$z^a$ Circle Pattern

Jürgen Richter-Gebert, Jan Techter

Media


Description


The Cinderella Applet [$z^a$ Cinderella Exhibit] shows the circle pattern that corresponds to a discrete version of the complex map $z \mapsto z^a$ for $0 \lt a \lt 2$.

The centers of the circles and their intersection points are given by the discrete map $f:\mathbb{Z}^2_+ \rightarrow \mathbb{C}$ on the first quadrant $\mathbb{Z}^2_+$ of the lattice $\mathbb{Z}^2$ that is characterized by the initial values $$ f_{0,0} = 0, \quad f_{1,0} = 1, \quad f_{0,1} = e^{i\frac{\pi}{2}a} \tag{IV} $$ and the two conditions $$ \frac{ (f_{m,n} - f_{m+1,n}) }{ (f_{m+1,n} - f_{m+1,n+1}) } \frac{ (f_{m+1,n+1} - f_{m,n+1}) }{ (f_{m,n+1} - f_{m,n}) } = -1 \tag{1} $$ $$ a f_{m,n} = 2m \frac{ (f_{m+1,n} - f_{m,n})(f_{m,n} - f_{m-1,n}) }{ f_{m+1,n} - f_{m-1,n} } + 2n \frac{ (f_{m,n+1} - f_{m,n})(f_{m,n} - f_{m,n-1}) }{ f_{m,n+1} - f_{m,n-1} } \tag{2} $$ Condition (1) means that all elementary quadrilateral are conformal squares, i.e. the cross ratio of its vertices are equal to $-1$. This condition is Möbius invariant and allows to compute all values of $f$ from the values $f_{m,0}$ and $f_{0,n}$ on the axes.

Condition (2) is compatible with condition (1) and describes some isomonodromic solutions of the latter. It can be used to compute the values $f_{m,0}$ and $f_{0,n}$ from the initial values (IV). Its smooth limit corresponds to the smooth $z^a$.

Move the y-axis to adjust (IV) in terms of the angle $\varphi = \frac{\pi}{2}a$ between the axes. For $\varphi \in (0,\pi)$ the power $a$ runs between $0$ and $2$ and the so obtained discrete map has no overlapping neighboring quadrilaterals, i.e. is an immersion, which was shown in [1]. Besides, one can adjust $\varphi$ to obtain any $a \in (-\infty,\infty)$, but the corresponding circle patterns are no longer immersed.

The discrete $z^a$ map for $0 \lt a \lt 2$ was introduced in [2]. A detailed description can also be found in [1] and [3]. The asymptotic behaviour at infinity was proven in [4]. The first implementations of the $z^a$ circle pattern were done by Tim Hoffmann, see also [5].

References


Prof. Dr. Dr. Jürgen Richter-Gebert   +

Projects: CaP, Z02
University: TU München, Zentrum Mathematik, M10, 02.06.054
Address: 02.06.054
Tel: +49 (89) 289 18354
E-Mail: richter[at]ma.tum.de
Website: http://www-m10.ma.tum.de/bin/view/Lehrstuhl/RichterGebert


Jan Techter   +

Projects: A02
University: TU Berlin
E-Mail: techter[at]math.tu-berlin.de