Seven-sided sigils and 21

Many of the Sareoso diagrams incorporate seven-sided sigils, shapes made by connecting seven points equally spaced around a circle in different ways. The diagrams below shows three of the simplest shapes, which I call the three mothers:

1. Made by connecting each point to the next one around the circle.

2. Made by connecting each point to the next-but-one on the circle.

3. Made by connecting each point to the ‘next-but-two’ point around the circle.

As you can see, the three are made by spacing out the points that one line joins together. In the first diagram, two adjacent points are joined, in the second one there is a gap of one point, and in the third one a gap of two points. If we were to carry on, leaving a gap of three points, we’d find the shape would be the same, but drawn backwards.

There’s an interesting effect when we superimpose all three diagrams. We get a seal made of 21 connections.

Seal of 21 and the three mothers.

Looking at the seal you can see that each point is connected to the six other points in the circle. Seven points connected to 6 other points gives 7×6=42 ways of going between points, but because each path is counted forwards and backwards, the total number of connections is 42÷2 = 21.

This connection of the three sigils with the 21 is hinted at in diagrams 14 and 22.

If you want to draw your own seven-sided sigils, you can use the construction method hinted at in diagrams 12 and 13 to make a quite accurate seven-sided shape (heptagon), using the approximation attributed to Albrecht Dürer. The method is described here:

constructing-a-heptagonDownload

Seven-Sided Sigil Pair-swaps

To make a seven-sided sigil, mark seven points equally spaced around a circle, and then join the points together with a single line which visits each point once and once only. Here are some examples of the shapes you get:

   

It turns out that there are exactly 39 different shapes you can get – although they can appear rotated or reflected. Seven-sided sigils appear in a number of the Sareoso diagrams.

The seven-sided sigils have their own relationships and characteristics, and can be categorised in different ways. For example, three of the sigils always look the same no matter how they are rotated or reflected. Another 21 have just one axis of symmetry, and the other 15 have no symmetry.

One way of considering the relationships between different sigils is to consider pair-swaps, where you switch two points on a sigil and see what other sigil results. For example:

The table below shows all the possible pair swap connections between the sigils.

The table is complex, but there are patterns. For example, looking at the right hand side, we can see that there are:

3 sigils that swap to 3 others (9 swaps in total)
3 sigils that swap to 21 others (63 swaps in total)
3 sigils that swap to 10 others (30 swaps in total
18 sigils that swap to 11 others (198 swaps in total)
12 sigils that swap to 16 other (192 swaps in total)

The pair swaps can be used to further categorise the sigils, as discussed in this article.