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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 108o
corners cannot be fitted together to form 360o
The 108o
corners cannot be fitted together to form 360o
It stands to reason then that if the corners do
not add up to 360o the shapes will not
tessellate.
The fact is …
only three regular shapes will tessellate
The equilateral triangle
The square
The regular hexagon
The regular hexagon
Extensions
Extension 1
Do all pentagons with one pair of parallel lines tessellate?
Extension 2
Are there other pentagons which tessellate?
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
Extension 2
The second quadrilateral was obtained by rotating
the first through 180o about O, the midpoint of a
side.
Extension 2
Extension 2
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 …
Extension 3
Extension 3
Extension 3
Why does this method always work?
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