What is this?

• A series of scripts to create tessellation patterns.
• Scripts to produce tessellations for laser cutting.

This project was born when we built the stairs of our house/office. We had 60 steps to cover. Each step tells the story of a neat mathematical problem: The discovery of the Einstein or aperiodic monotile.

Each riser in the stair (the vertical part) is either laser-cut or laser-engraved with a pattern. The stairs are divided in 7 sections:

1. Regular tessellations (made of a single regular polygon)

Only three regular polygons can tessellate the plane without gaps or overlaps: equilateral triangles, squares, and regular hexagons. This is because only these shapes, when placed edge to edge, have interior angles that are exact divisors of 360°, allowing them to fit together perfectly in a repeating pattern.

The last example illustrates how pentagons leave star and rhombus holes in the plane.

2. Semi-regular tessellations (made of two or more regular polygons)

Kepler discovered that out of the miriad of combinations of regular polygons, only 8 of them can tesselate the plane.

3. Symmetric tessellations (made of a single polygon with isometries of the plane)

4. Girih tiles with persian patterns

5. Socolar Aperiodic Tilings

### 6. P2 Penrose Aperiodic Tilings (with kites and darts) With a script created by [tangentstorm](http://tangentstorm.com/)

which was used to create a room divider with a Penrose tiling:
Manufactured by Carlos & Alberto Corrales. It generates these beautiful shadows:

7. Hat monotile aperiodic tiling (with a pattern)