Tessellations

Tessellation Definition

A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.

Another word for a tessellation is a tiling.

Tiling Definition

When you fit individual tiles together with no gaps or overlaps to fill a flat space like a ceiling, wall, or floor, you have a tiling.

What are Tessellations

The word 'tessera' in latin means a small stone cube. They were used to make up 'tessellata' - the mosaic pictures forming floors and tilings in Roman buildings
The term has become more specialised and is often used to refer to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps.

Symmetry

If you look at a completed tessellation, you will see the original motif repeats in a pattern. One mathematical idea that can be emphasized through tessellations is symmetry.

There are 17 possible ways that a pattern can be used to tile a flat surface or 'wallpaper'.

Polya: 17 ways to tile a surface

Escher read Pólya's 1924 paper on plane symmetry groups.Escher understood the 17 plane symmetry groups described in the mathematician Pólya's paper, even though he didn't understand the abstract concept of the groups discussed in the paper.

Between 1936 and 1942 Escher produced 43 colored drawings with a wide variety of symmetry types while working on possible periodic tilings. He adopted a highly mathematical approach with a systematic study using a notation which he invented himself.

Four Types of Symmetry in a Plane

There are 4 ways of moving a motif to another position in the pattern. These were described by Escher.

Translation
Reflection
Rotation
Glide Reflection

Translation

Reflection

Rotation

Glide Reflection

A translation is a shape that is simply translated, or slid, across the paper and drawn again in another place.

The translation shows the geometric shape in the same alignment as the original; it does not turn or flip.

A reflection is a shape that has been flipped. Most commonly flipped directly to the left or right (over a "y" axis) or flipped to the top or bottom (over an "x" axis), reflections can also be done at an angle.

If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical "mirror" images. To reflect a shape across an axis is to plot a special corresponding point for every point in the original shape.

Rotation is spinning the pattern around a point, rotating it. A rotation, or turn, occurs when an object is moved in a circular fashion around a central point which does not move.

A good example of a rotation is one "wing" of a pinwheel which turns around the center point. Rotations always have a center, and an angle of rotation.

In glide reflection, reflection and translation are used concurrently much like the following piece by Escher, Horseman. There is no reflectional symmetry, nor is there rotational symmetry.