Mathematik | Informatik
Caroline Dulay, 2006 | Zürich, ZH
In this project, the English tradition of change ringing is explored historically, mathematically, and with Python programming. Change ringing is the ringing of church bells by a group of ringers in a prescribed pattern, so that a set of bells is rung in all possible sequences, without moving the position of any bell in the sequence more than one position forwards or backwards. Ideally, all bells change positions regularly in order to keep the sequence interesting for the bell ringers. This tradition originated in England in the 17th century and is still practiced today. Change ringing sits at the intersection of music and mathematics, and both of these aspects will be explored here. In addition, the process of creating a Python program that was able to find the 24 valid change ringing sequences for a set of four bells is described.
Introduction
Change ringing is the ringing of bells by a group of ringers in a prescribed pattern, so that a set of bells is rung in all possible sequences. The goal of this project was to describe the history and mathematics of change ringing and to explore computational methods for finding valid ringing sequences.
Methods
The historical and mathematical parts of this project were largely descriptive and a variety of sources was used, including the bachelor thesis Campanology – Ringing the Changes by Fabia Weber.
In the computational part, I used python to find all of the valid change ringing sequences for a set of 4 bells and compared my results to traditional change ringing sequences. For 5 bells, valid sequences were found by creating so-called leads of 20 changes which when repeated, create a full extent.
To simulate a change ringing sequence, I recorded the bells of the church in Maschwanden, ZH and integrated them into the code such that a random resulting sequence could be “rung” by the computer.
Results
For the set of 4 bells, 24 valid sequences of 24 changes were found, which are all shifts and reversals of the same core sequences corresponding to the traditional “Plain Bob Minimus”, “Reverse Bob Minimus” and “Double Bob Minimus” methods. This method consists of three repeating sequences of eight changes. For sets of 5 bells, 716 different valid leads of length 20 were found – excluding reverses and shifts, these correspond to 84 unique valid leads.
Discussion
An interesting aspect of this problem is that the practice of change ringing started well before the mathematics (group theory, combinatorics) that describes it was established. Similarly, exploring the space of possible sequences methodically, as I did in this project, quickly becomes prohibitively computationally expensive as the number of bells increase. Although I was able to find some valid extents for five bells composed of six repeating leads of length 20, more computing power and/or programming skill would be required even to investigate leads of length 30, let alone full non-repeating extents of 120 changes or extents for more than 5 bells.
Conclusions
Change ringing is a mathematically and computationally interesting problem at the intersection of music and mathematics, with plenty of opportunities for future work.
Würdigung durch den Experten
Dr. Lorenz Halbeisen
In der Arbeit wird die englische Tradition des sogenannten «Change Ringing» zuerst aus historischer Sicht eingeführt und dann mit Hilfe der mathematischen Gruppentheorie (insbesonders Cayley-Graphen) theoretisch untersucht. Der theoretische Teil wird durch zahlreiche Python-Programme veranschaulicht und hörbar gemacht. Mit viel Engagement hat sich Caroline Dulay in die Mathematik, die hinter dem Change Ringing steckt, eingearbeitet und sich die entsprechenden mathematischen Methoden angeeignet. Schliesslich ist sie auch eigenen Fragestellungen nachgegangen und hat die Antworten praktisch umgesetzt.
Prädikat:
Silber
Kantonsschule Limmattal, Urdorf
Lehrer: Andreas Pfenninger