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Semi-regular Tessellations

Age 11 to 16
Challenge Level Yellow star
Secondary curriculum
  • Problem
  • Getting Started
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Goeun from Bangok Patana School in Thailand sent in this solution, which includes 8 semi-regular tesselations. Can Goeun be sure to have found them all?
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.
Students from Cowbridge Comprehensive School in Wales used this spreadsheet to convince themselves that only 3 polygons can make regular tesselations.

Goeun continued:
We can prove that a triangle will fit in the pattern {3, 4, 3, 3, 4} [given in the problem] because 360 - (90 + 60 + 90 + 60) = 60 which is the internal angle for an equilateral triangle.

There are 8 semi-regular tessellations in total. We know each is correct because again, the internal angle of these shapes add up to 360.
For example, for triangles and squares, 60 $\times$ 3 + 90 $\times$ 2 = 360.

- 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
 
See this article for more on the notation introduced in the problem, of listing the polygons which meet at each point.

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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