Tessellation

A pattern of shapes that fit perfectly together!

A Tessellation (or Tiling) is when we cover a surface with a pattern of flat shapes so that there are no overlaps or gaps.

Examples:

tessellation rectangles
Rectangles
tessellation octagons and squares
Octagons and Squares
tessellation pentagons
Different Pentagons

Regular Tessellations

A regular tessellation is a pattern made by repeating a regular polygon.

There are only 3 regular tessellations:

tessellation regular triangles
Triangles
3.3.3.3.3.3
tessellation regular squares
Squares
4.4.4.4
tessellation regular hexagons
Hexagons
6.6.6

Look at a Vertex ...

tessellation vertexA vertex is just a "corner point".

What shapes meet here?

Three hexagons meet at this vertex,
and a hexagon has 6 sides.

So this is called a "6.6.6" tessellation.

tessellation notation 6.6.6
 

For a regular tessellation, the pattern is identical at each vertex!

Semi-regular Tessellations

A semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same!

There are only 8 semi-regular tessellations:

tessellation 3.3.3.3.6
3.3.3.3.6
tessellation 3.3.3.4.4
3.3.3.4.4
tessellation 3.3.4.3.4
3.3.4.3.4
tessellation 3.4.6.4
3.4.6.4
tessellation 3.6.3.6
3.6.3.6
tessellation 3.12.12
3.12.12
tessellation 4.6.12
4.6.12
tessellation 4.8.8
4.8.8
right arrow

To name a tessellation, go around a vertex and write down how many sides each polygon has, in order ... like "3.12.12".

right arrowAnd always start at the polygon with the least number of sides, so "3.12.12", not "12.3.12"

 

questionQuestion 1: For the tessellations above, is the pattern the same at each vertex?
questionQuestion 2: One of those patterns becomes different when we make a mirror-image of it ... which one?

Other Tessellations

There are also "demiregular" tessellations, but mathematicians disagree on what they actually are!

And some people allow curved shapes (not just polygons) so we can have tessellations like these:

tessellation curvy
Curvy Shapes
tessellation circles lines
Circles
tessellation eagles
Eagles?

Tessellation Artist

All these images were made using Tessellation Artist, with some color added using a paint program.

You can try it too - maybe you will invent a new tessellation!