#
Mathematical

Salad

## Paper view

20 April 2018

Origami (from the Japanese for ‘to fold’ + ‘paper’) is used the world over to produce beautiful creations that sit in the fascinating intersection of mathematics and art. I’ve been in a multitude of sessions and workshops where I am in awe of not only the beautiful constructions made possible but also the wonderful maths behind them.

The recent British Congress of Mathematics Educators (BCME 9) was no different. Amongst the sessions I attended I was able to take part in a variety of beautiful tile constructions with Dr Jennie Golding.

One of the shapes we produced was a regular hexagon folded from a piece of A4. It’s a simple process but opens up a world of *what ifs?* and *whys?*

Folding the shapes is great for dexterity and hand-eye co-ordination, and then you can investigate the properties of the created shapes and experience the mathematics enacted in a creative activity. Recognising that 3D shapes (and the boundaries of those that are possible) can be created from a 2D material is a useful idea, particularly when you start to consider nets. Visualisation skills are enhanced as you start to predict what is being constructed. Shapes are met in a wide range of orientations and transformations; corresponding vertices, angles and edges can be identified.

As pupils meet various angle properties, congruency, similarity, trigonometry and different-but-equally-valid constructions can be investigated and justified. Initial constructions can be communicated in a variety of ways: written or spoken instructions, diagrams, a video or a combination. Getting pupils to communicate their own constructions in any format also gives them an opportunity to experience the challenges of designing clear instructions for others.

Now there are lots of ways to construct a regular hexagon, but I particularly liked this one as it didn’t rely on any use of trigonometry or the ratio of side of A4 in the explanation of why it worked. In fact, I’m going to go several steps back and show how to construct an angle of 60º using any rectangular sheet of paper.

Here’s how to do it…

Getting someone to explain why they know this angle has to be 60º just by looking at the final construction starts an important mathematical conversation. You can see that the angle constructed has been made with three layers of paper overlapping exactly and therefore each angle is 1/3 of 180º. But why does this work? How does this work? Do you think it will be challenging to prove?

Well actually – no. Surprisingly, the only facts that the explanation requires are that the interior angles in a triangle sum to 180º and side-angle-side congruency. If you’d like to pause at this point and see if you can construct the proof yourself, have a go!

Here’s my explanation – in diagrams, with as few words as possible: