**Polyhedra used to illustrate bell ringing methods**

**by Fortran Friends**

## Although this is a site for learners, we have included some very technical stuff for those of you who are brilliant mathematicians and are fascinated by numbers. If you are not one of those people, don't be surprised and discouraged if you can't understand this. You can be a really good ringer and never even encounter Polyhedra!

Bells were originally just rung in order from the smallest (highest note, usually called the |

We can leave out bell 6, because it is always in the same place and if we write out only lead ends we can also leave out bell 1 which is always first; look at the orange bars across the left diagram to see this.

A Plain course of Bob Doubles can be written as:

2 3 4 5

3 5 2 4

5 4 3 2

4 2 5 3

2 3 4 5

To show we were carrying on we could write more similar rows underneath but since these are always the same as those above, the numbers can be thought of as the corners of a square, shown at the right of the diagram below. The arrows show which way to move and the 'P' indicates a Plain lead.

If a 'Bob' is called we move along a line without an arrow on it, from a corner of the square to another square of Plain leads.

In the example of Bob Doubles the *Plain course* has four lead-ends giving the vertices of a square, with the direction of movement shown by the arrows; only two *Bob* lead-ends are needed to get back to the beginning so the polygon of these just form a line along which you can move in either direction, so these lines have no arrows. Because there are 24 lead-ends in all there must be 6 such squares joined by *Bob* lines. The Bob lines join up with Plain lead squares to form hexagons like this:

This figure shows part of what is possible with Bob Doubles - only three of the six squares of Plain leads are shown. There are eight hexagons altogether made from alternating *Plain* and *Bob* lead-ends.

Folding up the complete plan of squares and hexagons makes a 3D solid (called a Truncated Octahedron by mathematicians), where the 24 possible lead-ends are arranged at the vertices of a polyhedron and the edges of the polyhedron show how to get from one lead-end to the next. If you visit all the vertices in turn obeying the directions given by the arrows you will have completed all the possible combinations on 5 bells ringing Bob Doubles.

**A Truncated Octahedron**

This is the plain model without the bell sequences or arrows printed on it.

Similarly the *Grandsire Doubles* method can be represented by a **Truncated Tetrahedron**.

You can download templates with instructions to make both these models either as zipped Draw files (10K) or as a PDF file (10K). These polyhedra have the bell numbers for that lead printed on them, and arrows drawn along the edges which are plain leads.

You can find out more about polyhedra by looking at some of our other pages.

Web address: http://fortran.orpheusweb.co.uk/Bells/