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**Tantalising Tessellations**

Everybody knows that squares tessellate. What exactly is meant by this? Welcome to ‘Tantalising Tessellations’ : This investigation is introduced as part of the Maths Extension Programme for Years 4,5 & 6 in Chigwell School. ©DDimoline, Chigwell School, 2001

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

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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

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This is a tessellation

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**This is not a tessellation**

Can you spot the odd tiles out?

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**This is not a tessellation**

Can you spot the odd tiles out?

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

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

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

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**If you think of it, a parallelogram can be made from two triangles.**

Finding other patterns could be more worthwhile.

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

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

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

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

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**only three regular shapes will tessellate**

The fact is … only three regular shapes will tessellate

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**The equilateral triangle**

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The square

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The regular hexagon

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The regular hexagon

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Extensions

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**Do all pentagons with one pair of parallel lines tessellate?**

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

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**Are there other pentagons which tessellate?**

Extension 2 Are there other pentagons which tessellate?

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2

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Extension 2 The second quadrilateral was obtained by rotating the first through 180o about O, the midpoint of a side.

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Extension 2

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

Extension 2 Keep doing this and you will soon see a tessellation of quadrilaterals

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

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Extension 3

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Extension 3

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**Why does this method always work?**

Extension 3 Why does this method always work?

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