Music of the Spheres Introduction

I do not know what I appear to the world; but to myself I seem to have been only like a boy playing on a seashore, and diverting myself now and then by finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. -Sir Issac Newton

Introduction to the Music of the Spheres Exhibit

This Exhibit has two goals:
1. To demonstrate a previously undiscovered connection between solid geometry and music.
2. To inspire the viewer to follow his curiosity. Have you found an unusual pattern between two things? Do you wonder why something is the way it is? Then investigate! In other words, question the world in all its aspects.

The second goal I will address with a brief history of this exhibit. Several years ago I was entranced by the keyboard arrangement of the piano. The relation of the seven white keys to the five black keys in an octave struck me as very odd and mysterious. Why the alternating series of two and three black keys? Why seven
white keys? Why the lack of black keys between the white keys B-C and E-F?

Being a visually oriented person, I wondered if these relationships could bedemonstrated in another visual context besides the keyboard. I experimented with basic 2-d circle diagarams at first, but then hit upon the idea of applying the twelve tone names spanning an octave (the musical term for the distance from note A to when it repeats itself either above or below) to a three dimensional solid called the icosahedron, because it too has twelve parts (vertexes in this case) that are equally spaced in sequence, as well as equally spaced from its center point.

As you shall soon see, this combination produced some amazing results. It was tantamount to shining a beam of light through a prism and seeing the resulting spectrum, except in this case, the interaction illuminated both factors and showed a mysterious similarity between them.

Ultimatly my questions on the nature of the keyboard's arrangement would be answered more conventionally in books on western music theory and history. However, this relationship between the twelve tones and the Icosahederon is something I have continually persued, and is the reason why I became involved in computers and VRML in the first place. It showed me the world still has new mysteries lying in places we take for granted and I hope to show this particular
mystery to you.

Icosahedron? What's That?

It's one of a family of geometric solids (3 dimensional) called regular polyhedra. A regular polyhedron is composed of identical faces of equal area and it's vertex are equi-distant form its center point. In the case of the icosahedron, it's composed of twenty equilateral triangles, that collectively create twelve vertexes and thirty edges. The other four members are the tetrahedron, cube, octahedron and dodecahedron. They are collectively shown in the embedded vrml model below.

Click the red arrows on the left for dual relationships and the white circle on the right to show relationships with spheres.

The regular polyhedra have a trait known as dual relationship. This means that one polyhedra can define another by its vertexes being the center points
of another's faces. The cube's six faces and eight vertexes are in a dual relationship with the octahedraon's eight faces and six vertexes. The tetrahedron is in a dual relationship with itself. If it is in inverted, it's four vertexes become the center points of its four faces. Finally, the icosahedron's twelve vertexes and twenty faces are in a dual relationship with the dodecahedron's twelve faces and twenty vertexes. All the relationship structures you will see in the following links can be applied to the dodecahedron, but the icosahedron was chosen, because from a strictly pragmatic point of view, the resulting structures were more focused and visually easier to grasp.

The five regular polyhedra are also called "Platonic solids", because the ancient Greek philosopher Plato described them at length in his dialogue Timaeous. In an earlier dialogue , The Republic, Plato describes the famous Music of the Spheres, the supposed music sung by sirens on the wheels/ gears of the orbiting heavenly bodies. Because the icosahedron, as well as the other regular polyhedra, can be put inside a sphere with all their vertexes touching the circumfrence, I have given this exhibit its title: "Music of the Spheres".

Start here: A Simple Premise