Circles and spheres. Spheres and circles. My mind is spinning. Circles seems to be more free than spheres. Two circles can entangle each other without touching. Spheres can't do such things. But sphere have inside and outside. Circles do not know about such concepts. Well, maybe circles don't know, but they also have inside (and outside). 1D circles may carry a group structures, 2D spheres can't. There are also planes. They can be thought as spheres of infinite radius. Planes have two sides, but neither of them deserves to be called outside. Or both can be called such. Straight lines can be considered as circles of infinite radius. Points are spheres of zero radius - poor degenerate beings. So far talking about space I was dealing with points and linear concepts - like multivectors. It is time to get beyond linearity, and we will start our new adventure. It will be about the music of spheres. We will delve into the Lie sphere geometry. Wikipedia has an article on this subject. You can find some relevant references there. Wikipedia has also an article on Music of the Spheres. There we read:
The musica universalis (literally
universal music), also called music of the spheres or harmony of the
spheres, is a philosophical concept that regards proportions in the
movements of celestial bodies—the Sun, Moon, and planets—as a form of
music. The theory, originating in ancient Greece, was a tenet of
Pythagoreanism, and was later developed by 16th-century astronomer
Johannes Kepler. Kepler did not believe this "music" to be audible, but
felt that it could nevertheless be heard by the soul.
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| The music of the spheres |
My soul (if I have one) is mathematical. So that is how we will be proceeding - through the math, I am only learning this subject. I wanted to learn it long ago, having a deep feeling of its importance, but I was always postponing it - for later. And now it comes to the fruit. Whatever I learn, I will write about it. Little by little, step by step. At first it will be just geometry. With time algebra (geometric algebra) will join in. Spinors will reappear in new clothes. So, for a while, we leave aside Spin Chronicles. For those curious I will start with the approach described in the paper "Lie Sphere Geometry in Hilbert Spaces" by Walter Benz, Result. Math. 40(2001), 9-38. It is a real poetry.
In the article on Musica Universalis Wikipedia quotes William Shakespeare:
Sit, Jessica. Look how the floor of heaven
Is thick inlaid with patines of bright gold:
There's not the smallest orb which thou behold'st
But in his motion like an angel sings,
Still quiring to the young-eyed cherubins;
Such harmony is in immortal souls;
But whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it.
Sit, Jessica. Look how the floor of heaven
And indeed heaven outside, soul inside, and yet they are one. Like in Lie sphere geometry that will come.
