Activity 1-14: Wallpaper Patterns

www.carom-maths.co.uk

Wallpaper patterns all have one thing in common; they are all produced by repeating a single fundamental tile by translation.

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.)

They may have ROTATIONAL symmetry; the pattern may rotate onto itself before rotating 360o back to itself.

An order 6 example…

Fundamental Tile

They may have one or more REFLECTION symmetries; there may be mirror lines in the pattern.

Fundamental Tile

They may show GLIDE REFLECTIONS; the pattern may reflect in a ‘mirror’ line before translating

(parallel to the mirror line) onto itself.

Fundamental Tile

And a tiling may contain a combination of such symmetries.

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…

A big question: how many possible different combinations of symmetries are possible?

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 in opposite directions 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.

But…

For certain values of n, this does not happen.

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.

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.

If you click below, you will find pictures of the seventeen patterns,drawn by Miranda Lundy, author of the excellent Sacred Geometry.

Task: cut these out and draw up a poster showing the different patterns,

using the grid given earlier.

Note that each of the patterns has a name,given in the grid.

http://www.s253053503.websitehome.co.uk/

carom/carom-files/carom-1-14.pdf

Seventeen Wallpaper Patterns

http://www.amazon.co.uk/Sacred-Geometry-Wooden-Books-Gift/dp/1904263046/ref=la_B001KHN7TI_1_1?s=books&ie=UTF8&qid=1403356030&sr=1-1

Sacred GeometryBook link

There is a famous urban myth that the Alhambra Palace in Granada features all seventeen patterns within it. John Jaworski’s excellent ebook on the subjectcan be bought below.

There is a fun applet below that enables you to create the seventeen patterns

from a given image.

http://www.amazon.co.uk/Mathematicians-Guide-Alhambra-John-

Jaworski-ebook/dp/B00EIADK5K

A Mathematician’s Guide to the Alhambra

http://www.singsurf.org/wallpaper/wallpaper.php‘Create the 17 Patterns’ Link

With thanks to:Wikipedia, for a brilliant article on Wallpaper groups,

and to John Jaworski. Miranda Lundy and Wooden Books.

Carom is written by Jonny Griffiths, hello@jonny-griffiths.net