Geometry and numbers

From Fine Art America

14 Nov 2019

Don Redmond

Polygonal numbers as differences of squares

Apolloniana

31 October and 7 November 2019

Jerzy Kocik 

Lattices, spinors, and classification of Apollonian disk packings.  Here are the first 21 irreducible classes of lattices.

More on "geometric Apolloniana".
(Simply "tinyURL.com/jkocik", if you need to retype it)

Here is Don's derivation of parametrization of \(x\), \(y\), \(z\), and \(w\) satisfying  $$x^2+y^2=zw$$ by three parameters \(\alpha\), \(\beta\), and \(\gamma\): $$ \begin{array}{rl} x=&\beta\gamma-\alpha^2\\ y=&\beta\gamma+\alpha^2-2\alpha\gamma-\alpha\beta \qquad\quad(1)\\ z=& -\alpha^2 -(\beta-\alpha)^2\\ w=&-\gamma^2-(\gamma-\alpha)^2\\ \end{array}$$ This would suggest the following parametrization of spinors: $$\mathbf a=\begin{bmatrix}\alpha\\\alpha\!-\!\beta\end{bmatrix},\quad \mathbf b=\begin{bmatrix}-\gamma\\ -\gamma\!+\!\alpha\end{bmatrix}$$ with   \(x=\mathbf b\times\mathbf a\),  \(y=\mathbf a\cdot\mathbf b\),  \(z=-\|\mathbf a\|^2\),  and  \(w=-\|\mathbf b\|^2\).  Another arrangement: $$\mathbf a=\begin{bmatrix}\alpha\\\alpha\!-\!\beta\end{bmatrix},\quad \mathbf b=\begin{bmatrix}\gamma\!-\!\alpha\\ -\gamma\end{bmatrix}$$ with   \(x=\mathbf a\cdot\mathbf b\),  \(y=\mathbf a\times\mathbf b\),  \(z=-\|\mathbf a\|^2\),  and  \(w=-\|\mathbf b\|^2\). 

Question:   Does \(2^2 + 8^2 = 4\cdot 17\) admit a representation by some \(\alpha\), \(\beta\), and \(\gamma\) of  Eq. (1)?

Energy

10 and 17 October 2019

K.V. Shajesh 

How does quantum vacuum energy gravitate?

Numbers

Fordon James, "Primary spaces"

22 September and 3 October 2019

Don Redmond 

On representing integers as sums of four squares

Here are Don's notes on this subject

Trick and math

18 September 2019

Dinush Lanka 

Tricks, patterns and math behind it, revealed

Periodic Table

5 and 12 September 2019

Punit Kohli 

Periodic Table: patterns of elements



Here is the paper Mike has mentioned:
(Click here)

Title: How the modified Bertrand theorem explains regularities of the periodic table I. From conformal invariance to Hopf mapping 

Authors: Arkady L. Kholodenko, Louis H. Kauffman

Also:  a popular podcast "Battle of elements". 

Spider-web

29 August 2019

Jerzy (Jurek) Kocik

Unexpected fractal -- a case of experimental mathematics

The question of the three dimensional case is addressed here:

"3d spiderweb"

C. Rose -- 3. 6. 9

18 April 2019

Christian Rose

The significance of 3, 6, and 9

Mathematics of time

Time and again

Free discussion.  Among the topics: mathematics of time in languages; causality, and more.

Plus plums in chocolate.

Sugar

Promised long time ago.  Here is a small selection of articles:

> This is your brain on sugar
> Eating Sugar Makes You Stupid
> How sugar literally destroys your health and makes you stupid

I will bring some cookies for the next meeting...

K.V. Shajesh - retarded time

28 March and 4 April,  2019

K.V. Shajesh
Retarded Time

Shajesh kindly shares his notes with us.  And here is  a worksheet that illustrates some counter-intuitive features of retarded time, which he also used as a homework problem in his class.

Also, Punit sends a message:

Please read in the middle of the document about Pauli's comment on Einstein, american physicists, and american journalism.

https://physicstoday.scitation.org/doi/10.1063/PT.3.4183

Albert Einstein, celebrity physicist: Physics Today: Vol 72, No 4

Paul Halpern is a professor of physics at the University of the Sciences in Philadelphia. He is the author of Einstein’s Dice and Schrödinger’s Cat (2015) and The Quantum Labyrinth: How Richard Feynman and John Wheeler Revolutionized Time and Reality (2017).

M. Sayeh: time

21 March 2019

Mohammad Sayeh
Time and the hazardous F-word

This will be a non-mathematics seminar on time, freedom, and related issues.  I understand we need to review I. Kant, Bergson, etc. before the meeting.