Euclidean tilings by convex regular polygons

Imagine you have a bunch of shapes, like squares or triangles or hexagons, that are all the same size and shape. These shapes are called "convex regular polygons."

Now, you can take these shapes and arrange them in a certain way so that they fill up a space without any gaps or overlaps. This is called a "tiling."

In a "euclidean tiling," the shapes are arranged in a way that follows the rules of Euclidean geometry. This means that the shapes are all flat and the angles between them all add up to 180 degrees.

So, when you have a euclidean tiling of convex regular polygons, you have a bunch of shapes that all fit together perfectly without any gaps. They form a pattern that repeats over and over again, like a puzzle piece that fits into itself endlessly.

This type of tiling can be found in many places in nature and art, and has been studied by mathematicians for centuries. It's a fascinating way to explore the beauty and complexity of geometry!