*From: Jess Tauber
Location: New Jersey
Date: 05/16/2011*

Hi- a big fan. Anyway, this might be of interest/use, and is all true, all discovered in last 2 years (see http://tech.groups.yahoo.com/group/tetrahedronT3)- periodic table relations often directly out of Pascal Triangle, both electronic and nuclear: Best physical 2D electronic table is Janet (see second illustration at http://en.wikipedia.org/wiki/Alternative_periodic_tables). Periods here end with alkaline earths. I found every other alkaline earth atomic number identical to every other Pascal tetrahedral number, and counting leftwards from alkaline earths Pascal triangular numbers always landed on positions where quantum number ml=0. Also from Pascal, Fibonacci numbers as atomic numbers map nonrandomly (to leftmost positions in the orbital half rows), as do related sequences such as Lucas numbers (rightmost). In the nucleus differences between relevant period-like positions mostly map to double triangular numbers. Lots more if intrigued- might eventually lead to better ability to manipulate matter on an informational level, something I know you're interested in. There are relations also to the Golden Ratio (been known for a long time that the N/P ratio in the nucleus converges on it with higher atomic numbers), and such things also show up apparently in DNA, protein structure, and human language structure. All seem to show, in addition, modifications that increase combination and its control at the expense of transparency at the lower levels of structure (hierarchy/systems management). In human languages such changes go both ways, with the morphosyntactic historical cycle, and there is some evidence for this also in genomes- whether or not one might find it at more fundamental levels of material reality remains a tantalizing possibility. Would love to hear from you. Thanks. JT

**From: Greg Bear**

Date: 05/25/2011

Interesting congruencies. Thanks, Jess.

*From: Jess Tauber
Location: New Jersey
Date: 05/30/2012*

In the past year have found many more Pascal-based relationships with the electronic and nuclear periodic systems, including being able to simply read off quantum number n, l, ml, ms information from a simple stack of generalized Fibonacci-like sequences, at least for idealized tabular structure ignoring effects of relativity. For the Janet table this works exactly, and forever. To account for relativistic effects it looks as if we need to broaden the perspective to include not just the Golden Mean, but the other Metallic Means as well- each in turn acts like a kind of depictive default, controlling a different aspect of the total, which implies modular organization. For ex. the early nuclides have neutron/proton (N/P) hovering around 1, which is the first Metallic Mean- this works for stable elements til calcium, Z=20. After this you can still have N/P=1 but not stable til Z=40. In the mid-range of the nuclides the ratio N/P= very close to the Golden Mean, for stable elements up to lead, Z=82. Again this stays here but elements unstable til the end of the known elements. If the above trends are based on shifts between Metallic Means, then the next *should* be the Silver Mean, 1+sqrt2.

**From: Greg Bear**

Date: 07/06/2012

I'm certainly not competent to judge your work on this, but perhaps we have readers who can?

*From: Craig Knecht
Location: Tenn
Date: 06/09/2015*

I am looking at a physical model of Pascal's triangle.

This model generates a derivative triangle from Pascal's triangle. This triangle describes the height of water retention between uniform diameter cylinders whose height and arrangement are specified by Pascal's cylinder

(see Water Retention on Mathematical Surfaces)

(see OEIS A258445

Specifically rearranging Pascal's triangle values to achieve minimum retention for the triangle shows a configuration that mimics the stability peak of Fe at 7 rows.

Thanks for any thoughts or ideas.

Craig