a short tale about two small tiles
TRANSCRIPT
A Short Tale about Two Small TilesAuthor(s): Bill RichardsonSource: Mathematics in School, Vol. 29, No. 1 (Jan., 2000), pp. 16-17Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212062 .
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A short tale about two small tiles
by Bill Richardson
Tiling is a very visual and attractive aspect of classroom mathematics. The object of this article is to look at some tilings generated by a pair of quadrilaterals. Each quadrilateral is a kite and they are such that their sides are actually equal, as is one of their angles. As shown in the diagram, one of the kites is re-entrant (or convex).
Small tile Bigger tile
At first, it does not matter precisely what size the angles are, other than the angles marked actually being equal.
Individually, the two tiles will cover a plane. For example:
It is easy to create tilings which involve both tiles.
However, by resisting the temptation to tile 'horizontally' and also by a careful choice of the angles indicated in the first pair of diagrams, a very spectacular tiling can be achieved. The angle is made to be 720 and the other angles in the 'small kite' to be 360, 360 and 2160 and in the 'large kite' they are 72o, 720 and 1440.
~6~
/72a o
~6~
/72
1440 o\ 72 o z
,72",
16 Mathematics in School, January 2000
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The result of this is:
which is an example of a 'Penrose Tiling'. Such tilings are named after Professor Roger Penrose (Rouse Ball Professor of Mathematics at the University of Oxford) who discovered them in the 1960s. As well as being truly spectacular, this tiling has five-fold symmetry which was thought to be impossible. They are mentioned in the book 'The Emperor's New Mind' by Penrose, published by OUP (ISBN 0 19 851973 7). The idea has been taken up commercially by Pentaplax Ltd. (01484 401122 http://www.yorkshirenet.co.uk/pentaplax). This company has produced a 'jigsaw puzzle' in which the simple kite tiles are metamorphosed into two bird shapes. The product is called 'Perplexing Poultry'. In developing the tiling above, Imade use of the bird tiles as well as a
computer diagram. You should be aware that the tile arrangements near the edges may be a bit speculative. It reminds me of pictures of old maps - Terra Incognito. So what next? One could try to continue the pattern but that could be quite difficult. Or perhaps try values other than 72o for the angle - a task for the future!
References Penrose, R. The Emperor's New Mind, OUP (ISBN 0 19 851973 7). Perplexing Poultry, Pentaplax Ltd. (01484 401122 http://www.yorkshirenet.co.uk/pentaplax). Keywords: Tessellations; Penrose; Tiling.
Author Bill Richardson, Elgin Academy, Elgin IV30 4ND. e-mail: [email protected]
Mathematics in School, January 2000 17
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