Alhambra: translational symmetry

In mathematics there are 17 different types of translational symmetry, all of them can be found in the tiles at the  Alhambra Palace Granada.

The 17 "Wallpaper Patterns" of the Alhambra

The Alhambra Palace in Granada, Spain, is famous for its stunning Islamic tilework. Hidden in these patterns is a mathematical secret: they showcase all 17 possible types of repeating symmetry (called "wallpaper groups") that can exist in flat surfaces! Here's why that's cool:

What is Translational Symmetry?

Imagine sliding a tile pattern sideways, up, or down without rotating it. If it repeats perfectly, it has translational symmetry. The Alhambra's patterns take this further by combining slides with rotations, flips, and reflections.

Examples from the Tiles:

• Hexagonal Symmetry: Some tiles repeat like honeycombs (6-fold rotation).
• Checkerboard Patterns: Squares that flip colors when slid (like a chessboard).
• Spiral-like Designs: Patterns that rotate 90° or 180° while repeating.
• Mirror Tricks: Tiles that reflect like a kaleidoscope when shifted.

Why 17

Mathematicians proved only 17 combinations of slides, flips, and rotations work for infinite repeating patterns. The Alhambra’s artists discovered them through craft 500 years before the math was formalized!

Fun Fact:

Some argue the Alhambra "almost" has all 17 types, as a few patterns are debated. But it’s still a masterpiece of math-meets-art!

Next time you see tiles, wallpaper, or fabric patterns, look for these hidden symmetries!