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:


Rectangles

Octagons and Squares

Different Pentagons

Regular Tessellations

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

There are only 3 regular tessellations:


Triangles
3.3.3.3.3.3

Squares
4.4.4.4

Hexagons
6.6.6

Look at a Vertex ...

A 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.

 

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:


3.3.3.3.6

3.3.3.4.4

3.3.4.3.4

3.4.6.4

3.6.3.6

3.12.12

4.6.12

4.8.8

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

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

 

Question 1: For the tessellations above, is the pattern the same at each vertex?
Question 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:


Curvy Shapes

Circles

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!