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. We should investigate it all no matter where it leads us.
1. This illustration found within Wikipedia demonstrates a process that begins with pi and emerges with the basics of 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.
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.
 Emails and letters to experts-scholars: We will need help so we are seeking out experts within this field of study.
much more to come…