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Tantalising Tessellations Everybody knows that squares tessellate. What exactly is meant by this?

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Presentation on theme: "Tantalising Tessellations Everybody knows that squares tessellate. What exactly is meant by this?"— Presentation transcript:

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2 Tantalising Tessellations Everybody knows that squares tessellate. What exactly is meant by this?

3 A tessellation originally was the result of covering an area with tesserae – the small square blocks used by the Romans to make mosaics.

4 Nowadays the word tessellation is used to represent any tiling of a plane surface by a regular pattern of one or more congruent, non-overlapping shapes

5 This is a tessellation

6 This is not a tessellation Can you spot the odd tiles out?

7 This is not a tessellation Can you spot the odd tiles out?

8 Can you show that a tessellation can be made from any parallelogram?

9 If you can think of parallelograms forming strips, you can also see that they fit together very easily.

10 Can you show that a tessellation can be made from any triangle?

11 If you think of it, a parallelogram can be made from two triangles. Finding other patterns could be more worthwhile.

12 How would you show that no tessellation is possible from a regular pentagon?

13 The 108 o corners cannot be fitted together to form 360 o

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15 It stands to reason then that if the corners do not add up to 360 o the shapes will not tessellate.

16 The fact is … only three regular shapes will tessellate

17 The equilateral triangle

18 The square

19 The regular hexagon

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21 Extensions

22 Extension 1 Do all pentagons with one pair of parallel lines tessellate?

23 Extension 2 Are there other pentagons which tessellate?

24 Extension 2

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36 The second quadrilateral was obtained by rotating the first through 180 o about O, the midpoint of a side.

37 Extension 2

38 Keep doing this and you will soon see a tessellation of quadrilaterals

39 Extension 3 Try the method used in Extension 2 for re-entrant quadrilaterals such as …

40 Extension 3

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42 Why does this method always work?


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