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The numbers that intrigued the Pythagoreans the most were those found in musical ratios. The only harmonious musical intervals to them were those produced, for example, by dividing a string with a bridge so that the two resultant lengths were in the simple ratios of 2:1, 3:2, 4:3, or 5:4. They called these intervals the octave, the fifth, the fourth and the third. Thus the numbers 1, 2, 3, 4 and 5 could produce all of the musical intervals they considered pleasing. So what does all this have to do with astrology or astronomy? It turns out that another significant contribution of the Pythagoreans was their discovery of the fifth regular solid, the dodecahedron. For background, polygons (Greek "many sides") are closed plane figures with straight sides, and regular polygons are polygons all of whose sides are equal in length, like the faces of the solid figures above. A polyhedron (pl. polyhedra) is a 3dimensional solid whose faces are all polygons, and regular polyhedra, also called "regular solids" or "platonic solids" (see picture below) are those whose faces are all regular polygons. And, before I forget, Hippasus of Metapontum, credited with discovering the fifth regular solid, was the guy later drowned by the Pythagoreans for divulging the existence of irrational numbers, like √2. Amazingly enough, there can only be five regular solids (the ones pictured above), as was proven by Euclid (c.325  c.270 B.C.) in the last proposition of his Elements. That there are only five is also a direct result of Euler's Polyhedron Theorem establishing the relationship between the faces (F), the edges (E), and the vertices (V) of all convex polyhedra as: V  E + F = 2. (This relationship was first stated by Descartes in 1639, rediscovered by Euler about 1750, and later proven by the mathematical genius Augustin Louis Cauchy (1789  1857) at the age of 20). Before Euclid, the five regular solids were known to and described by Plato (c. 427  347 B.C.), who equated the tetrahedron (four equilateral triangular faces) with the element fire, the cube (six square faces) with the element earth, the octahedron (eight equilateral triangular faces) with the element air, the dodecahedron (12 regular pentagons as faces) with the stuff of which the heavens and the constellations are made, and the icosahedron (20 regular triangular faces) with the element water. And 1000 years before Plato they were known to the Neolithic peoples of Scotland who created stone models of them. For reasons very important to western culture and about to be explained, it also turns out that if you fit any regular solid just inside a sphere, all of its vertices touch the inside of the sphere. And conversely, if you fit a sphere just inside any regular solid, it simultaneously touches all the faces of that regular solid. 
Kepler’s Model of the Universe, Robert Lawlor, Sacred Geometry.
Now here’s why it matters. The Pythagoreans believed that the universe consisted of a central, spherical earth surrounded by one of the five regular solids, in turn surrounded by a crystalline sphere surrounded by another regular solid, and so on. Five concentric spheres in all surrounded the earth, each circumscribed about a regular solid. The planets and the stars were attached to these crystalline spheres, and as they rotated they created wondrous musical harmonies. The Pythagoreans believed most people couldn’t hear this "harmony of the spheres" because they had grown accustomed to it from birth, but that Pythagoras alone could indeed hear it. Over 2000 years later Johannes Kepler, most remembered for his three laws of motion, would spend most of his life trying to discover the harmony of the spheres. This Pythagoras and the Five Regular Solids page and much of this 600page website are excerpted from You and the Universe, a handmade, individualized fine art book on astrology, mythology and astronomy through which is woven each recipient's complete astrological reading. Continue to Pythagoras' proof that irrational numbers exist. 
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© Carl Woebcke: The Five regular SolidsThe Five Platonic Solids, 19912016. All rights reserved.