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)?
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