1. Tessellations Objective:
Wednesday, 09 November 2016
Tessellations
Objective:
To understand and construct tessellations using polygons
MC Escher´s work
Create a non-regular shape and tessellate

2. What shapes are used to make up the honeycomb?
Can these shapes be arranged so that there are no gaps between them?
A regular tessellation is a repeating pattern of a regular polygon, which fits together exactly, leaving NO GAPS.

3. What does this have to do with tessellations?
A regular tessellation is a repeating pattern of a regular polygon, which fits together exactly, leaving NO GAPS.
So the bees honeycomb… is a regular tessellation of hexagons

4. Which regular polygons tessellate?

5. Equilateral Triangles:
Do tessellate 

6. Which regular polygons tessellate?

7. Squares:
Do tessellate 

8. Which regular polygons tessellate?

9. Regular Pentagons:
Don’t tessellate

10. Which regular polygons tessellate?

11. Regular Octagons:
Don’t tessellate:
This is called a semi-regular tessellation since more than one regular polygon is used.

12. Non Regular Tessellations
A non-regular tessellation is a repeating pattern of a non-regular polygon, which fits together exactly, leaving NO GAPS.

13. Drawing tessellations
example
Does this shape tessellate?

14. Drawing tessellations
example
Does this shape tessellate?

15. Drawing tessellations
How about this one?

16. Non Regular Tessellations
There are tessellations all around us!!!

17. Geometric shapes & Art
Maurits Cornelis Escher (usually referred to as M. C. Escher) born in 1898 in Leeuwarden in Holland, was a Dutch graphic artist. He is developed Tessellating shapes as an art form
Let’s take a look at some of his work…

18. MC ESCHER
BY DISTORTING BASIC SHAPES HE CHANGED THEM INTO ANIMALS, BIRDS AND OTHER FIGURES 18

19. Your turn
Step 1: Draw a line between two adjacent corners on one of the sides of the square. Your line can be squiggly or made up of straight segments. Whatever its shape, your line must connect two corners that share one side of the rectangle.
Step 2: Cut along the line you drew. Take the piece you cut off and slide it straight across to the opposite long side of the rectangle. Line up the long, straight edges of the two pieces and tape them together.
Step 3: Now draw another line that connects two adjacent corners on either side of the shape. Cut along this new line. Take the piece you cut off and slide it straight across to the opposite side of the shape. Line up the straight edges and tape them together.

20. Your turn
Step 4: On your grid paper, carefully trace around your pattern shape. It may help to position the squared-off corner (formerly the edge of the index card) in one corner of the grid.

21. Plenary 360 n What’s the size of an exterior angle of a regular:
What’s the size of an interior angle of a regular:
a) square?
b) pentagon?
c) hexagon?
a) square?
b) pentagon?
c) hexagon?
180 – 90 = 90o
360 4
= 90o
360 5
= 72o
180 – 72 = 108o
360 6
= 60o
180 – 60 = 100o

22. Consider the sum of the interior angles about the indicated point.
There are only 3 regular tessellations. Can you see why?
60o
60o
120o
90o
90o
Consider the sum of the interior angles about the indicated point.
120o
6 x 60o = 360o
4 x 90o = 360o
3 x 120o = 360o
108o
135o
36o
90o
2 x 135o = 270o
3 x 108o = 324o

23. Regular Polygon
Size of each exterior angle
Size of each interior angle
Does this polygon tessellate?
Equilateral Triangle
Square
Regular Pentagon
Regular Hexagon
Regular Octagon
Regular Decagon
360 3
= 120o
360 60
= 6
Yes
180 – 120 = 60o
360 4
= 90o
360 90
180 – 90 = 90o
= 4
Yes
360 5
= 72o
360
108
= 3.33 No
180 – 72 = 108o
360 6
360
120
= 60o
= 3
Yes
180 – 60 = 120o
360 8
360
135
= 45o
= 2.67 No
180 – 45 = 135o
360 10
= 36o
360
144
= 2.5
180 – 36 = 144o No