Paving the Plane

2. Planet of the Shapes

Paving the Plane

From the tiles you see on bathroom floors or walls to the designs found on Native American pottery or in palace decorations in India, repeating geometric patterns have been inspiring artists all over the world for thousands of years.

Seed pot with a geometric design created by Native American artist Sandra Victorino, New Mexico.

Intricate design on the ceiling of a palace in Jaipur, India.

Such patterns also fascinate mathematicians.

Tiling (or tessellation) is the process of fitting together flat geometric shapes so that the pieces cover a flat surface—what mathematicians call a plane—without overlapping one another or leaving any gaps. The result is a kind of jigsaw puzzle that stretches off to infinity.

No one knows how many different combinations of tile shapes can fill a plane. There are an enormous number of possibilities, but only the simplest ones have been completely identified and cataloged.

Suppose you start with straight-sided figures called polygons. A regular polygon has all its sides the same length and all its angles the same size. A square is an example of a four-sided regular polygon, with 90-degree angles. An equilateral triangle is an example of a three-sided regular polygon, with 60-degree angles. A regular pentagon has five sides, and each angle between its adjacent sides is 108 degrees.

Checkerboard City, Triangle Terrace, and Honeycomb Haven are examples of how equilateral triangles, squares, and regular hexagons can fit together to cover a flat surface (see "Planet of the Shapes"). If you tried to use regular pentagons, however, you would end up with gaps in your pattern.

A regular pentagon has five sides of equal length and five equal interior angles (left). Tiles of this shape can't fit together without creating overlap or leaving gaps (right).

In fact, there are only three ways to tile a plane with a single type of regular polygon.

The next step is to consider patterns in which two or more regular polygons are fitted together corner to corner, so that the same tiles, in the same order, surround each corner, or vertex.

There are exactly eight ways to do this using various combinations of regular polygons—triangles, squares, hexagons, octagons, and dodecagons (twelve-sided polygons). Any of these eight combinations would make a nice floor tiling.

Two of the eight ways to fit together different types of regular polygons involve dodecagons. One tiling pattern requires dodecagons and equilateral triangles (above) and the other dodecagons, squares, and regular hexagons (below).

Change the tiling rules a little bit to allow other shapes and combinations, and you readily end up with an enormous, possibly unlimited, number of possibilities.

The polygons don't have to be regular, for instance. Lifting the restriction that all sides have to be equal in length and all angles between sides must have the same measure means that you can cover a surface with certain types of pentagons—even though you can't do it with regular pentagons.

Dividing a regular hexagon in half produces two pentagons. Unlike regular pentagons, these pentagons can be used to tile the plane.

Honeycombs

When honey bees mold wax to construct honeycombs for storing honey, they create a grid of hexagonal cells.

USDA ARS

Mathematicians have proved that a hexagonal grid represents the best way to divide a surface into equal areas with the smallest total perimeter (length of the outside edges). When it comes to using wax, bees certainly know how to economize.

NEXT: Sorting Patterns