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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 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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