We've been looking at simple abstract patterns that you get from connecting points together. The latest set we've been playing with are based on connecting seven points together to form a closed shape where each point is visited once.

The Numbers Game

Mathematics tells us that there are 360 of these shapes: starting from one point there are six choices, and from that point 5, then 4, 3, 2 and 1. This gives 6 x 5 x 4 x 3 x 2 x 1 = 720, but that counts drawing the same shape backwards and forwards, so only half of the shapes (360) are different.

When you draw them out, you find that a lot of the shapes are rotated or mirror images of other ones. If you count the ones which are unique, there are 39 of them.

Why is this? Is there a mathematical law in operation? If you have any thoughts, we'd be very interested to hear them.

Some of the shapes are more symmetric than others. The more symmetric shapes appear less often in the 360, because a rotated or mirrored version of a symmetric shape is more likely to be exactly the same as itself. An unsymmetric shape looks different every time it is rotated or mirrored. In fact, the 360 are made up as follows:

3 shapes are very symmetric, and each appears once

21 shapes are fairly symmetric, and each appears 7 times

15 shapes are unsymmetric, and each appears 14 times

But What Does It All Mean?

Is there a meaning to these patterns? We've looked in the past at meanings for smaller patterns, like the 12 shapes that come from six points treated in the same way, but 12 is easier to comprehend than 39!

We have a few ideas, and maybe you have some you could add as well.

One possibility is to start from the fact that 39 = 3 x 13. In fact, the 39 shapes seem as if they could be grouped into threes. For example, there are three highly symmetric shapes. If this is so, then what is the significance of 13? One example is the thirteen lunar months in a year. Perhaps we could look at the 39 shapes as a kind of calendar with astrological connections?

Another possibility comes from Judaism, where there are 'forty minus one' activities which are forbidden on the Sabbath: 39 activities forbidden on the 7th day - 39 and 7. Get it? Perhaps there is symbolism here, the 39 activities symbolising all the creative activities that God rested from on the Sabbath? If so, then each of the 39 shapes might represent one of these activities.