Proportions, polygons, and pieces of paper

So here’s a lovely idea I found in Developing Thinking in Geometry (Johnston-Wilder and Mason, 2005).

Imagine a dozen or so sheets of A4 paper. Lay the first sheet down in portrait orientation. Now lay the second over the first, corner-to-corner, so that two triangles are created. Can you see anything interesting happening? Can you predict what’s going to happen?

Now try it yourself with sheets of A4. Keep layering on the next sheet. Keep going! Can you see something emerging… Now that’s clever!

Was it what you had predicted? Why does that happen?

Now the real question is will it always be the case? What if I use a different size of paper?

Take the first two pieces of paper and trace around the second piece on the first. Fold along these lines and consider what happened. You could angle-chase and then put the two pieces back together again to convince yourself that you are indeed looking at the vertex of a regular octagon.

Here’s a rectangle of paper. Its width is a and height a + b.

We lay a second rectangle over the first so that the top-left corner of the upper piece coincides with the top-right corner of the lower piece and the bottom-left corner of the upper piece coincides with the left edge of the lower pieces.

This results in a triangle of the first sheet being left uncovered in the top left.

Let’s take a look at this triangle.

We know its base (at the top :-) ) is a and its hypotenuse is a + b.

So its height, h, is √[(a + b)² – a²]

When we complete this with A4 paper this height is a….but why? What’s the ratio of a+b to a?

Now you could either type in A4 paper ratio sides into a search engine or work through the maths yourself (much more fun.)

For A4 paper (a + b)² – a² = a² and we want to know a + b : a

(a + b)²  = 2a² meaning a + b = √2a and therefore (a+b)/a = √2 and a + b : a is √2 : 1

Investigating this a little further, you can discover that A4 paper (and in that case all A paper) is indeed designed in the ratio of √2 : 1. This is called the Lichtenberg ratio and allows for the scaling property of the different A sizes - the way that two A4 sheets join to form A3 and so on.

Now my mind starts to want to know how do I form other regular polygons?

What beautiful spirals can I produce?

What if you are reading this and live in the US, Canada and even parts of Mexico and can’t put your hands on A4 paper straight away? What if I use envelopes, magazines, maths exercise books? I’ll leave you to ponder.