Ribbons, Friezes and Mathematics  

This is a time of year when I pay attention to ribbons and wrapping paper.

When I saw this  oak leaf ribbon, I had to buy some – just because of its beautiful symmetry.

But what kind of symmetry is it?  Thinking about this question reminded me of Bob Burn, a mathematics teacher educator, who introduced me to reasoning about algebra-geometry in a pleasurable, visual way . So that’s the thread of this blog, which is more about mathematics I enjoy than mathematics education.

 

Ribbons are like friezes, or decorative borders that appear on walls.  They become mathematical when they have a regular repeat in them. There is a section (the ‘motif’) that is ‘translated’ – let’s say moved horizontally – again and again to make the whole repeating pattern. It needs to be translated by the same distance each time: on this ribbon it’s about 3cm. Of course this is a sensible way of manufacturing them.

Every mathematical frieze has a basic translation symmetry defined by the pattern repeat; let’s call it T.  If I translate the frieze using T it looks the same, since all I have done is moved it along by exactly one repeat.  From now on I won’t worry about where the ends of the frieze are:  I am just going to think about its middle, or say it is infinitely long. Then I can say T ‘leaves the frieze invariant’.  In our OU mathematics education modules that’s called finding invariance in the midst of change: the translation is the change, but the overall appearance has not varied.

Here’s another ribbon with a repeating pattern of red and white checks. This time T is a translation of about 3.2cm. I can also see this ribbon has a horizontal line of reflective symmetry along its length. That means there is a horizontal reflection (call it H) that would just swap the top and bottom in the line of symmetry but again leave it looking the same overall.

      

It has other symmetries as well: I can see a vertical line of symmetry through the central one of each group of 3 white stripes, and through the middle of each red space.  So there are also vertical reflections V that leave this pattern invariant.

And there is rotational symmetry R as well.  Every frieze has rotational symmetry of 0ᵒ or 360ᵒ – that is not interesting – but it is possible to rotate this ribbon just by 180ᵒ and leave the pattern invariant.  What I have to do is choose the centre of rotation to be in the very centre of a red rectangle, or the centre of a middle white stripe. That is exactly where the vertical lines of symmetry meet the horizontal one (coincidence?).

This ribbon has four symmetries that leave it invariant: T and also H, V and R.  It’s got so much symmetry that it looks neat and crisp, but may be a bit boring.

There’s one more useful thing to notice about it though: in principle you could choose the motif as being any section of ribbon that is 3.2 cm long. I’ve put four example sections in the picture; each of these works since translating that section gives the whole ribbon. But only two of those motifs have got the same other symmetries as the whole ribbon, that is they have V,H and R. (Its the middle two)

It was Bob Burn who first told me about frieze symmetries. He often wrote about them in the ATM journals. For example, he gave Logo instructions to generate them in Micromaths (Spring 1995).  Bob was a dedicated scholar of mathematics education who worked at Homerton College. This was before it became a college of Cambridge University, and when it specialised in teacher training for undergraduates and postgraduates.  He later moved to Exeter University and was influential in thinking about the pedagogy of undergraduate mathematics education. At Homerton I taught several courses for teachers that Bob had helped design, and they always had two features.

First, they were advanced mathematics courses – but the mathematics included was the kind that underpinned the school curriculum. Teachers learnt about number systems, geometry and symmetry, how to categorise change and identify structures that stayed invariant, how to justify their reasoning and what was considered mathematically beautiful in those fields. It was not the same mathematics as in the school curriculum but it was intended to be the mathematics that would help teachers later, when they or their pupils raised questions about maths. (Now I realise we should have set up a research project to investigate this claim!) This is not a new idea: Felix Klein’s work in mathematics education before 1910 was titled ‘Elementary Mathematics from a Higher Standpoint’ and is definitely undergraduate-level mathematics.

Secondly, these courses for teachers were based on a pedagogic principle of suggesting just enough mathematics for people to start doing mathematics and asking their own questions.  I link that to the OU phrase ‘manipulate; get a sense of; articulate’.

When Bob drew the friezes he used a rather dull basic shape, like a flag, to make his motifs.  I decided to be more seasonal and see what motifs I could come up with. So that year a group of prospective mathematics teachers worked on identifying motifs and symmetries in these seven friezes:








Activity:

Which friezes have Horizontal symmetry? Vertical symmetry? Rotational symmetry? Which have more than one of these? None of these?

Can you find a motif for each frieze? Where there is a choice of motif can you find one that has exactly the same symmetries as the whole frieze?

Can you draw a motif that has V and R symmetries but not H?


One of the reasons I like this activity is that it creates – in me at least – a need for some new mathematics.   While I can ‘see’ horizontal and vertical symmetry (unlike some students) it always takes me a bit of time to appreciate rotational symmetry.  It helps to try and draw my own motifs.

I can also see that the reindeer pattern has some sort of symmetry, but it is not one of the ones that I know about – not H, V or R.  It’s the same kind of symmetry that makes my oak leaf ribbon beautiful, a horizontal reflection that is slightly out of step. Because I had noticed something and needed a name for it, I was delighted when Bob told me about the new bit of mathematics I needed: a glide reflection. First you reflect then you translate (glide) parallel to the reflection line.

The glide reflection G must fit perfectly with the basic repeat translation. The basic motif includes two reindeer, one facing up and one facing down, and I need to translate it through its full width for a pattern repeat. But if I do a glide reflection with the ‘glide’ part being exactly half the length of the motif, then it lands perfectly on itself, leaving the pattern invariant.  So this frieze has a glide reflection G and it has the basic symmetry T, but it does not have H or V or R.

The symmetry in the oak leaf ribbon is also a glide reflection: each leaf is reflected to the other side of the ribbon and moved down by 1.5cm, which is half the pattern repeat. There is also one in the frieze of angels, but that is not the only symmetry there. Once alerted to the existence of glide reflections, you will find you see them quite often. It’s a very attractive design feature, with a lovely sense of direction: a bit like footprints. I challenge you to find it in other decorations.

 

Last thought and activity: There’s a lot about friezes on the internet, usually pointing out that there are seven frieze groups. So: why seven? And particularly, why only seven?  It’s been easy to find friezes with no symmetry apart from T, and ones with just H, just V, just R, just G – that’s five types without even trying. But what about patterns that have H and V, G and R, … or three of these … or even all four? That is the kind of question about mathematical reasoning that I (and I guess Bob Burn) would want teachers to ask and encourage.

Activity

Can you prove that there are only seven distinct types of frieze? 

One approach is to consider all the systematic variations; try to draw an appropriate motif for each one, and see what happens. 

Bob’s articles in the ATM journals show some ways of constructing these motifs with flags although I prefer the creativity of finding images and playing with them in computer  ‘art’ or geometry packages.   Of course he gives no answers, because the point is to enjoy working it out. (If you can access a university library, there is an article by Belcastro and Hull that does). 

References

Belcastro, S.-M., & Hull, T. C. (2002). Classifying Frieze Patterns without Using Groups. The College Mathematics Journal, 33(2), 93–98.

Burn, R. (1995) Friezes with Logo. Micromaths 11:7-8.

Burn, R. (2008) Friezes. Mathematics Teaching 211:21.

Klein, F. (1908, 2016). Elementary Mathematics from a Higher Standpoint. Berlin, Heidelberg: Springer Berlin Heidelberg.

 

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