Hermetic Geometry (Spheres II)

Objects in space can be either separate, intersecting, or touching. Spheres always intersect in circles, and the most fundamental case is when the angle of intersection is 90º.

Let’s look at intersecting circles first. Somewhat surprisingly, two circles that meet at one point at a right angle also meet at the other intersection point at a right angle. This observation allows to construct two families of circles such that any two circles from both families meet at right angles. The first family consists of all circles that touch the x-axis at the origin, and the second family consist of all circles that touch the y-axis at the origin.

For what’s to come, let’s also look at these circles just in the first quadrant.

The same procedure works in space with now three families of spheres. Each family consist of all those spheres that touch one of the coordinate planes at the origin. Viewing these in the first octant only and head-on, i.e. in the direction of the main space diagonal, shows patterns that disguise their origin quite well.

Again, like in Spheres I, the alienation becomes extreme when we pretend that the objects are part of some large piece of architecture. The walls of the facade show the same pattern as the intersecting circles.

This way geometry becomes hermetic. Instead of explaining a theorem, the statement is disguised in pure appearance.