Message Boards Message Boards

GROUPS:

[GIF] Limits (Möbius transformations of the triangular tiling)

Posted 3 years ago
3605 Views
|
1 Reply
|
4 Total Likes
|

Möbius transformations of the triangular tiling

Limits

To make this, I'm applying a family of Möbius transformations $z \mapsto \frac{Z_\infty z - \gamma_1 \gamma_2}{z - z_\infty}$ to the tiling of the plane by equilateral triangles. Specifically, the fixed points of all transformations are $\gamma_1 = 2$ and $\gamma_2 = -2$, and the pole is at $z_\infty = \tan(\pi (t - 1/2))i$ as $t$ varies from 0 to 1. Consequently, the inverse pole (the point to which infinity is sent), is $Z_\infty = \gamma_1 + \gamma_2 - z_\infty = - \tan(\pi (t - 1/2))i$.

In other words, the point infinity gets mapped to just varies along the imaginary axis.

(Actually, there's a slight lie above: I reparametrize $t$ using the smootheststep function in order to get it to pause nicely at the beginning/end).

Of course, I can't really tell Mathematica to transform infinitely many triangles, so the animation actually only shows $1001^2$ triangles, which is why there's a hole in the middle. In some ways the hole annoys me, but I also kind of like it: it's a good reminder of the limits of computation (as opposed to imagination).

Of course, transforming a million triangles doesn't really work in a Manipulate[] (in fact, the animation took many hours to render), so the code below only shows $21^2$ triangles, which is why it has a much larger hole:

smootheststep[t_] := -20 t^7 + 70 t^6 - 84 t^5 + 35 t^4;

DynamicModule[{?1 = 2, ?2 = -2, z?, 
  Z?, cols = RGBColor /@ {"#eaeaea", "#0D2C54"}},
 Manipulate[
  z? = Tan[? (smootheststep[t] - 1/2) ] I;
  Z? = ?1 + ?2 - z?;
  Graphics[
   {cols[[1]],
    Table[
     If[Abs[3/4 y - Tan[? (smootheststep[t] - 1/2)]] < 1/2 && x == 0, Nothing,
      Polygon[
       Flatten[
        Table[
         ReIm[(Z? # - ?1 ?2)/(# - z?) 
           &[Sqrt[3]/2 (x + 1/4 (-1)^Mod[y, 2]) + 3/4 y I + 1/2 ((1 - s) Exp[I (?)] + s Exp[I (? + 2 ?/3)])]],
         {?, ?/2., 2 ?, 2 ?/3}, {s, 0., 1, 1/10}],
        1]
       ]
      ],
     {x, -10, 10}, {y, -10, 10}]},
   PlotRange -> {{-((15 Sqrt[3])/8), (15 Sqrt[3])/8}, {-3.25, 3.5}}, ImageSize -> 540, Background -> cols[[-1]]],
  {t, .001, 1 - .001}]
 ]

enter image description here - Congratulations! This post is now a Staff Pick as distinguished by a badge on your profile! Thank you, keep it coming!

Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract