the hyperbolic partition equation
The hyperbolic partition equation encodes the allowable transformations of the figure-eight knot complement, hyperbolically partitioning the 288 constants of Nature inside the Planck mass gap. It's 4 solutions,
,
and
—the hyperbolic partition constants—possess the following product, sum and quadrance.
The first of those quadrances is the fine structure constant .
In polar coordinates and
are expressed as
and
, where:
In addition to their product, sum, and quadrance, these hyperbolic partition constants possess the following algebraic-geometric symmetries.