Here we introduce the simplest self-persistent stage in topology—the hyperbolic figure eight knot. This topology is defined as a double-cover of the simplest possible 3-manifold ( the Gieseking Manifold ) defining the complement with the smallest possible volume.

Videos by Jeff Chapple, based on work by Henry Segerman.

The constructive equation for the hyperbolic figure eight knot volume is:

Where = the polylogarithm of order 2 (the dilogarithm ), = the imaginary unit, = the imaginary golden ratio, , and = Gieseking's constant for the minimum 3-manifold.

The real conjugate of the hyperblic figure eight knot defines a double covered sphere, divided into 288 pieces—each associated to a unique constant of Nature.

Where = Archimedes' constant.

These constructions pull apart into two dilogarithm pieces, made of identical real and inverse imagainry parts.

Where = the gamma function.

Switching the arguments of that construction, from the imaginary golden ratio to the imaginary unit, yields twice Catalan's constant—.

These constructions naturally divide the world up into electron, proton, and neutron radii.

Where = the neutron radius, = the classical electron radius, and = the proton rms charge radius.