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Semi-regular Tessellations

Stage: 3 Challenge Level: Challenge Level:1

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:
 regular tessellations

Can you convince yourself that there are no more?

 
Semi-regular tessellations (or Archimedean tessellations) have two properties:
  • They are formed by two or more types of regular polygon, each with the same side length
  • Each vertex has the same pattern of polygons around it.
Here are two examples:
 two semiregular tessellations

In the first, triangle, triangle, triangle, square, square {3, 3, 3, 4, 4} meet at each point.
In the second, triangle, hexagon, triangle, hexagon {3, 6, 3, 6} meet at each point.

Can you find all the semi-regular tessellations?
Can you show that you have found them all?
 
The interactivity below can be used to test your ideas.
 
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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