by Bruce Camber
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 smallscale sciences — physics, chemistry and biology– as well as within the humanscale sciences — population systems, government systems, and global systems. The mostsimple system is that it is all starts within the infinitesimal, beyond the reach of measuring devices, and it starts with sphere stacking of the mostsimple, primordial spheres, building systems along the way.
We should investigate it all no matter where it leads us.
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 tetrahedraloctahedral complex begins and also doubles as observed in the most simple models.
References
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/
3. Another study, possibly derivative of the first, is called perioddoubling bifurcation^{1}
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.
A credible source will be the articles within ArXiv^{2}. Yet, the Google Scholar Citations^{3} are also being reviewed.
4. Emails and Letters^{4}. 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.
5. New resources come online all the time. Scholarpedia^{5} 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 doubling^{6} and multiscale modeling^{7} are also believed to hold special keys to unlock the concept of doubling.
More references and source materials
[1] Wikipedia’s perioddoubling bifurcation article
[2] ArXiv’s key articles on perioddoubling bifurcation

arXiv:1803.04903 [pdf, ps, other] Global secondary bifurcation, symmetry breaking and perioddoubling 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
[4] Emails and letters to expertsscholars: We will need help so we are seeking out experts within this field of study.
[5] Scholarpedia
[6] Nonlinear resonance period doubling
[8] Emergence
much more to come…