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

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.

Equilateral Triangles

Squares

Regular Pentagons

Regular Hexagons

Regular Heptagons

Regular Octagons

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 …

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

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

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

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

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

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

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

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

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.

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?

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?

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

Explain why a regular decagon will not tessellate

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

a) will not tessellate on their own

b) will tessellate with equilateral triangles

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.