Geometric art

From ancient civilisations to modern bathrooms and kitchens, mosaics and tiles have been more than just decorative elements. Behind their beauty lies a solid foundation of geometry. The repeating shapes, the patterns that fit together without leaving gaps, and the symmetries that catch the eye all respond to mathematical principles that combine aesthetics and functionality.

We walk into a kitchen and see a floor covered with tiles. They all fit together perfectly: there are no gaps or overlaps. In a bathroom, the tiles form patterns that seem infinite. In public buildings, underground stations and historical monuments, decorative mosaics surround us. Without noticing it, we interact daily with geometric tessellations that bring order, hygiene and visual harmony.

A tessellation is a way of covering a surface completely with geometric shapes, leaving no gaps or overlaps. Only certain shapes can tessellate the plane on their own: regular triangles, squares and hexagons, among others. These shapes allow for exact repetition, making it possible to cover large surfaces without cuts or waste.

In addition, tessellations allow for more complex combinations, such as semi-regular ones, which alternate different polygons with a stable pattern. Often, these patterns follow principles of rotational, translational or reflection symmetry, which are not only visually appealing but also help to modulate the space.

This use of geometric patterns dates back to cultures such as Islamic, which explored the decorative potential of pure geometry in religious and palatial architecture, such as in the Alhambra in Granada or the mosque in Cordoba (Martín-Molina & Sánchez, 2021).

In addition to their aesthetic value, mosaics and tiles serve practical purposes:

• They facilitate cleaning: their flat, uniform surfaces are easy to maintain.
• They allow for modularity: by following a regular pattern, they are easy to replace or expand.
• They save material: the geometric shapes that tessellate minimise waste in production and installation.

In urban design and modern architecture, the conscious use of tessellations allows us to play with the perception of space. A pattern can make a room appear larger, give a wall more depth, or make a space more welcoming.

This use of geometry shows us how art, science, and everyday life are intertwined. Every tile we walk on or contemplate is a small piece of a large mathematical framework, designed to withstand time and use without sacrificing beauty.

References:

Martín-Molina, V., & Sánchez, J. (2021). La geometría de la Alhambra: simetría y teselaciones en el arte islámico. Revista de Educación Matemática, 19(2), 55–68.

Pérez-Gómez, A. (2019). Formas que encajan: teselaciones geométricas en arquitectura y diseño. Madrid: Ediciones Técnicas.

Stewart, I. (2007). Las matemáticas de la belleza. Barcelona: Crítica.