# Random Tessellation Exploration

From EscherMath

**Objective:**
Produce tilings using randomized methods

### Materials

- Graph paper
- Domino rotation grid

### Procedure

- Using a piece of graph paper, make a 5x5 grid of squares, each one of which contains four graph paper squares. For each big square, flip a coin and use the result to divide into two dominoes either horizontally or vertically. Continue until you have a random tessellation by dominoes.
- Another way to generate a random tiling with dominoes is to set up all dominoes facing the same direction, then randomly select a neighboring pair to rotate by 90°:

Lay eight dominoes (or strips) on the domino rotation grid, and use random numbers (generate at random.org) to select pairs of dominoes to rotate by 90°. Perform about 20 rotations to produce a random tiling by dominoes. - Run this applet: Random Square Tiling a couple of times to produce random tilings of a square, and compare with your hand-made results.
- This plus-shaped region is a simple example of a shape called an Aztec diamond.

Draw all eight possible ways to tile the region with dominoes. - If you pick one of the eight tilings from question 4 at random, what is the probability that the top two squares are covered by a single domino, as in this picture:
- This applet uses random pair rotations to produce a Random tiling of an Aztec Diamond. Does it seem likely that the top two squares will be covered by a single horizontal domino?
- Try other similar applets here: Tiling Applets
- Experiment with Paul Harrison's Ghost Diagrams Applet for many interesting random tilings.

**Handin:**
A sheet with answers to all questions.