By Bruce Camber (Initiated in December 2017)

In 2006 Ari Lehto refocused his work as he explored period doubling at the Planck scale and in 2014 Charles Tresser added insights regarding its universality. Yet, mechanisms to trigger period doubling have not been well understood. Our simple solution: consider sphere stacking to be the doubling mechanism for period doubling bifurcations, particularly at or near the *Planck scale*.

Henri Poincaré was perhaps the first to provide a detailed description of period doubling bifurcation with his section, maps, and metrics. Mitchell Feigenbaum’s discovery of the precise values for when there is a bifurcation, a ratio of convergence (4.669201609…), was postulated to be the result of the dynamics and symmetries within sphere stacking. In 2008 Ari Lehto opened his exploration of period doubling at the Planck scale and in 2014 Charles Tresser opened explorations about its universality.

Explorations of period doubling: *Transformation*, August 2019.

## References & Resources

**Period doubling**: Nine key concepts within this section will be further developed. We have barely smudged the surface. The anticipation, perhaps a hope or a dream or fantasy, is that all those disciplines (and all their scholarship) that have been off the grid and in their own silos will eventually discover a place within particular notations or range of notations.

- The December 2013 chart will eventually be updated so every notation has its own box and that box will click through to an ongoing analysis
- Plato’s concept of
*forms*was instantiated between notations 2-10. - A Bourbakian sense of structure was engaged for notations 11-20.
*More.*.. - Aristotle’s concept of
*Ousia*or*Substances*, a precursor of the Periodic Table was intuited for Notations 21-30. - Perhaps the essence of qualities,
*Qualia*(pure qualities) are between Notations 31-40. - The concept of
*Relations*(versus the*subject*–*object*) were intuited for Notations 41-50. - This model anticipates that at some one of these notations (and probably not one of the primes), a geometry of imperfection most simply begins with the simple object created by five tetrahedrons with its gap of 0.12838822 radians (7.356103 degree gap). The question should be asked, “Are there precursors of quantum fluctuations?”
- The next group of notations, 51-60, is designated,
*Systems*, and opens up systems theory, including all the studies of consciousness and brain-mind relations. - Ari Lehto’s work gives me some hope that we are on a the right path.

### Universal cascades of period doubling bifurcations

### Topological Universality and other precursors and asides

“Before metric universality was discovered, combinatorial and topological forms of universality were described for real maps, for instance in a theorem by Sharkovskii (1964) that remained too long mostly unknown [see also Metropolis (1973); Milnor and Thurston (1988), and RG-related *ad hoc* theories were formulated to understand some of that Gumowski and Mira (1975); Derrida et al., 1978).”

**Link chemical bonding and cellular development:**

Consider as many things as possible that are in a continuous state of doubling, what do you see?

**Perhaps these definitions are too confining:**

“…a continuous family fμ, a *period-doubling* is a “bifurcation” whereby a τ-periodic orbit O0,μ looses its ‘stability* as the parameter μ crosses the *critical value*μc of μ (we will assume, w.l.o.g. that μ crosses from below), and at which point either*:

- a stable 2τ-periodic orbit emerges (
*supercritical period doubling*); or - an unstable 2τ-periodic orbit coalesces with O0,μ and is destroyed (
*subcritical period doubling*).”

**Research**

Enough is enough. It’s over. The old Big Bang Theory is dead. The one birthed in part by Friedman in 1922, then *Lemaître*, then Hawking and Guth.