Stephen Hill's website

The Box of seven joys and binary fractions

I was at the Colchester convention and Michael was attempting to teach me Philip Chapman-Bell's excellent Box of Seven Joys model, along with a number of other folders including Ian, James and Lyn. I'd encountered this model before when Michael had tried to teach it to me at the last Birmingham convention and I'd failed utterly. I also failed in Colchester, although the other folders managed to fine. Maybe I'm just particularly bad at point to point folding. Sigh..

The "Box of Seven Joys" model is hard to fold for a couple of reasons.

The first problem is that it needs to be folded from a circle. It turns out the only pre-cut circular paper that's large enough to fold the model comfortably is Lakeland's baking paper which is far from being ideal for origami.

The second problem is that the model requires the circle to be divided into seven equal-sized segments and that's not easy to do with any kind of accuracy.

After my repeated failures to fold this model, I set myself the goal of solving the following problems:

1. To discover a reliable way of cutting standard paper into circles
2. To discover an easier way of dividing a circular sheet into sevenths
3. To completely and utterly master the "Box of Seven Joys"

I've now solved the first two problems to my own satisfaction and I'm well on my way to solving the third.

Creating the circles turned to be trivial, because I just bought a cheap circle-cutter from "The Works". You may be concerned that the cutter will leave a hole in the paper, but the one from "The Works" comes with a paper-protector that will stop that happening. You do need to fix the paper-protector thing onto the paper with a bit of masking tape to stop it slipping, but I can confirm that the cutter will produce circles that are more than adequate for the purposes of origami. Oh, and it costs just £2.

The problem of dividing the circle into sevenths was a tad harder, but after a bit of experimentation, I discovered that it could be solved with the application of binary fractions.

Any decimal fraction can be calculated by adding a series of "binary fractions" together. A binary fraction is a number such as a half, an eighth or a sixteenth. It's essentially one divided by a chosen power of two e.g. 1/2, 1/4, 1/8, 1/16 ...

A seventh can be calculated to a decent level of accuracy by adding 1/8 and 1/64. This means that 2/7 can be calculated using the formula 1/4 + 1/32 which turns out to be surprisingly easy to fold.

Alternative way of folding a circle into sevenths
Alternative way of folding a square into sevenths

If you're making the box of seven joys, fold the remaining six segments in half and joyfully proceed to step 11 of Philip Chapman-Bell's model. If you've created a regular heptagon, fold the points into the centre as if they were the arcs of the circle used in step 11 and continue with the rest of the proceedure. You won't get quite as good a result as if you used a circle but it's still a cool model.

Oh, and you can exploit the binary fractions trick to divide paper into thirds, fifths or pretty much any number of segments you like. You can also use it to create grids.

All you need is the required binary fractions.

A third is approximately 1/4 + 1/16 + 1/64

A fifth is approximately an eighth plus a sixteenth plus a sixty-forth.

The power of this technique is potentially rather huge. I was experimenting last night and I accidently folded a perfect decagon out an A5 rectangle. There's no fiddly angles to worry about and few additional creases. I'm astonished it's not one of the standard techniques.

If you want to learn more about binary fractions, visit The Lab and experiment with my online division calculator.