Activity 1-14: Wallpaper Patterns

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www.carom-maths.co.uk
Activity 1-14 Wallpaper Patterns
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Wallpaper patterns all have one thing in common
they are all produced by repeating a single
fundamental tile by translation.
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That one basic rule gives rise to a range of
possible patterns, each with their distinctive
symmetries (the edges of the fundamental tiles
are not taken as part of the pattern.)
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They may have ROTATIONAL symmetry the pattern
may rotate onto itself before rotating 360o back
to itself. An order 6 example
Fundamental Tile
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They may have one or more REFLECTION symmetries
there may be mirror lines in the pattern.
Fundamental Tile
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They may show GLIDE REFLECTIONS the pattern may
reflect in a mirror line before translating
(parallel to the mirror line) onto itself.
Fundamental Tile
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And a tiling may contain a combination of such
symmetries.
A big question how many possible different
combinations of symmetries are possible?
To start with we might ask, what orders of
rotational symmetry are possible?
It turns out that the pattern cannot have order
of rotational symmetry greater than 6.
The proof is surprisingly easy
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Given a tiling with rotational symmetry order
n,pick two centres of rotational symmetry that
are a minimum distance apart.
Now rotate the pattern 360/n degrees about each
centre. What happens to these two centres?
It appears that the two rotated centres are
closer than the originals, contradicting the
minimum distance apart rule.
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But
For certain values of n, this does not happen.
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So a wallpaper pattern can only have rotational
symmetry of order 1, 2, 3, 4 or 6.
For all other n, however, the contradiction comes
into play.
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It transpires that when you add reflection and
glide reflection,there are only seventeen
different possible patterns.The grid below shows
how to classify these.
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If you click here, you will find pictures of the
seventeen patterns, taken from the wonderful
book Sacred Geometry by Miranda Lundy.
Note that each of the patterns has a name, given
in the grid.
Task cut these out and draw up a poster showing
the different patterns, using the grid given
earlier.
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There is a famous urban myth that the Alhambra
Palace in Granada features all seventeen
patterns within it. John Jaworskis excellent
ebook on the subject can be downloaded here.
There is a fun applet below that enables you to
create the seventeen patterns from a given
image.
http//www.singsurf.org/wallpaper/wallpaper.php
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With thanks toWikipedia, for a brilliant
article on Wallpaper groups, to Miranda Lundy,
and toJohn Jaworski and mathMedia Ltd.
Carom is written by Jonny Griffiths,
mail_at_jonny-griffiths.net