Fundamentals and History of Geometric Tessellation

Definition and Basic Polygons

Tessellation is the practice of covering a surface with one or more shapes so that no gaps or overlaps remain. Any straight-edged three-sided shape, such as a triangle, or any four-sided shape can tessellate perfectly. A common method to achieve this is to rotate each subsequent shape $180^\circ$ from the previous one placed on the surface.

Geometric Principles of Tessellation

For shapes to tessellate, the interior angles meeting at any shared vertex must sum to exactly $360^\circ$. If the sum is different, the shapes will either overlap or leave an uncovered gap. Because the interior angles of any quadrilateral always sum to $360^\circ$, arranging the shape so that each of the four different interior angles meets at a single vertex allows it to cover a plane entirely.

The Study of Pentagons

Regular pentagons cannot tessellate because their interior angles add up to $540^\circ$. However, there are currently 15 known types of convex pentagons that can tessellate. In 1918, Carl Reinhart discovered the first five types. In 1968, a scientist named Kirschner found three additional types and published a proof claiming only eight types existed. This was disproven in 1975 when a computer scientist found a ninth type after reading about the topic in Scientific America.

Following these discoveries, Marjorie Rice, a mother with a high school math education, identified four more types using her own unique notation at her kitchen table. Two more types were discovered later, with the most recent being found in 2015. It remains unknown if more than 15 types of convex pentagons exist.

Limitations of Higher-Order Polygons

In addition to his work on pentagons, Carl Reinhart discovered in 1918 that there are only three types of hexagons capable of tessellating. Mathematical research currently suggests that no convex polygon with seven or more sides can tessellate a surface.