The sandpile identity element on an ellipse

Figure 1.  Figures 1a) and 1b) correspond to the identity elements for the ellipses $ E_{A,k} $, with $ A = \begin{pmatrix}\frac{10}{9} & \frac{1}{3} \\[0.15cm] \frac{1}{3} & 1\end{pmatrix} $ and $ \begin{pmatrix}\frac{4}{3} & \frac{1}{2} \\[0.15cm] \frac{1}{2} & \frac{3}{4}\end{pmatrix}, $ respectively, with $ k = 18^2 $ in both. Figures 1c) and 1d) show the patterns $ p_A $ corresponding to the ellipses in 1a) and 1b) respectively. A black square represents a vertex with 3 grains of sand; dark patterned (parallel lines), 2; light patterned (cross), 1; white, 0.

Figure 2.  The identity elements for the ellipses given by $ E_{A,k} $, with $ A = \begin{pmatrix}\frac{5}{4} & \frac{1}{2} \\[0.15cm] \frac{1}{2} & 1\end{pmatrix} $ and various $ k $. A black square represents a vertex with 3 grains of sand; dark patterned (parallel lines), 2; light patterned (cross), 1; white, 0.

Figure 7.  A demonstration of r-goodness. The white point $ x_1 $ is not r-good, while the light gray point $ x_2 $ is $ r $-good

Figure 3.  The identity element for the graph $ E^p_{A,k}, $ with} $ A = \begin{pmatrix}\frac{10}{9} & \frac{1}{3} \\[0.15cm] \frac{1}{3} & 1\end{pmatrix} $, $ k = 18^2, $ and p = (.47,.5)

Figure 4.  [5] A portion of the Apollonian circle packing between the lines $ \{x = 0\} $ and $ \{x = 2\} $

Figure 5.  [5] A portion of the boundary of the set $ \Theta $

Figure 6.  Identity elements of $ E_{A_i,k} $, $ A_i \notin \Gamma^+, $ $ k = 18^2 $ $ A_1 \in \Gamma \backslash \partial \Gamma, $ $ A_2 \notin \Gamma $

Figure 8.  $ x \in \partial E $ and the corresponding $ y_x \in \partial \tilde{F}_1$

Figure 9.  A schematic for Lemma 4.5. $ x \in B - A $

Figure 10.  The identity element of $ B_{1152}(0) \cap \mathbb{Z}^2 $