Mathematical Groups in CampanologyAuthor(s): B. D. PriceReviewed work(s):Source: The Mathematical Gazette, Vol. 53, No. 384 (May, 1969), pp. 129-133Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/3614536 .Accessed: 05/08/2012 20:14

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13TH DAY OF THE MONTH TTKELRY TO BE A FRIDAY 129 13TH DAY OF THE MONTH TTKELRY TO BE A FRIDAY 129

Then one can use these frequencies to amplify our previous table. Then one can use these frequencies to amplify our previous table.

0 0

1 1 2 2 1 2 3

1 1 2 2 1 2 3

1 1 3 1 2 2 2

1 1 3 1 2 2 2

Frequencies

I

3 1 1 2 2 1 2

Frequencies

I

3 1 1 2 2 1 2

2 2 1 1 3 1 2

2 2 1 1 3 1 2

2 4 6 2 2 4 4

2 4 6 2 2 4 4

4

2 4 6 2 2

4

2 4 6 2 2

From this one can see that Friday is the most therefore the most probable day.

From this one can see that Friday is the most therefore the most probable day.

Eton College Eton College

frequent, and frequent, and

S. R. BAXTER (when age 13) S. R. BAXTER (when age 13)

MATHEMATICAL GROUPS IN CAMPANOLOGY

BY B. D. PRICE

Groups have for several hundred years been used in the composition of long peals in change ringing, as was mentioned on p. 399 of the Gazette. A brief outline of their use for the non-ringing mathe- matician may be of some interest.

The English art of change-ringing is founded on the sounding of consecutive permutations of the bells being rung (usually 6, 8, 10 or 12 in the diatonic scale descending to the tonic), the bells being swung through a full circle or nearly so, and small differences in energy induced by pulling or checking the rope cause small changes in the highest point reached at consecutive swings, and hence large differences in the period of swing, so that bells up to 4 tons in weight may be controlled by one person, and permutations created by advancing or retarding the several swings. It is an invariable rule that a bell may move only one place in successive permutations,

B

MATHEMATICAL GROUPS IN CAMPANOLOGY

BY B. D. PRICE

Groups have for several hundred years been used in the composition of long peals in change ringing, as was mentioned on p. 399 of the Gazette. A brief outline of their use for the non-ringing mathe- matician may be of some interest.

The English art of change-ringing is founded on the sounding of consecutive permutations of the bells being rung (usually 6, 8, 10 or 12 in the diatonic scale descending to the tonic), the bells being swung through a full circle or nearly so, and small differences in energy induced by pulling or checking the rope cause small changes in the highest point reached at consecutive swings, and hence large differences in the period of swing, so that bells up to 4 tons in weight may be controlled by one person, and permutations created by advancing or retarding the several swings. It is an invariable rule that a bell may move only one place in successive permutations,

B

Mondays Tuesdays Wednesdays Thursdays Fridays Saturdays Sundays

Mondays Tuesdays Wednesdays Thursdays Fridays Saturdays Sundays

E 0

13 13 15 12 16 +- 12 15

E 0

13 13 15 12 16 +- 12 15

THE MATHEMATICAL GAZETTE

and usually may not stay in the same position more than twice, hence change ringing consists of a series of interchanges thus:

12345678 21436587 241356 78 42316587 24361578

The bells are numbered in descending pitch order, and the starting order 12345678 (called " rounds ") gives way to a succession of permutations, no two of which may be alike, finally ending with rounds again. On eight bells all the 40,320 permutations take about 24 hours to ring even at a fast rate of ringing on light bells. The normal " peal " is a length of at least 5,000 permutations.

The composing of such long lengths cannot be haphazard, and the rows are organized firstly into short lengths called " methods ", and periodic or semi-periodic alterations at previously agreed points produce fresh rows without repetition. The most exacting peals to compose are those which employ all the possible permutations, which happens when 7 bells are being permuted (the 8th bell in this case always ringing last) and here all 5,040 rows must be rung. On higher numbers of bells musical considerations may limit the permutations available.

We need to define certain entities because of the confusion of terminology existing. "Change " and " permutation " will be avoided.

A row is an order of the bells as rung, and consists on paper of a row of digits.

A transposition is an operation carried out on a row, which pro- duces another row. For example in

42631857 62417538

regarded as a transposition, note is taken that whatever digit is in the first position moves to 3rd position, and so on. If the same transposition is applied to " rounds " taken as the standard reference row,

12345678 32158746

results, and the operation is named the transposition 32158746.

130

MATHEMATICAL GROUPS IN CAMPANOLOGY

A transdigit is the dual of a transposition, in which " digit " and " place " are exchanged. Thus

42631857 62417538

regarded as a transdigit replaces 4 by 6, in whatever position the 4 may be, and so applying this transdigit to " rounds ",

12345678 72163485

this operation is the transdigit 72163485. The duality of these two concepts is completed by the simple but

important lemma that if the operation from row A to row B is the same transposition as from C to D, then A to C and B to D exhibit the same transdigit, and vice versa. For instance

A 426153 B 624135 C 215634 D

Here, A B and C are three random rows on 6 digits. If D is trans- posed from C as B from A, the result is 512643, and this is the same result as if D is transdigited from B as C from A. Alternatively, if D is transdigited from C as B from A the result is 213456, and this gives A to C and B to D as the same transposition.

This lemma is fairly obvious when examined, and is important as it enables a Group to be used for the simplification of composition of peals. A Group of rows is defined as such that if any three rows A, B and C are chosen (not necessarily distinct) and a fourth row D constructed as above, by either method, then the fourth row D is always a row in the Group. It is not necessary for " rounds " to be a row in the Group, as it is the set of operations linking the rows which really constitutes the Group.

Consider a simple case with 4 digits in each row:

A1 1 2 3 4 A2 2 1 4 3 A3 3 4 1 2 A4 4 3 2 1 B1 1 3 2 4 B2 B3 B4 C1 etc.

Here A1 A2 A4 form a Group of rows of order 4, being in fact an isomorphism of the non-cyclic Abelian Group. B1 is any other row from the extent of 4! rows. B2 is constructed from A2, B3 from A3 etc. by applying the transposition A1--B1. Then by the above lemma the set of rows B1 B2 B3 B4 have the same transdigit rela-

131

THE MATHEMATICAL GAZETTE

tions between them as the set of A's and hence form a Group of rows also. The B's must be distinct from the A's, otherwise the whole Group would be identical contrary to the choice of B1.

Then CQ is chosen, as a row not already obtained, and the process eventually gives the extent in 6 rows and 4 columns, each column having the same transposition structure, and each row forming a Group.

Evidently the same result could be achieved by making the columns have transdigit structure, but in fact it is transposition structure which is memorised by change ringers, largely because the rule about a digit moving only one place in consecutively rung rows leads to this type of pattern. A very simple instance familiar to the ringer as " Plain Bob Minimus " is

1234 1342 1432

2143 3124 4132 2413 3214 4312 4231 2341 3421 4321 2431 3241 3412 4213 2314 3142 4123 2134 1324 1432 1243 1342 1423 1234

Here the operative Group is the cyclic Group 234, 342, 423 and the columns form a repeating transposition pattern like plaiting, known as " plain hunting ".

This technique extends to the longest peals. A transposition structure, which may be thousands of rows in length, is found such that the use of a Group of rows, one row at the head of each such structure, will give a desired extent. This process by successive reduction becomes easier than it may seem at first thought. Firstly a " method " is agreed between the ringers, so if no order is given by the conductor, a series of rows is rung, ending in " rounds ", about 100-200 rows in length according to difficulty, and having a Group structure in itself similar to the example Plain Bob Minimus above. The " composer " has examined previously where to make altera- tions in the ringing, dealing with the rows in sections of from 14 to 32 rows (according to method) and using a Group structure on a larger scale. The " conductor " then issues orders at intervals to effect these alterations and it may be necessary to give only a few dozen orders in three hours' ringing. Thus a periodic sequence of transposition structure by irregular alterations becomes part of a

132

MATHEMATICAL GROUPS IN CAMPANOLOGY MATHEMATICAL GROUPS IN CAMPANOLOGY

longer sequence, itself periodic, and a long peal becomes possible to memorise because of its recurrent patterns.

This leads to a highly sophisticated technique in which a Group is chosen which is compatible with the method structure, the actual rows are put in blocks and lose their identity, being coded by selected rows, then the Group sets up a structure of co-set Groups as explained above, and a certain amount of trial and error results in a peal which involves no repetition of rows. The trial-and-error process may often be reduced to a topological problem of finding a complete circuit around a line diagram. These techniques were thoroughly explored by W. H. Thompson in the last century. Much use has been made, since his day, of the cube and dodeca- hedron in the graphical solution of change ringing problems.

King's College B. D. PRICE Taunton

PERSPECTIVE DRAWING BY NUMBERS

BY F. L. CARTER

FIG. 1. Diirer wood cut showing how an artist can copy on ruled paper the image of his subject as " seen through " (projected on) a

string grid.

The use of matrices in connection with geometric transformations is now well established in the literature devoted to the teaching of " modern mathematics " in schools. A possible way of motivating an interest in projective transformations is to be found in the problem of constructing an accurate perspective drawing of an object as seen from a selected viewing point. One of the first artists to master the technique was Albrecht Diirer, and the woodcut by him (Figure 1) shows clearly the basic idea, that such a drawing is

longer sequence, itself periodic, and a long peal becomes possible to memorise because of its recurrent patterns.

This leads to a highly sophisticated technique in which a Group is chosen which is compatible with the method structure, the actual rows are put in blocks and lose their identity, being coded by selected rows, then the Group sets up a structure of co-set Groups as explained above, and a certain amount of trial and error results in a peal which involves no repetition of rows. The trial-and-error process may often be reduced to a topological problem of finding a complete circuit around a line diagram. These techniques were thoroughly explored by W. H. Thompson in the last century. Much use has been made, since his day, of the cube and dodeca- hedron in the graphical solution of change ringing problems.

King's College B. D. PRICE Taunton

PERSPECTIVE DRAWING BY NUMBERS

BY F. L. CARTER

FIG. 1. Diirer wood cut showing how an artist can copy on ruled paper the image of his subject as " seen through " (projected on) a

string grid.

The use of matrices in connection with geometric transformations is now well established in the literature devoted to the teaching of " modern mathematics " in schools. A possible way of motivating an interest in projective transformations is to be found in the problem of constructing an accurate perspective drawing of an object as seen from a selected viewing point. One of the first artists to master the technique was Albrecht Diirer, and the woodcut by him (Figure 1) shows clearly the basic idea, that such a drawing is

133 133