the hyperbolic partition equation
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 solutionszhe_1, zhe_2, zhe_3 and zhe_4—the hyperbolic partition constants—possess the following product, sum and quadrance.

Hyperbolic Partition Product
Hyperbolic Partition Sum
Hyperbolic Partition Quadrance
zhe_1.svg= 0.0854245431533304 ...
zhe_2.svg= 3.66756753485501 ...
zhe_3.svg= -1.87649603900417 ... + 4.06615262615972 ...i
zhe_4.svg= -1.87649603900417 ... - 4.06615262615972 ...i

The first of those quadrances is the fine structure constant zhe_squared=alpha.

In polar coordinates zhe_3 and zhe_4 are expressed as zhe_3_polar and zhe_4_polar, where:

zhe_r= 4.47826244916751 ...
zhe_theta= 2.00316562310924 ...

In addition to their product, sum, and quadrance, these hyperbolic partition constants possess the following algebraic-geometric symmetries.

zhe big productzhe big sumzhe big quadrance
zhe_3 zhe_4 productzhe_1 zhe_2 product
zhe -1zhe inversion
zhe bi product sum