Friday, December 15, 2023


Happy Holidays!

Last few topics for the next semester

(1)  The $\hbox{SL}(2,\mathbb Z)$ tessellation of the uper half-plane $\mathbb C^{+}$, when viewed as a system of circles rather then tiles, and cosequently as vectors (points) in the Minkowski space, reveals a very simple algebraic algorithm for drawing it emerges, without tracing a group action!  The Dedekind tessellation (somewhat less known) is an effortless addition to this result.

(2)  "Superintegrality":  May every integral solution to Descartes' formula be a source of a geometric Apollonian disk packing such that all coordinates $(\dot x,\dot y, \beta,\gamma)$ and spinors be integral?  With the use of a certain "kaleidoscope group" this claim is confirmed. 

(3)  Depth function, defined on the set of all possible Descartes configurations, gives rise to an interesting fractal, and an interesting extension of group $\hbox{SL}(2,\mathbb Z)$ to a subgroup of  $\hbox{SL}(2,\mathbb Z[i])$.

(4)  Lens sequences: pleasant 4-term recurrence bilateral integer sequences (some of which may be observed in the Apollonian Window), accompanied with mysterious "underground" sequences that await geometric interpretation.

(5)  Plus other minor but interesting mathematical pieces (e.g., generalizations to higher dimensions).  Also, intriguing questions, unsolved problems, and possible projects.

Tuesday, December 5, 2023

Friday, December 1, 2023


1.  Descartes' Formula from the spinor theorems

2. Towards the integrality of Apollonian disk packing 

Thursday, November 16, 2023


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 $\mathbf b = [3, -4]$. 

Thursday, November 9, 2023


Stern-Brocot tree and the magic quipu

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

Thursday, November 2, 2023


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. 

Thursday, October 26, 2023


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?

Friday, October 20, 2023


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]

Friday, October 13, 2023


 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.

Thursday, October 5, 2023

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. 

Thursday, September 28, 2023

Week 4

Inversions of disks are reflections in the Minkowski space!
$$\hbox{Ref}_{\mu(K)} \mu(D) \ = \ \mu\left( \hbox{Inv}_K \, D\right)$$

Simply:$$\hbox{Inv}_K \, D = D+2\langle K,D\rangle \, K$$ where $D$ is a disk (vector) and $K$ is a circle through which the disk is inverted.

Mini-challage:  Find the matrix representation of the inversion through $K$ and show it is a Lorentz transformation (belongs to $O(1,3)$).

A quick introduction to inversions in 2D (classical geometry).

Thursday, September 21, 2023

Week 3

1. The crystalline structure of "apollohedron".

2. The homomorphism from the Euclidean group to Lorentz group, and their actions on the set of disks and the Minkowski space, respectively  $$\frac{\hbox{Eucl} (2)}{\mathcal D_2}  \quad \longrightarrow\quad \frac{\hbox{SO}(3,1)}{\mathbb M^{1,3}}$$  Just translations for now.

PS.  The above diagram is "approximate".  We shall extend it to the conformal group and take care of some details.  

[Click to enlarge]