## A deep yearning for submission.

te can comprehend complex variables and differential geometry?

Toy Euler: It is your challenge to explain.

Faustus Mortal: Hmm. Let me describe the picture. Consider a ball, a solid sphere, in three-dimensions (3D). Imagine a solid, hard gray ball. It may not be a perfect sphere but as far as the eye can tell it is. Now there is a tiny "correction factor" but forget that for the moment.

Toy Euler: Oh.

Faustus Mortal: Now sit the ball on a plane. Let it touch at the origin and the axes be "x" from right to left, "z" from back to front, and "y" from the bottom to the top. "x" turns into "y" to drive "z" out of the blackboard. A "right-hand coordinate system."

Faustus Mortal: Sit the ball on the "x-z" plane. Let the point it sits on be the origin, O, or (0,0,0).

Faustus Mortal: Make the ball have radius of one. (r = 1).

Toy Euler is confused.

Faustus Mortal: OK take a cue ball from a billiard table.

Toy Euler: Got it.

Faustus Mortal: Chalk in two lines on the floor. "x" from origin to the right and "z" from the origin forward.

Toy Euler: And "y" points upwards. That's not the way we learned it in high school.

Faustus Mortal: Forget high school.

Toy Euler: No way.

Faustus Mortal: OK, good bye.

Toy Euler: Wait, wait.

Faustus Mortal: Pretend that "y" goes from the origin through the ball and points upwards.

Toy Euler: So a ball sits at (0,0,0) for (x,y,z)?

Faustus Mortal: The floor is the x-z plane. Now think of a line drawn from the origin tangent to the ball at the origin of length 4*PI().

Toy Euler: "4pi"?

Faustus Mortal: Yes.

Faustus Mortal: In the x-z plane see how the line can "spin out" a circle of radius 4*PI() without moving the ball? Just spinning it on its y-axis.

Toy Euler: OK.

Faustus Mortal: That is the equator.

Toy Euler: A billiard ball sitting on a flat disk with radius about 12 times the radius of the ball.

Faustus Mortal: So, if the billiard ball is two inches in diameter then the flat disk it's on is about twenty-five inches in diameter.

Toy Euler: I can cut that out of cardboard.

Faustus Mortal: Now comes the difficult part. Rotate the y-z axis around the x-axis. Keep the ball tangent to the line.

Toy Euler: It spins out a torus.

Faustus Mortal: Precisely. And the cross-section is four, the radius is two.

Toy Euler: So, this cuts out two concentric circles: One of radius 4*PI() and another of radius two. And this means?

Faustus Mortal: Everything in the universe outside of the ball of radius 4*PI() can map into an inner concentric circle of radius = 1/PI(), by the inversion of the spheres. 4 = 2 * 2 = (1/PI())*(4*PI()). And everything between the ball of radius 1/PI() and the ball of radius 4*PI() maps back into itself.

Toy Euler: Now I'm lost.

Faustus Mortal: Look in Wikipedia under Inversion of the Spheres.

Toy Euler is still confused.

Faustus Mortal: Trust me, the inner sphere has radius 1/PI() and maps to the whole universe from the region outside of the ball of radius 4*PI().

Toy Euler: Can't see it.

Faustus Mortal: The inner ball collapses giving a new ball in the y-z plane of radius 4*PI()-1/PI().

Toy Euler: If you say so.

Faustus Mortal: We did the x-z and the y-z, now what's left?

Toy Euler:?

Faustus Mortal: Come on x-z, y-z, then?

Toy Euler: x to y?

Faustus Mortal: Exactly, the x-y plane. So you have a ball in the x-z plane of radius 4*PI() and a ball in the y-z plane of 4*PI()-1/PI().

Toy Euler: OK. I guess...

Faustus Mortal: Now look at the x-y plane. It slices the torus to make two circles of radius one each. This spins out three circles... one of radius 4*PI() and two with radius each of two units.

Toy Euler: Oh.

Faustus Mortal: This doubles the mapping, sending the whole universe from the region outside the ball of radius 4*PI() to a ball of radius 2/PI().

Toy Euler: And this means?

Faustus Mortal: In the x-y plane the ball with center removed collapses to a solid ball of radius 4*PI()-2/PI().

Toy Euler: So you have three numbers, one from each coordinate system.

Faustus Mortal: (4*PI()