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 ...




For a regular tessellation, the pattern is identical at each vertex! 
Semiregular Tessellations
A semiregular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same!
There are only 8 semiregular 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 all the tessellations above, is the pattern the same at each vertex?  
Question 2: One of those patterns becomes different when we make a mirrorimage 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!
World Tessellation Day
World Tessellation Day is June 17. Celebrate!