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Tantalising Tessellations Everybody knows that squares tessellate. What exactly is meant by this?
A tessellation originally was the result of covering an area with tesserae – the small square blocks used by the Romans to make mosaics.
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
This is a tessellation
This is not a tessellation Can you spot the odd tiles out?
This is not a tessellation Can you spot the odd tiles out?
Can you show that a tessellation can be made from any parallelogram?
If you can think of parallelograms forming strips, you can also see that they fit together very easily.
Can you show that a tessellation can be made from any triangle?
If you think of it, a parallelogram can be made from two triangles. Finding other patterns could be more worthwhile.
How would you show that no tessellation is possible from a regular pentagon?
The 108 o corners cannot be fitted together to form 360 o
It stands to reason then that if the corners do not add up to 360 o the shapes will not tessellate.
The fact is … only three regular shapes will tessellate
The equilateral triangle
The regular hexagon
Extension 1 Do all pentagons with one pair of parallel lines tessellate?
Extension 2 Are there other pentagons which tessellate?
The second quadrilateral was obtained by rotating the first through 180 o about O, the midpoint of a side.
Keep doing this and you will soon see a tessellation of quadrilaterals
Extension 3 Try the method used in Extension 2 for re-entrant quadrilaterals such as …
Why does this method always work?
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What is a tessellation? A tessellation is a pattern of repeating figures that fit together with NO overlapping or empty spaces. Tessellations are formed.
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