Which of the shapes below tessellate What about

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? Complete the tables in your pairs …

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

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

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

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

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

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

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

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

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

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

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?

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.