37 min 45 sec
Cosmos: A Personal Voyage (1990 Update), The Backbone of Night [Episode 7]
Context: There can be an infinite number of polygons, but only five regular solids. Four of the solids were associated with earth, fire, air and water. The cube for example represented earth. These four elements, they thought, make up terrestrial matter. So the fifth solid they mystically associated with the Cosmos. Perhaps it was the substance of the heavens. This fifth solid was called the dodecahedron. Its faces are pentagons, twelve of them. Knowledge of the dodecahedron was considered too dangerous for the public. Ordinary people were to be kept ignorant of the dodecahedron. In love with whole numbers, the Pythagoreans believed that all things could be derived from them. Certainly all other numbers.
So a crisis in doctrine occurred when they discovered that the square root of two was irrational. That is: the square root of two could not be represented as the ratio of two whole numbers, no matter how big they were. "Irrational" originally meant only that. That you can't express a number as a ratio. But for the Pythagoreans it came to mean something else, something threatening, a hint that their world view might not make sense, the other meaning of "irrational".
