This article was started in April 2018. Updated: March 21, 2021- BEC
Introduction. The doubling process is deep and wide throughout the foundations of mathematics, the small-scale sciences — physics, chemistry and biology– as well as within the human-scale sciences — population systems, government systems, and global systems. The most-simple system is that it is all starts within the infinitesimal, beyond the reach of measuring devices, and it starts with sphere stacking of the most-simple, primordial spheres, building systems along the way.
Discussion. Perhaps the very basis of doubling resides within the very first notation where planckspheres are naturally being closely packed and the emergence of the tetrahedral-octahedral complex begins and also doubles as observed in the most simple models.
Working on it
1. This illustration found within Wikipedia demonstrates a process that begins with pi and emerges with basic Euclidean geometry. It is both symbolic and functional. For more: https://81018.com/emergence/
2. Another study, possibly derivative of the first, is called period-doubling bifurcation1
however, it has a rather specialized language and mathematics. Notwithstanding, references throughout our study will be to many of the leading scholars within this work.
3. Emails and Letters4. Within all these articles, the authors have references to their source materials that are very helpful. A note of thanks is sent to many of these scholars along with questions.
4. New resources come online all the time. Scholarpedia5 is an example and an article by Dr. Charles Tresser of IBM Watson Research Lab has caught our attention. The studies within nonlinear resonance period doubling6 and multiscale modeling7 are also believed to hold special keys to unlock the concept of doubling.
- arXiv:1803.04903 [pdf, ps, other] Global secondary bifurcation, symmetry breaking and period-doubling by Rainer Mandel
- arXiv:1801.05730 [pdf, ps, other] A Complete Analytical Study on the Dynamics of Simple Chaotic Systems by G. Sivaganesh, A. Arulgnanam, A. N. Seethalakshmi, January 2018
 Emails and letters to experts-scholars: We will need help so we are seeking out experts within this field of study.
More to come…