WEEK 8

Seven principal theorems
(some serendipitous)
Proofs, proofs, proofs...

We reviewed the proofs of all basic theorems that reveal the secrets of tangency spinors.  They relate to the tangential arrangements of two, three, and four disks.  And they have the taste of "linearization" of the quadratic nature of the features ruling the curvatures,  much like Dirac equation is a linearization of the Klein-Gordon equation.  

The "comix" of the theorems is presented below.  One of the theorems may be viewed as the "square root of the Descartes Formula".

CHALLENGE:  Which is it? Can you derive the Descartes Formula from it?
Would squaring it be enough?...

[Click on to enlarge]

PS.  Is the following figure better?

WEEK 7

Descendance to the spinor spaces....

 

We proved the "curl u = 0" theorem.  Below is a selection of the "tangency spinors" in the Apollonian Window.  See this theorem at work.  Detect some other patterns... 

[Click on to enlarge]

WEEK 6

 A couple of "Apollonian gems":

1.  Pencils of lines and circles turn out to be points of intersection of 2D subspaces of $\mathbb M^{1,3}$ with the unit sphere (actually, hyperboloid).

2.  The Barning matrices that form a semigroup the orbit of which through [3,4,5] recovers all irreducible Pythagorean triples may be viewed as derived from certain three inversions in the Apollonian window.

Week 5

1.  Theorem (formula) for 4 disks in general configuration.

2.  Special case: Descartes' formula, and its extended version, both for tangent disks.

Define data matrix  $\mathbf D=[C_1|C_2|C_3|C_4]$  where $C_i = \begin{bmatrix}\dot x_i\\ \dot y_i\\ \beta_i\\ \gamma_i\end{bmatrix}$ 

Define the 4x4 configuration matrix $\mathbf F$ as a Gramian of inner products of pairs of disks: $$F_{ij} = \langle C_i , C_j \rangle$$  The general (matrix) formula is $$\boxed {\quad D\, f D^T \ = \ g \phantom{\Big|^2}\ }$$ where $f = F^{-1}$ (needs to be calculated from $F$ for a particular configuration), and $g=G^{-1}$, i.e., $$g = \begin{bmatrix} -1&\ 0&0&0\\ \ 0 & -1 &0&0\\ \ 0& \ 0&0 & 2 \\  \ 0&\ 0&2&0 \end{bmatrix}$$  Teaser: Draw four mutually perpendicular circles.