There are some equations I’m working on.^{[1]} They are intended to describe a projection from the outside surface of a sphere to the inside surface of a concentric spherical shell. This paradigm corresponds to a hypothetical gravitational field that has a net zero gravitational force in a universe with a single black hole.^{[2]}

In two dimensions, the equations are scaled and rotated logarithmic spirals, which preserve the angles between its tangent and the intersecting circle — the polar slope angle is constant, which is consistent with flat planar projections. The equations in three dimensions will be published in a later post.

*There exists at least one r*_{1}(φ_{1}) and a scaled inverse r_{2}(φ_{1} + φ_{2}), where every product of r_{1}(φ) with r_{2} (φ) is constant. See the proof at the bottom of the page. [Click the image to enlarge.]

See equations (1-5) below, which include:

(1) a logarithmic spiral in polar form,

(2) the inverse of equation 1,

(3) equation 2 scaled and rotated, and

(4) domain, scaling and rotational factors.

(1)

(2)

(3)

(4)

(5)

The variable *P* is the subtended projection angle in units* 1=π (or 1=1/2 turn),* and the variable *b* is the radius of the neutral surface, where all projection lines meet.

Below is an example drawn to represent the projection in three dimensions (not to scale) in a universe containing only a single black hole — a very small universe with a very large black hole!

* [Click the image to enlarge.]*

[1] For a more dynamic view of these equations, visit desmos.com.

[2] Here is where I am on all this speculation. Click the link to see a recently reported-measurable experiment inside **(see page 18).** This link has a graphic on page 24 that might be helpful.

Happy Pi Day, everyone!

**PROOF:**