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)?