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Semi-regular Tessellations

Stage: 3 Challenge Level: Challenge Level:1

Students from Cowbridge Comprehensive School in Wales took a look at this problem. They convinced themselves that there are only 3 regular polygons that can be used to create regular tessellations. Some used this spreadsheet to justify their conclusion.

Catherine, from St. Michael's School, explained why it is possible to produce a semi-regular tessellation with triangle, hexagon, triangle, hexagon (or 3,6,3,6) meeting at each point:

In equilateral triangles angles are all $60^\circ$.
In regular hexagons angles are all $120^\circ$.
$2$ x $120^\circ$ = $240^\circ$ and $60^\circ$ x $2$ = $120^\circ$
$120^\circ$ + $240^\circ$ = $360^\circ$!

She then offered another semi-regular tessellation:

$4, 8, 8$ (square, octagon, octagon) meeting at each point.

You can use the interactivity provided with the problem to check that this combination works.

Classes 9S and 9W from Aylesbury High School tackled this problem.

We found that there were $7$ semi-regular tessellations. We made a list of all the interior angles for regular polygons. We tried all the combinations which add up to $360^\circ$.

Here are the solutions we found:

$150^\circ$ + $150^\circ$ + $60^\circ$: $12, 12, 3$
$150^\circ$ + $120^\circ$ + $90^\circ$: $12, 6, 4$
$135^\circ$ + $135^\circ$ + $90^\circ$: $8, 8, 4$
$120^\circ$ + $120^\circ$ + $60^\circ$ + $60^\circ$: $6, 3, 6, 3$
$120^\circ$ + $90^\circ$ + $90^\circ$ + $60^\circ$: $6, 4, 3, 4$
$120^\circ$ + $60^\circ$ + $60^\circ$ + $60^\circ$ + $60^\circ$: $6, 3, 3, 3, 3$
$90^\circ$ + $90^\circ$ + $60^\circ$ + $60^\circ$ + $60^\circ$: $4, 4, 3, 3, 3$
$150^\circ$ + $90^\circ$ + $60^\circ$ + $60^\circ$: doesn't work
$144^\circ$ + $108^\circ$ + $108^\circ$: doesn't work 
If there are more than $12$ sides, then because the angles are so big the shapes will overlap. So we decided we didn't need to consider any shapes with more than $12$ sides. We did test some out, but they didn't work.

Sue pointed out that actually there are $8$ semi-regular tessellations, because you can have both $4, 4, 3, 3, 3$ and $3, 4, 3, 4, 3$. You can see pictures of them all here.