Escher and Tessellations: Artistic geometry

A tessellation is a pattern of shapes that fit together perfectly without gaps or overlaps.

 If you’ve ever looked at a tiled floor, you’ve seen a tessellation. While regular tessellations follow strict mathematical rules, irregular tessellations are often used, following some but not all rules.

One artist who turned this idea of tessellations into unforgettable art was Escher. He was said to be inspired by the tessellations in the tiles at Alhambra in Spain.

In his famous work Reptiles, Escher took a flat tessellating pattern and transformed it. The reptiles in his art interlock perfectly.

In Rptiles, each shape is identical in structure and then rotated or shifted across the plane. This is called a translation in geometry, caused by sliding a shape without changing its size or orientation. In other tessellations, he used reflections and rotations, which are also mathematical operations.

What makes tessellations concrete is that they can be proven. For example, a regular hexagon tessellates because each interior angle is 120 degrees. Three hexagons meeting at a point give you 120 + 120 + 120 = 360 degrees. The space closes exactly. Squares work because 90 + 90 + 90 + 90 = 360. But a regular pentagon, with 108-degree angles, does not tessellate on its own, because 108 does not divide evenly into 360.

Escher would start with a square or hexagon, then cut shapes out of one side and attach them to another. As long as the edges still matched perfectly, the tessellation held. The underlying math stayed intact, even though the shapes became birds or lizards.

That’s the key idea: tessellation is geometry you can test. You can trace one unit, copy it and see if the plane fills. The beauty comes from repetition, symmetry and transformation.