Sorting Patterns

2. Planet of the Shapes

Sorting Patterns

To study tilings, mathematicians can sort them into different categories.

One way to classify tiling patterns is by their type of symmetry. Symmetry often involves balance, harmony, and proportion. A checkerboard design is symmetric, but random splashes of paint are not. Many natural objects are symmetric, from flowers and butterflies to crystals and leaves.

This moth is said to be symmetric because its right side matches its left side.

There are many different types of symmetry. A design has mirror, or reflectional, symmetry if it can be folded along a line so that all the points on one side of the fold exactly coincide with corresponding points on the other side of the fold.

This tiling pattern has mirror symmetry. The left half is a mirror image of the right half, and the top half is a mirror image of the bottom half.

A design has rotational symmetry if it can be traced and rotated less than a complete cycle about a point so that the tracing matches the original design.

A regular pentagon and a starfish both have fivefold rotational symmetry, because you can turn either object one-fifth of the way around without changing its appearance. An equilateral triangle has threefold symmetry.

The pattern inside the square has threefold rotational symmetry. Turning the pattern 120 degrees (one-third of the way around) matches the original design.

Many symmetric tilings are also periodic, which means that a basic pattern of one or more tiles is repeated throughout the entire tiling design.

Mathematicians have found that there are seventeen different types of symmetry among tiling patterns that repeat themselves. These periodic patterns appear in artwork all over the world, from the intricate tiles on Islamic buildings to the geometric patterns on African textiles.

These examples represent four of the seventeen different types of symmetry possible in a repeating wallpaper pattern.

TRY IT!

Look around in your home or neighborhood to identify designs with different types of symmetry.

Find examples of the different types of repeating wallpaper patterns.

Examples:

A version of the recycling symbol with threefold rotational symmetry.

A star design with fivefold rotational symmetry.

NEXT: Special Tiles