Semi-regular Tessellations

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find all the semi-regular tessellations?

A semi-regular tessellation has two properties:
  • It is formed by two or more regular polygons, each with the same side length
  • Each vertex has the same pattern of polygons around it.
    (more information is provided below)

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Regular tessellations use identical regular polygons to fill the plane. The vertices of each polygon must coincide with the vertices of other polygons.

You can produce exactly three regular tessellations:

with equilateral triangles
tessellation of equilateral triangles


with squares

tessellation of squares


and with regular hexagons

tessellation of hexagons

Can you convince yourself that there are no more?


You can also use regular polygons to make semi-regular tessellations (or Archimedean tessellations) .

A semi-regular tessellation has two properties:
  • It is formed by two or more regular polygons, each with the same side length
  • Each vertex has the same pattern of polygons around it.

For example, triangle, hexagon, triangle, hexagon (or 3.6.3.6) meeting at each point produces

triangles and hexagons tessellations

whereas triangle, triangle, triangle, square, square (or 3.3.3.4.4) meeting at each point produces

tessellation of squares and triangles

Can you find all the semi-regular tessellations?
Can you show that you have found them all?

To help you when you are working away from the computer, click below for multiple copies of the different polygons. You can print them, cut them out and use them to test which polygons fit together: 3 4 5 6 8 9 10 12


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Published May 2006.