Aperiodic tiling is a concept in mathematics, specifically in geometry and tiling theory, that deals with covering an infinite plane using shapes (called tiles) in a way that never repeats periodically, yet still fills the plane completely without gaps or overlaps.


What is Tiling?

In general, a tiling (or tessellation) is a way to cover a flat surface using one or more geometric shapes, called tiles, such that:

1. There are no gaps between tiles.

2. There are no overlaps—tiles don't stack on top of each other.


Most common tilings we see in floors and walls are periodic tilings, meaning they repeat in a regular, repeating pattern. If you shift the whole tiling in certain directions, the pattern looks exactly the same.


What is Aperiodic Tiling?

An aperiodic tiling is a tiling that never repeats itself in a periodic way, no matter how far you go in any direction. Yet:

1. It still fills the plane completely.

2. It follows specific matching rules (no randomness or chaos).

3. It's ordered but non-repetitive.


A set of tiles that can tile the plane but only aperiodically is called an aperiodic set of tiles.

How are the models being calculated?

Notebook written by Hans Jakob Rivertz

Repository

This project is open-source, and the code for the models and their visualization can be found on GitLab.