Fibonacci numbers

9 Apr 2020

Jerzy (Jurek) Kocik

Fibonacci sequence and Ford circles


For more, see  arXiv:2003.00852

Can get it here

5 Feb 2020

Don Redmond

When are Rectangles Squares?

27 Feb 2020

Kalpa Madhawa

Quaternionic eigenvalues

From Arts Bathsheba

20 Feb 2020

Duston Wetzel

On gyroid

Gyroid was discovered by our own Alan Schoen and is enjoing an increasing general popularity in sience and in in general math culture

Ford and scattering

6 and 13 Feb 2020 

K.V Shajesh

Rayleigh scattering off Ford circles

Math from physics

23 and 30 Jan 2020

Philip Feinsilver

Eigenvectors from eigenvalues

>> Relevant paper on Arxiv
>> Quanta Magazine on this subject

  <<  see the original image

  Philip's presentation  (NEW)
 plus this page.

Symmetry

21 November and
and 5 December 2019

Thushari Jayasekera

Symmetry in Materials Physics

← click to view details.

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