1. Which of the shapes below tessellate?
What about a regular heptagon and a regular octagon?

2. Defining Tessellation
A tessellation can be defined as the covering of a surface with a repeating unit consisting of one or more shapes so that:
There are no spaces between, and no overlapping.
The covering process has the potential to continue indefinitely.

3. Equilateral Triangles

4. Squares

5. Regular Pentagons

6. Regular Hexagons

7. Regular Heptagons

8. Regular Octagons

9. So why can some shapes tessellate while others do not
So why can some shapes tessellate while others do not? Complete the tables in your pairs …

10. Which regular polygons tessellate?
Size of each exterior angle
Size of each interior angle
Does this polygon tessellate?
Equilateral Triangle
Square
Regular Pentagon
Regular Hexagon
Regular Octagon

11. Which regular polygons tessellate?
Size of each exterior angle
Size of each interior angle
Does this polygon tessellate?
Equilateral Triangle
Square
Regular Pentagon
Regular Hexagon
Regular Octagon
360 3
= 120o
Yes
180 – 120 = 60o
360 4
= 90o
180 – 90 = 90o
Yes
360 5
= 72o
180 – 72 = 108o No
360 6
= 60o
180 – 60 = 120o
Yes
360 8
= 45o
180 – 45 = 135o No

12. There are only 3 regular tessellations. Why?
Let’s look at the tessellations in more detail.
What is the size of the interior angle of an equilateral triangle?
60o
60o

13. There are only 3 regular tessellations. Why?
Let’s look at the tessellations in more detail.
What is the size of the interior angle of a square?
90o
90o

14. There are only 3 regular tessellations. Why?
Let’s look at the tessellations in more detail.
What is the size of the interior angle of a pentagon?
108o

15. There are only 3 regular tessellations. Why?
Let’s look at the tessellations in more detail.
What is the size of the interior angle of a hexagon?
120o
120o
120o

16. There are only 3 regular tessellations. Why?
Let’s look at the tessellations in more detail.
What is the size of the interior angle of an octagon?
135o

17. There are only 3 regular tessellations. Why?
3 x 120o = 360o
6 x 60o = 360o
60o
60o
90o
90o
120o
120o
4 x 90o = 360o
108o
135o
3 x 108o = 324o
2 x 135o = 270o

18. What conditions must exist for a polygon to tessellate?
In your pairs:
What conditions must exist for a polygon to tessellate?
A polygon must have an interior angle that is a factor of 360o in order for it to tessellate.

19. How does the following table show that these shapes do not tessellate?
Discuss:
How does the following table show that these shapes do not tessellate?
What is this number actually telling us?

20. How does the following table show that these shapes do not tessellate?
Discuss:
How does the following table show that these shapes do not tessellate?
The result of dividing 360° by the interior angle is
not an integer.
What does this tell us?

21. How does the following table show that these shapes do not tessellate?
Discuss:
How does the following table show that these shapes do not tessellate?
Therefore, for any of these shapes it is impossible for a whole number of them to meet at a point on the surface in order for it to be covered

22. Explain why a regular decagon will not tessellate

23. Explain why regular dodecagons will not tessellate on their own Challenge: Explain why regular dodecagons can tessellate with equilateral triangles
Super Challenge:
Can you find a combination of 3 regular polygons that can tessellate

24. a) will not tessellate on their own

25. b) will tessellate with equilateral triangles

26. The Dutch graphic artist M C Escher became famous for his tessellations in which the individual tiles are recognisable images such as birds and fish.