Torus under the Identity Map

Torus after the First Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right] \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus after the Second Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^2 \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Third Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^3 \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Fourth Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^4 \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Fifth Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^5 \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Sixth Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^6 \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Seventh Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^7 \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Eighth Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^8 \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Ninth Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{10} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Eleventh Iteration

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{11} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 12

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{12} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 24

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{24} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 48

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{48} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 94

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{94} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 96

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{96} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$ Note! You should also see iteration 94.

Torus under Iteration 190

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{190} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 192

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{192} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 193

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{193} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 194

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{194} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 200

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{200} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 350

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{350} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 370

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{370} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 380

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{380} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 381

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{381} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 382

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{382} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 383

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{383} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 384

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{384} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 385

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{385} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 386

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{386} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 750

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{750} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 760

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{760} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 765

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{765} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 766

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{766} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under Iteration 767

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{767} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$

Torus under the Iteration 768

$$\left[\begin{array}[c]{rr} 1 & 1 \\ 1 & 2 \end{array}\right]^{768} \begin{bmatrix} x \\ y \end{bmatrix} ( \mbox{ mod } 1024 ) $$