1. Tessellations
Year 8

2. Eisteddfod Competition
Tessellations
Create a tessellating pattern suitable for tiles in a bathroom or kitchen.
Computer generated images are acceptable.
A5 size is recommended (that is half A4)
Please ensure that your name, form and house name are put on the back of the design.
For further information please see your teacher or Mrs Thomas (Maths).
9/19/2018

3. Eisteddfod 2015
All entries must be submitted to your Maths teacher by Tuesday 2nd March, 2015
Maths Eisteddfod Lunch Time – Room 25 Mrs Thomas
1.30 – 2.00 every day
See your maths teacher for a lunch pass

4. What are Tessellations?
The word 'tessera' in Latin means a small stone cube.
They were used to make up 'tessellata' - the mosaic pictures forming floors and tilings in Roman buildings
Tessellations refers to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps.
A Roman floor mosaic

5. M. C. Escher
Escher is known as the father of tessellations. He was born in the Netherlands in 1898.
He is known as for being an artist as well as a mathematician.
Escher created thousands of tessellating shapes in the form of fish, birds, insects, and other beasts.

6. M. C. Escher Escher – regarded as the 'Father' of modern tessellations
Famous for his impossible depictions like Waterfall
Master of lino and wood cuts and produced many superbly crafted landscapes as well.
'Waterfall'

7. M. C. Escher
Escher produced '8 Heads' in a hint of things to come. Turn the picture upside-down if all the heads are not apparent.
He took a boat trip to Spain and went to the Alhambra.
There, he copied many of the tiling patterns.
'8 Heads'

8. The Alhambra Palace, Granada, Spain, 12th-13th Century

9. Escher’s Last Tessellation
His last tessellation was a solution to a puzzle sent to him by Roger Penrose, the mathematician. Escher solved it and, true to form, changed the angular wood blocks into rounded 'ghosts'.
Penrose 'Ghosts'

10. Tessellation by M. C. Escher
China Boy, 1936
Tessellation by M. C. Escher

11. Tessellation by M. C. Escher
Squirrels, 1936
Tessellation by M. C. Escher

12. Tessellation by M. C. Escher
Fish, 1938
Tessellation by M. C. Escher

13. Design for Wood Intarsia Panel for Leiden Town Hall, 1940
Tessellation
transitions
by M. C.
Escher

14. Tessellation by M. C. Escher
Horsemen, 1946
Tessellation by M. C. Escher

15. Tessellation by M. C. Escher
4 Motifs, 1950
Tessellation by M. C. Escher

16. Tessellation by M. C. Escher
Scarabs, 1953
Tessellation by M. C. Escher

17. Tessellation mural by M. C. Escher
Fishes, 1958 Mural
Tessellation mural by M. C. Escher

18. Tessellation by M. C. Escher
Pegasus, 1959
Tessellation by M. C. Escher

19. Tessellation by M. C. Escher
Birds, 1967
Tessellation by M. C. Escher

20. Realism & Tessellations Combined
Sometimes, M. C. Escher would combine realism and tessellations.
Reptiles is an example of this combination.
'Reptiles'

21. by M. C. Escher Realism & Tessellation Combined
Metamorphosis I, 1937
by M. C. Escher Realism & Tessellation Combined

22. Realism & Tessellation Combined
Cycle, 1938
by M. C. Escher
Realism & Tessellation Combined

23. by M. C. Escher Realism & Tessellation Combined
Day and Night, 1938
by M. C. Escher Realism & Tessellation Combined

24. A Full Life Escher died on March 27, 1972. He had produced
448 woodcuts, linocuts and lithos and
over 2,000 drawings.
M. C. Escher

25. What is a tessellation?
Any repeating pattern of shapes that cover a plane without overlap is considered a tessellation. The pattern must be able to go on forever.
Link
Tessellations and tilings can be found in many cultures, both ancient and modern.
Japanese
Egyptian
Islamic
Link

26. Patchwork quilts (strong tradition in Wales)

27. Tessellations are all around us.
Can you think of think of some examples?

28. Tessellations cover many of our school subjects.
Geography PE
Art RE

29. Tiles

30. Pavements or roads
Roman tessellated pavement found in Mumbles in Swansea.

31. Fishing nets

32. A honeycomb Each hexagonal section tessellates with the rest.

33. Fish scales

34. Footballs

35. Soles of trainers etc.

36. The Giant’s Causeway in Northern Island

37. Eaglehawk neck tessellated pavement
Natural salt pools
Connecting the Tasman Peninsula to Tasmania is covered in a pattern of regular rectangular saltwater pools.

38. The Eden Project Cornwall

39. Tessellation planning
In Tessellation Planning, the pattern and colours are not just for visual effect, they represent spaces and their uses.

40. 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

41. Equilateral Triangles:
Do tessellate 

42. Squares:
Do tessellate 

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

44. Regular Hexagons:
Do tessellate 

45. 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.

46. 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

47. 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

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

56. In fact any quadrilateral will tessellate.

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

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

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

67. 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

68. 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

69. 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

70. Do it yourself!

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