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2. Tessellations
A tessellation is a repeating pattern that tiles the plane without leaving any gaps.
Which of these will tile the plane?
A regular tessellation is made from a single regular polygon.
Square 4
Regular Pentagon 5
Equilateral Triangle 3
Regular Hexagon 6
Regular Octagon 8

3. Equilateral Triangles:
Do tessellate 

4. Squares:
Do tessellate 

5. Regular Pentagons:
Don’t tessellate
They make a nice ring.

6. Regular Hexagons:
Do tessellate 

7. Regular Octagons: Don’t tessellate:
This is called a semi-regular tessellation since more than one regular polygon is used.
Regular Octagons:
Don’t tessellate:
But they make a great kitchen floor when put with squares.

8. 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

9. Interactive There are 8 distinct semi-regular tessellations.
The tiles for 4 of them are shown below.
Click button to view
Click to exit
Interactive

14. Non-Regular tessellations.
Other shapes such as rectangles will tessellate.

15. In fact any quadrilateral will tessellate.

16. Drawing Tessellations
Show that the kite tessellates. Draw at least 6 more on the grid

17. Drawing tessellations
Show that the hexagon tessellates. Draw at least 6 more on the grid
Drawing tessellations

18. Drawing tessellations
Show that the trapezium tessellates. Draw at least 8 more on the grid
Drawing tessellations

19. Investigating Polygon Tiles
6 x 60o = 360o
4 x 90o = 360o
3 x 120o = 360o
By considering the interior angles about the indicated node, what are the conditions needed to produce a tile.
135o
135o
90o
A tessellation can be described by its Schlfli symbol.
2 x 135o + 90o= 360o

20. By completing the table below for regular polygons and by considering the different combinations of interior angles can you find other tiles. Only 8 of these tiles will form semi-regular tessellations. Can you find these? A semi-regular tessellation is one in which the same polygons in the same order appear at every node.
Dodecagon
Hendecagon
Decagon
Nonagon
Octagon
Heptagon
Hexagon
Pentagon
Square
Triangle
Interior
Exterior
Sides
Shape

21. By completing the table below for regular polygons and by considering the different combinations of interior angles can you find other tiles. Only 8 of these tiles will form semi-regular tessellations. Can you find these? A semi-regular tessellation is one in which the same polygons in the same order appear at every node.
Shape
Sides
Exterior
Interior
Triangle 3
120o
60o
Square 4
90o
Pentagon 5
72o
108o
Hexagon 6
Heptagon 7
51.42…o
128.57…o
Octagon 8
45o
135o
Nonagon 9
40o
140o
Decagon 10
36o
144o
Hendecagon 11
32.72…o
147.27…o
Dodecagon 12
30o
150o
2 x 135o + 90o

22. There are 8 Semi-Regular Tessellations
There are 8 Semi-Regular Tessellations

23. The 3 Regular and 8 Semi-Regular Tessellations

24. Master of Tessellations
M. C. Escher
M.C. Escher
Master of Tessellations

25. Maurits C. Escher was a Dutch artist, draftsman, book illustrator, tapestry designer and printmaker. While living in Rome from 1922 to 1935, he spent the spring and summer months travelling throughout Italy to make drawings. Later, in his studio in Rome, Escher developed these into prints. Whether depicting the winding roads of the Italian countryside, the dense architecture of small hillside towns, or details of massive buildings in Rome, Escher often created enigmatic spatial effects by combining various -- often conflicting -- vantage points, for instance, looking up and down at the same time. He frequently made such effects more dramatic through his treatment of light, using vivid contrasts of black and white. After Escher left Italy in 1935, his interest shifted from landscape to something he described as "mental imagery," often based on theoretical premises. This was prompted in part by a second visit in 1936 to the fourteenth-century palace of the Alhambra in Granada, Spain. The lavish tile work adorning the Moorish architecture suggested new directions in the use of colour and the flattened patterning of interlocking forms. Replacing the abstract patterns of Moorish tiles with recognizable figures, in the late 1930s Escher developed "the regular division of the plane.“ N National Gallery for Art. Washington DC

34. Use different coloured card for each polygon
Polygon cut outs
Use different coloured card for each polygon