Happy Holidays!

From bookbuilder.cast.org |

**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.