###### 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 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.

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 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 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.

3. **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.

4. 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.

## **Source materials**:

1. Wikipedia’s *period-doubling bifurcation* article

2. ArXiv’s key articles on period-doubling bifurcation

- 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

4. Emails and letters to experts-scholars: We will need help so we are seeking out experts within this field of study.

5. Scholarpedia

6. * Nonlinear resonance period doubling*

8. Emergence

More to come…