The Platonic solids: a.k.a. THE DICE of the GODS

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The patterns:

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The tetrahedron or "regular pyramid".

The cube, sometimes called a "hexahedron".

The octahedron.

The icosahedron.

The famous dodecahedron.

**The HISTORY:**

This excerp is taken from the book

String, Straightedge and Shadow : The...Story of Geometry

By Julia e. Diggins

The Viking Press Inc.- New York 1965

As time went on, the Pythagoreans made even more exciting discoveries-and gave them strange cosmic meanings. This curious blend was characteristic of Pythagorean geometry. For the initiates of the Brotherhood were seeking a special key to the universe in this wonderful new realm of numbers and abstract forms: triangles, circles, squares, spheres, and the more elaborate forms they made themselves. And their search had
a thrilling climax. After long and painstaing experiments, they discovered
the The full tale of these five solids can only be guessed at from bits of legend and history, for all the experiments were top secrets, of course. To impress this on newcomers, perhaps the first thing they were shown was how to make a mystic "pentagram," the emblem that members of the Order wore on their clothing. By means of a secret device ( which we will explain later) a five-sided figure, or pentagon, was traced on cloth. Then its points were connected with diagonals to make a five-pointed star. Finally, around the five points of the star were placed the letters of the Greek work for health, (hygeia), from which we get the word "hygiene". This was the sacred symbol of the Pythagorean Order - the "magic pentacle" that remained a favorite device of sorcerers and conjurors for many centuries. But it was also an experimental discovery: The first known use of letters on a geometric figure. Possibly the next experiment shared with newcomers was a basic one with tiles. Ordinary floor tiles had yielded the easiest example of the Pythagorean theorem. So the Secret Brotherhood went on with a painstaking study of these close-fitting forms that covered many Greek floors. They made loose tiles
of variious shaped and placed them in patterns on the ground. And they
reached a striking conclusion. There were only If they tried pentagons, they got a beautiful blossomlike design, but there were gaps between the tiles, and tiles of more than six sides would always overlap. No other regular geometric forms of the same size and shape could be so combined. They explained this mystery to the newcomers: "Since there are four right angles (360 degrees) around a point, you can only use forms whose corner angles together will make that total. There are just three possibilities: six equilateral triangles with 60 degree angles, four squares with 90 degree angles, and three hexagons with 120 degree angles." From this simple experiment came the fascinating idea of making "solid angles" by fastening tiles together with mortar, or gluing together shapes of wood, or sewing together pieces of leather. And this led to building shapes with the solid angles. They called them
Finally, someone
got the right inspiration-five equilateral triangles for the top, and
five triangles for the bottom, and then a center band of ten more tianles
based on the old Babylonian pattern. They had made an The Dodecahedron:
This last form was made with the pentagons so dear to the Pythagorean
Order. They used that flowerlike pattern of a single pentagon surrounded
by five others-the tiles that would not fit together on the flat floor.
But if the surrounding ones were lifted up, then all six pentagons fitted
perfectly in a solid cuplike shape. This could be capped by an inverted
one just like it to yield the most difficult and beautiful of the five
regular solids: the These five shapes created a great stir among the first geometers who studied them. Men examined them in fascination and awe, handling them, turning them around in different positions, looking through them as if they were glass. And it was inevitable that the Pythagoreans should finally give them mystical meanings. By the time the Secret Brotherhood had spread to many towns and islands of Sicily and southern Italy. The Sicilian members were friendly with another strange teacher who lived near Mount Etna- Empedocles, who dressed all in purple, gave away his money, and did scientific experiments. Empedocles taught that the world was made of earth, air, fire, and water, and the first four regular solids came to be identified with these "elements." We know the identification from a famous passage in one of Plato's Dialogues, where it is made by a Pythagorean from Locri in the south of Italy. About the fifth solid, there were many weird stories. Its existence was kept secret as it seemed to require a fifth "element." The esoteric reasoning, as repeated later, went something like this: "The cube, standing firmly on its base, corresponds tot he stable earth. The octahedron, which rotates freely when held by its two opposite corners, corresponds to the mobile air. "Since the regular pyramid has the smallest volume for its surface, and the almost spherical icosahedron the largest, and these are the qualities of dryness and wetness, the pyramid stands for fire and the icosahedron for water." As for the last-found regular solid, with its twelve faces, "Why not let the dodecahedron represent the whole universe, since the Zodiac has twelve signs!" Such notions were typical of that age. And more than two thousand years later, the famous astronomer Kepler was still so awed by the unique properties of the five regular solids that he tried to apply them as planetary orbits: he assigned the cube to Saturn, the pyramid to Jupiter, the dodecahedron to Mars, the icosahedron to Venus, and the octahedron to Mercury, and he designed a machine to show this! Of course, the attempt was a failure. Yet even today, these solids seem almost magical in their beauty and their interrelations. in the first place, it is quite startling that there are only five. An infinite number of regular polygons can be inscribed in a circle-their sides becoming so small that they approach the form of the circle itself. But it is not so with regular convex polyhedra inscribed in a sphere. There are only these five posible shapes, and no others. And these five shapes are connected with one another in a most remarkable way. All five can be fitted together, one inside the next, like the compartments of some magic box. And they are further linked by a strange inner harmony. They can be inscribed in themselves or each other, in certain endless thythmic alternations. So it's no wonder the five regular solids were long referred to as the "dice of the gods." |

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