How do things double and just what is a doubling?

Sphere to tetrahedron-octahedron couplet
Planckspheres, cubic-close packing, emergence

This article was started on Sunday, 22 April 2018.
It is still under construction. – Bruce 

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. 2.

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:

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.

A credible source will be the articles within ArXiv2. Yet, the Google Scholar Citations3 are also being reviewed.

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 to emerge and also double as observed in the most simple models.

References and source materials

[1] Wikipedia’s period-doubling bifurcation article

[2] ArXiv’s key articles on period-doubling bifurcation

[3] Google Scholar Citations

[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

[7]  Multiscale modeling

[8] Emergence

much more to come…