don steward
mathematics teaching 10 ~ 16

Monday, 23 July 2012

frieze patterns

these Maori designs are good examples of frieze or strip patterns
powerpoint: frieze pattern types

the topic seems an interesting way of following up work on symmetry

Heather McLeay (Bangor University, North Wales)  has championed including a study of frieze (strip) patterns in the curriculum; her photographs of the seven different types are here

a frieze is an infinite strip of a repeating pattern, extending in one direction only
the symmetries talked about refer to the whole strip rather than sections of it

considering the isometries (distance preserving transformations): reflection, translation, rotation and glide reflection
and which combinations of these are not redundant
you can establish that there can only be 7 types of frieze pattern in the world

a glide reflection is where you reflect and then move the shape along a bit (like footsteps in the sand)

the Illuminations program allows generating shapes to be modified (by dragging) which might help understanding

there are several notations for classifying the 7 strip patterns and, once appreciated, students can be asked to analyse the symmetries and so recognise the types in various examples

the symmetries are of the whole (i.e. infinitely long) strip rather than just a small 'window' section

some symmetries take precedence in the notations

this notation is due to the IUC (international union of crystallography, there are (annoying) variations: sometimes 'a' is used instead of 'g'and sometimes the positions of the three symbols are different in different texts)
[the convention in these resources is from Wikipedia]

a clear youtube clip, geese

John Conway devised his own notation, based on images of the symmetries as various ways of 'stepping' (students could act these out)

an identification key
which of the 7 types are which?

the 17 possible wallpaper patterns
a wallpaper key

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