Interlude

Two items:

The text Christian Rose mentioned (and currently reads) can be downloaded here.   (It contains a nice chapter on math of lattices.)    See also the lectures recorded at Oxford.  Thanks, Christian.

The lectures on Knot Theory started.  Here are recordings of the first four:

Lecture 1 (Roger Fenn)

Lecture 2 (Louis Kauffman)

Lecture 3  (Roger Fenn)
Lecture 4  (Louis Kauffman)
Lecture 5  (Roger Fenn, 1 May)

Yous probably remember Louis from his visits at Carbondale.

negative mass

Click on the image to see details

23 April 2020

Moses Gaither-Ganim

Minus matter: the possibility of negative mass

[Recording] - the first few minutes are missing, but can easily be reconstructed from the whiteboard. 

Origami geometry

16 April 2020

William Holt

Pentagons from paper, development of division in ancient art

  <<< click on the image to enlarge



Please prepare at least 5 sheets of letter size paper for folding. 

Here is the recording of the meeting.

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)?