# Generating Aperiodic Tilings with Plane Waves

• I
• cuallito
In summary, the conversation discusses the use of an applet for generating aperiodic tilings using the addition of plane waves at different angles. The code provided includes variables for graph resolution, symmetry, and the use of a clipped sine function to achieve odd symmetry. The problem of creating double symmetry with a regular sine function is also mentioned.
cuallito
Hello, I saw an applet awhile ago during a late-night mathematical web-surfing marathon that used the addition of plane waves at different angles to generate aperiodic tilings (like the penrose tiling.) I haven't been able to find it again. I'm trying to make my own version of it, using the same idea.

Code:
xmin = -10; (*Min X coord to graph*)
xmax = 10; (*Max X coord to graph*)
ymin = -10; (*Min Y coord to graph*)
ymax = 10; (*Max Y coord to graph*)
rez = 300; (*Graph resolution*)
sym = 8; (*Directions of symmetry*)
v = 2*Pi*(Sin[i*2*Pi]*x + Cos[i*2*Pi]*y);
Sinu = Cos[v];
P = N[Sum[Sinu, {i, 0, (sym - 1)/sym, 1/sym}]]
MatrixPlot[ Table[P, {x, xmin, xmax, (xmax - xmin)/rez}, {y, ymin,   ymax, (ymax - ymin)/rez}]]

It works like a charm for even numbers of symmetry (change the 'sym' variable to play around with different fold symmetries), but for odd numbers of 'sym' it generates patterns with 2*sym-fold symmetry. I've tried everything, like using Exp[i*v] instead of Cos[v], varying the phase between each of the component waves, and even using Sinu=Cos[v]*UnitStep[v] to stop from getting 'extra' interference, but none of them gives me the correct result for odd numbers :( Help!?

By construction, your pattern is symmetric for x-> -x, y-> -y, your v changes its sign and cosine is an even function. You cannot produce odd symmetry with it. Using Sinu=Sin[v] might work for odd symmetries.

For some reason, using a clipped Sine function seems to work:

Anyone know why? A regular sine function just creates the double symmetry problem again.

Code:
xmin = -10;
xmax = 10;
ymin = -10;
ymax = 10;
rez = 400;
sym = 5;
v = 2*Pi*(Sin[i*2*Pi]*x + Cos[i*2*Pi]*y);
Sinu = Max[Sin[ v], 0];
P = Sum[Sinu, {i, 0, (sym - 1)/sym, 1/sym}];
MatrixPlot[ Table[N[P], {x, xmin, xmax, (xmax - xmin)/rez}, {y, ymin,   ymax, (ymax - ymin)/rez}]]

Last edited:

## 1. What are Aperiodic Tilings?

Aperiodic tilings are arrangements of geometric shapes in a plane that do not repeat or have a predictable pattern.

## 2. What are Plane Waves?

Plane waves are continuous, uniform waves that propagate through space without changing shape or amplitude.

## 3. How can Plane Waves be used to generate Aperiodic Tilings?

By arranging plane waves at specific angles and frequencies, they can interfere with each other to create aperiodic patterns in the plane.

## 4. What is the significance of Aperiodic Tilings in science?

Aperiodic tilings have important applications in fields such as crystallography, materials science, and computer science. They can also be used to study the mathematical properties of tilings and wave patterns.

## 5. What are some real-world examples of Aperiodic Tilings generated with Plane Waves?

One example is the Penrose tiling, which has been observed in certain types of quasicrystals. Another example is the Ammann-Beenker tiling, which has been used in the design of architectural structures and decorative patterns.

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