans is the discovery that simple whole number ratios of string lengths produce harmonious tones when plucked, probably the first time a physical law had been mathematically expressed.

Numerous other discoveries such as ‘the sum of a triangle’s angles is the equal to two right angles’ and ‘the sum of the interior angles in a polygon of n-sides is equal to 2n-4 right angles’ were made. They also discovered irrational numbers, from the realisation that the square root of two cannot be expressed as a perfect fraction. This was a major blow to the Pythagorean idea of perfection and according to some, attempts were made to try to conceal the discovery.

## PLATONIC SOLIDS

To the Pythagoreans, the fifth polyhedron had monumental significance. Outnumbering by one the number of recognized elements, the dodecahedron was considered to represent the shape of the universe.
A omerta, or code of silence, was imposed regarding the dodecahedron and divulging this secret to outsiders could mean a death penalty.

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# THE PLATONIC SOLIDS

The properties of solid figures have kept mathematicians occupied for centuries. Regular polyhedra are formed from regular polygons such as squares or triangles and mathematicians have failed to find any more than five of them.

Although they were defined by Pythagoras two hundred years before Plato was born, they are known collectively as the platonic solids, named in honour of PLATO by the geometer Euclid.

“THE PLATONIC SOLIDS – The regular polyhedron is defined as a three-dimensional solid comprising regular polygons for its surfaces – and with all its surfaces, edges and vertices identical. The five regular polyhedra are the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces) and the icosahedron (twenty triangular faces).”

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