WEEK 11

Area of coronas and a Zeta funtion

$$A=\frac{\pi}{2}\; \sum_{\mathbf f\in \mathbb Z^2_o} \frac{1}{\left(\|\mathbf M\mathbf f \|^2 - B\right)^2}$$

where $\mathbf a$ and $\mathbf b$ are two adjacent spionors in a corona,  $\mathbf M= [\mathbf a|\mathbf b]$, and $B=\pm \,\mathbf a\times \mathbf b$.

The figure below shows spinors around the disk of curvature $B=23$ in the Apollonian Window.  Initial spins may be chosen $\mathbf a = [1,\, 5]$ and $\mathbf b = [3, -4]$. 

WEEK 10

Stern-Brocot tree and the magic quipu

Arrangements of spinors along a disk in an Apollonian packing has an interesting pattern...

WEEK 9

Can spinors be combed?...

We started with several simple reflections:

• The sum of the curvatures of two tangent disks is an Eulerian sum of squares due to the "Norm Theorem" (see the comix-like illustration from the previous meeting)
• Thm 6 = Thm 7  (visualized by continuous transformation) 
• If two adjacent spinors in an Apollonian disk packing are integral, so are all spinors in the packing.

 Then we defined "tuned spinors" and parallel transport of signs.