The binomial constructor encodes the two-part transformations of the hyperbolic figure-eight knot complement within its coherent external transform space—the 24-dimensional unit ball. In Erlangen terms, the constructor separates each transform into an external action, its boundary data, and a bounded internal action. Here, 
= the external geometric action of each transform, 
= the external boundary data on which that action is defined, 
= the internal geometric action, and 
= the hyperbolic inversion boundary. The inversion boundary is constructed from the normalized Planck length = 
, the normalized Planck mass = 
, and the square of the normalized Planck charge = 
.
Every constant of Nature is a transform built from the roots of the hyperbolic partition equation. These roots supply the primitive quantities from which the transform families are built. 2 families are polar (expressed in powers of 
and 
), while 6 are Cartesian combinations of the roots: the 2–part products, the 3–part products, the 2–part sums, the 3–part sums, the 2–part quadrances, and the 3–part quadrances.
Explore the Constants of Nature page to see how these roots construct each constant of Nature.