Bell on bell (math) | Science

Going through the history of mathematics leads to unexpected places, in this case to the steeples of 17th century England. At that time, groups of English bell ringers met regularly to practice the art of ringing bells. They set out to ring the bells of a bell tower in all possible orders without repetition. Such was the interest that several treatises were written about it, among them “Tintinnalogia”, by Richard Duckworth and Fabian Stedman. This problem anticipated a branch of mathematics called group theory.

The number of possible ways in which we can sort n bells is given by n! (read factorial) and is equal to n! =n. (n-one).(n– 2) … 3.2.1. Thus, in a bell tower with six bells, there are 6! = = 720 different ways to score them; But it is not enough to list these possible orders and play one after the other, you have to follow certain rules.

First of all, bell ringers cannot carry ‘scores’ describing the sequence of orders. Therefore, you have to find an easy to remember pattern, something that in itself is not too complicated. The problem arises from the restrictions associated with the very nature of the hoods. These are heavy, mounted on wheels, and their cadence is difficult to modify. In practice, this implies that between one configuration and the next, you can only change the order of the bells that have played in a row. For example, if we have four bells and call them A, B, C, D, the ACDB sequence indicates that we play A first, then C, then D, and finally B. So after ACDB we can ring CADB or ACBD or even CABD. However, after ACDB we cannot sound DCAB. In addition, for aesthetic reasons, you must start and end with the same configuration. So if you start with ABCD, you must go through all the configurations respecting the previous norm and without repetitions, to end again in ABCD.

It is possible, and not very complicated, to find a pattern that complies with the previous rules of recursively. That is, knowing a pattern for a certain number of bells, a pattern can be designed to play one more bell. Several solutions of this style have been proposed throughout history. For example, the computer scientist Donald Knuth, in his famous Art of Computing, quotes an algorithm by Peter Mundy from 1653 that allows it to be done.

The last condition on patterns has to do with loudness. Seventeenth-century fashion dictated that a bell could not be played in the same position three times in a row. Now we do have a challenge! Stedman found a solution for this problem with five bells, which is included in his “Tintinnalogia”. Its pattern, known as Stedman’s doubles, is closely related to how the regular pentagon group of isometries appears as a subgroup of the bell permutations group. If you don’t understand what we’re talking about, don’t worry, Stedman wouldn’t understand either. These are concepts from a branch of mathematics that was not formalized until two centuries later, Group Theory.

Figure 1: This graph represents all possible orders for four bells, and all the ways to exchange bells played in consecutive orders. The red lines represent the exchange of the first two positions, the golden ones of the second and third positions, the blue ones the last two and the green ones represent the first two and the last two simultaneously. The patterns sought correspond to closed paths, which pass through all the vertices, do not repeat any and when passing through a yellow edge, the one before and after must be green.

The art of ringing bells keeps its adherents in England. In 1963, eight English bell ringers played on the Loughborough Bell Foundry eight bells in all possible orders and without repetition (a total of 40320 different configurations). For this they used 17 hours and 58 minutes. The search for different musical patterns also continues. Currently it is approached from a perspective that connects group theory and graph theory. Specifically, the sequence in which the different orders of the bells are played corresponds to a path in a certain graph (the Cayley graph associated with the group of permutations) and the restrictions that are imposed on the pattern are equivalent to properties of the path sought. This approach already appeared in a work by Robert A. Rankin from 1948, in which he solved a problem about bells from 1741, and survives in current works like this or this.

Yago Antolin He is a PhD Assistant at the Autonomous University of Madrid and a member of ICMAT.

Carolina Vallejo Juan de la Cierva is a researcher at the Autonomous University of Madrid and a member of ICMAT.

Coffee and Theorems It is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which the researchers and members of the center describe the latest advances in this discipline, sharing meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.

Editing and coordination: Timon Agate (ICMAT).

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