What about aregular heptagon and a regular
octagon?
Which of the shapes below tessellate?
2.
Defining Tessellation
A tessellationcan 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.
So why cansome shapes tessellate while
others do not?
Complete the tables in your pairs …
10.
Regular
Polygon
Size of each
exteriorangle
Size of each interior
angle
Does this polygon
tessellate?
Equilateral
Triangle
Square
Regular
Pentagon
Regular
Hexagon
Regular
Octagon
Which regular polygons tessellate?
11.
Regular
Polygon
Size of each
exteriorangle
Size of each interior
angle
Does this polygon
tessellate?
Equilateral
Triangle
Square
Regular
Pentagon
Regular
Hexagon
Regular
Octagon
Which regular polygons tessellate?
180 – 90 = 90o
180 – 72 = 108o
180 – 60 = 120o
180 – 120 = 60o
180 – 45 = 135o
360
3
360
4
360
5
360
6
360
8
= 120o
= 72o
= 45o
= 90o
= 60o
Yes
No
No
Yes
Yes
12.
There are only3 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
60o
60o
60o
60o
13.
There are only3 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
90o
90o
14.
There are only3 regular tessellations. Why?
Let’s look at the tessellations in more detail.
What is the size of the interior angle of a
pentagon?
108o
108o
108o
15.
There are only3 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
120o
120o
120o
16.
There are only3 regular tessellations. Why?
Let’s look at the tessellations in more detail.
What is the size of the interior angle of an
octagon?
135o
135o
17.
60o
60o
60o
60o
60o
60o
6 x 60o
=360o
90o
90o
90o
90o
4 x 90o
= 360o
120o
There are only 3 regular tessellations. Why?
120o
120o
3 x 120o
= 360o
108o
108o
108o
3 x 108o
= 324o
135o
135o
2 x 135o
= 270o
18.
A polygon musthave an interior angle that is a
factor of 360o
in order for it to tessellate.
In your pairs:
What conditions must exist for a polygon to
tessellate?
19.
Discuss:
How does thefollowing table show that these
shapes do not tessellate?
What is this number actually telling us?
20.
Discuss:
How does thefollowing 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.
Discuss:
How does thefollowing 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 regulardodecagons 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
