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Firstly, there are only three regular tessellations which are triangles, squares, and hexagons. To make a regular tessellation, the internal angle of the polygon has to be a diviser of 360. This is because the angles have to be added up to 360 so it does not leave any gaps. For example, we can make a regular tessellation with triangles because 60 x 6 = 360.
We can prove that a triangle will fit in the pattern {3, 4, 3, 3, 4} because 360 - (90 + 60 + 90 + 60) = 60 which is the internal angle for an equilateral triangle.
There is 8 semi-regular tessellations in total:
- Triangles & Squares
- Triangles & Squares (but a different pattern )
- Hexagons & Triangles
- Hexagons & Triangles (but a different pattern)
- Hexagons & Triangles & Squares
-Octagons & Squares
- dodecagons & triangles
- dodecagons & Squares & Hexagons
We can know this is correct because again, the internal angle of these shapes add up to 360. For example, for triangles and squares, 60 x 3 + 90 x 2 = 360. There are some pictures of these below.