What are the conditions for a polygon to be tessellated? Upon one of my mathematical journey's (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate spheroid that we call a planet. I'm most interested in the tessellation of regular polygons and their 3D counterparts. I've noticed that simple shapes like squares or cubes can be tessellated but not circles or spheres. !tesselation example Somewhere after hexagons, shapes lose that ability to be tessellated by only themselves. Although it is intuitively clear to me when shape can be tessellated, I cant put it into words. Please describe to me, in a fair amount of detail, what the lesser sided shapes had that the greater sided shapes did not inorder to be tessellated.

A regular polygon can only tessellate the plane when its interior angle (in degrees) divides $360$ (this is because an integral number of them must meet at a vertex). This condition is met for equilateral triangles, squares, and regular hexagons.

You can create irregular polygons that tessellate the plane easily, by cutting out and adding symmetrically.