A number of the systems listed above fall into sets which would extend indefinitely on higher numbers of working bells. There are two clearly-defined such sets, as follows:

These were called 'palindromes analogous to the plain hunt' in Price 1989. A plain hunt, on three bells or more, is the simplest example of a palindrome and belongs to this set. All the backstroke rows of a plain hunt form a cyclic group, while all of the rows form a dihedral group. Geometrically, the dihedral group consists of the rotations of a regular prism. Think of a thin piece of card having its two faces as regular polygons, the corners being labelled with the numbers 1, 2, 3.... The cyclic group consists of the rotations in its plane which leave its position unaltered; while other rotations turn it over, thus swapping certain corners in pairs. If the order of the cyclic group is odd, one corner will stay still and all others swap in pairs; but if the order is even, then either all the corners swap in pairs, or else two numbers opposite one another will stay still, and the others swap in pairs. This of course is exactly what happens in the plain hunt. The replacing of the numbers on the corners by others as the result of a rotation is equivalent to a transfigure.

But further to this, the plain course of any treble-dominated method (plain or treble-bob) is such a palindrome. By tradition, "regular" methods have plain-bob leadheads, with coursing order 2-4-6-8-7-5-3 for major etc., and these numbers will occur in that order on the geometrical model. If one is composing a regular method starting at rounds as a leadhead, and choosing places to be made, when one arrives at the half-lead, the condition for the next leadhead to be a regular one is that the pairs of bells swapping at the half-lead lie on parallel lines across the coursing order, for those are the pairs which may swap when the polygon of the coursing order is turned over.

Take as an example, for instance, Plain Bob Minor:

Leadheads Leadends 2 3 4 5 6 3 2 5 4 6 transposed 2 4 6 5 3 3 5 6 4 2 3 5 2 6 4 5 3 6 2 4 into coursing 3 2 4 6 5 5 6 4 2 3 5 6 3 4 2 6 5 4 3 2 orders 5 3 2 4 6 6 4 2 3 5 6 4 5 2 3 4 6 2 5 3 6 5 3 2 4 4 2 3 5 6 4 2 6 3 5 2 4 3 6 5 4 6 5 3 2 2 3 4 6 5

When the leadheads and leadends are transposed into coursing order the cyclic nature becomes clearer. In the leadheads, the five bells rotate cyclically, and in the leadends they rotate backwards. The relation of any leadhead to any leadend is a transfigure of kind (2,2,1). The first column forms the cyclic group [5.05] of order 5, and the set of all ten rows the dihedral group [5.04]. When this system is used in Major on five working bells, the pair 7-8 may swap for the apical transfigures in which case the corresponding overall group is [7.15].

Irregular methods such as Woodbine obey the same rules, though the coursing order is different. In the cases of Minimus, Minor, Major, Maximus with 4, 6, 8, 12 bells the number of leads in the course is one less at 3, 5, 7, 11 and since these numbers are all prime, there is no trouble in getting a lead to extend to a course. But for Royal, 9 is not prime and some putative methods will give only a 3-lead course; such short courses are still palindromes, but they cease to be simple Cyclic-Dihedral ones (although they are still Cyclic-Dihedral from the mathematical point of view).

The progression of Cyclic-Dihedral palindromes from plain hunt to a treble-dominated method continues with a touch or peal of such a method, but the underlying fact is unaltered - if a cross-section of the rows be taken at any level (i.e. equidistant from the apices) the set of rows will form a dihedral group, and alternate ones of the set will be cyclic.

Another set of systems which extends indefinitely, but has only two examples above in their simple form, is the set of Rotating-Sets Palindromes.

Consider the six figures 2 to 7 arranged in a circle so that 2, 3 and 4 are alternate to 5, 6 and 7. This is the left-hand circle in the diagram below. Now consider the following two transformations:

2 5 4 5 7 2 4 6 5 -----> -----> 3 4 6 7 2 3 6 3 7

In the first, the three pairs 2x5, 3x6, 4x7 swap, which has the effect of rotating separately the triangles 234 and 567 through 60 degrees; in the second transformation the swapping pairs are 2x6, 3x7, 4x5. The overall result is that the triangle 234 has rotated 120 degrees anticlockwise, whereas 567 has rotated 120 degrees clockwise. Three such pairs of transformations will return the figures to their original positions. The corresponding rows for a 3-part palindrome are:

Generating group [6.33] Apical transfigures 2 3 4 5 6 7 7 6 5 4 3 2 * Overall 3 4 2 6 7 5 6 5 7 3 2 4 * group 4 2 3 7 5 6 5 7 6 2 4 3 * [6.16]

All three apical transfigures are of kind (2,2,2). Geometrically, the group [6.16] is the set of rotations of a triangular prism, the vertices being labelled 234 at one end and 567 at the other. The apical transfigures rotate the prism on axes which swap the ends.

When there are only two figures in each set, the system gives 2-part palindromes with the "pairs of pairs" group [4.04] as the overall group on the working bells. Refer to the system [4.07]-[6.27] listed above. The next case up would be that of 4-part palindromes on 8 working bells.

In Price 1996 a method of composing part-peals by means of groups was expounded; the paper attempted to explain the process, but did not claim it as new - in fact it is evident that many composers in the past have used it (see, for instance Snowdon's Stedman, first edition 1903 pp.197 et seq, where part-typing of sixes is explicit). In this paper, a method of composing palindromes is developed which may be original. But in both cases although the method is sufficient to produce peals of a certain class, it is not necessary. That the four examples cited below exist, prove this. The first two are concerned with group structure, the last two with palindromic structure.

5,040 Plain Bob Triples in Four Parts M H 23456 25643 42365 62534 -------------------------------------- - - 45236 64253 36425 53624 - 24536 26453 43625 65324 S 25436 24653 46325 63524 - 42536 62453 34625 56324 - 54236 46253 63425 35624 S 52436 42653 64325 36524 - - 43526 65423 32645 52364 - 54326 46523 63245 35264 S 53426 45623 62345 32564 - 45326 64523 36245 53264 - 34526 56423 23645 25364 S 35426 54623 26345 23564 - - 42356 62543 34265 56234 - 34256 56243 23465 25634 - 23456 25643 42365 62534 --------------------------------------

This example was created by Roger Bailey. There are four identically-called blocks which make up the extent of 5,040 changes, but the partheads do not form a group. They may be linked by asymmetric calls. Each block is constituted of fifteen whole courses. In the method, any lead exists in two forms which are mutually reverse, and one or the other must occur in any peal. A whole course may be reversed; but it does not follow that all the leads in a course must be rung in the same direction. However, in this peal they are.

The existence of such an assembly proves that the condition of using a group for part composition is not a necessary one.

It is interesting to analyse the structure of these blocks. A whole course is specified by a coursing order of the six bells 234567 arranged in a circle, the reverse course having its circle of six figures turned over. There are 120 coursing-circles, in 60 such mutually-false pairs. Consider the reduction of the above blocks:

[5 2 4] 3 6 7 [4 2 6] 5 3 7 [6 4 3] 2 5 7 [3 6 5] 2 4 7 [3 5 4] 2 6 7 [5 4 6] 2 3 7 [2 6 3] 4 5 7 [2 3 5] 6 4 7 (2 3 4) 5 6 7 (2 5 6) 4 3 7 (4 2 3) 6 5 7 (6 2 5) 3 4 7

These are the courses in the above blocks. Square brackets, such as [5 2 4] signify that the 3 bells inside permute in all 6 ways; whereas round brackets (2 3 4) that they rotate, giving only 3 ways.

The two bells adjacent to the 7, such as 36 in [5 2 4] 3 6 7, come on either side of the 7 in the coursing-circle, revealing by reversal (36 or 63) whether a coursing-circle might have been turned over (to give the false course). There are ten such combination-pairs of the figures 23456, and the only duplications in the set of courses above occur in the bottom (third) row, where falseness is avoided by the rotation, not permutation, of the three bells in brackets. This method of generating the extent has not been achieved by using groups.

5,040 Plain Bob Triples M W R 23456 ------------------ - - 45236 - 24536 - 52436 - - - 64523 - 56423 - 45623 ------------------ 10-part, S half-way and end.

This peal by Henry Hubbard is given in the Central Council Collection of Triples Methods (1933), although Plain Bob Triples itself is not mentioned, being beyond the pale as having adjacent places! The peal keeps the 7th unaffected, but is not a palindrome. The 5-part block given is doubled by swapping 2x3 with singles, and the resulting 10 partends do not form a group. How can we explain this anomaly? Any course (P-block) of P.B.T. has a 6-part structure so that half the leadheads are +ve and half -ve. But if the coursing order of P-blocks is examined, a parity structure emerges provided one bell (here the 7th) is unaffected by calls. For instance, the plain course has coursing order 7-6-4-2-3-5-7 and the five-bell order 64235 happens to be a +ve permutation of 23456. Calling bobs which do not affect the 7th will rotate three bells of the five but +ve parity will be unaffected. Hence of the 120 possible coursing orders, half are inaccessible; but the reverse coursing order 53246, which corresponds to the plain course inverted, is accessible as it is also +ve. The extent of Plain Bob Triples is thus partitioned into two independent sets, and following this the group structures of the separate halves of Hubbard's peal are also independent. This partition is a consequence of keeping the 7th bell unaffected by calls.

5,040 Plain Bob Triples in Six Parts M W H 23456 M W H 23456 M W H 23456 M W H 23456 -------------- -------------- -------------- -------------- - - 42635 - - 42635 - 43652 - - 42635 $ -* 62534 -* 62534 - - 64235 -* 62534 - 52436 - 36524 - - 26543 - - 56423 - 35426 - - 53462 - 42563 -* $ 25463 - - 43652 - - 45236 - - 54326 - 62453 - - 64235 - - 24653 - 34625 - - 46325 - - 26543 - 64352 $ -* 64523 - 36524 - 56342 - 56342 - - 56342 - - 53462 -* 45362 -* $ 45362 -* 45362 - - 45236 - - 34256 - - 34256 - - 34256 - 34256 -------------- -------------- -------------- -------------- No.1 Reverse No.1 No.2 Reverse No.2

I composed the above two peals as 6-part peals using the group [5.08]. The blocks are 3-part; turning calls, S for B, are needed at one of the positions marked *, in order to swap 5-6.

If extra bobs are called at positions $, in each case the bells 234 are rotated, and each block issues into rounds in one part. The blocks now become palindromes, and of course the reverses become identical; but furthermore, the palindromes are 2-part ones!

Solution 1 is given below, with the extra bob inserted. Across the top are the original six partheads of the 6-part peal, but there are now a further six partheads (also underlined) as each block is now a 2-part palindrome.

M W 23456 34256 42356 23465 34265 42365 --------------------------------------------------------------- - @ - 42635 - - 64523 - @54326 - 25346 - - 32654 43652 24653 32564 43562 24563 --------------------------------------------------------------- - @ - 63425 - - 46532 - @56234 - 35264 - - 23456 34256 42356 23465 34265 42365 ---------------------------------------------------------------

The first apex is at leadhead 1763425, swaps (7 3x6 2x4 5), and the second at home leadhead 1543267, swaps (5 3x4 2x6 7). Thus 5 and 7 are fixed in the part, and 2346 describe a Rotating-Sets 2-part palindrome with overall swaps 2x3, 4x6. Solution 2 behaves similarly.

Thus the extent is marshalled in twelve identically-called parts, but the partheads do not form a group!

The following palindrome is based on a peal by A.J. Cox. It was discussed in Price 1989, in the appendix on palindromes.

3,584 Superlative S. Major M W H 2 3 4 5 6 7 8 ---------------------------- - 5 2 4 3 6 7 8 -a1 4 2 6 3 5 7 8 - 6 2 5 3 4 7 8 -b1 S S@ 3 4 2 5 6 7 8 S -a2 5 6 2 4 3 7 8 - 4 5 2 6 3 7 8 -b2 6 4 2 5 3 7 8 - -@ 3 2 4 5 6 7 8 ---------------------------- Two-part Apices marked @

This palindrome of system [2.01]-[6.36] presents interesting enigmas for the composer. It may be doubled into a true 7,198 in several ways, each structured on the group [4.06] (two pairs of working bells swapping independently) of which [2.01] is a subgroup. But neither of the resulting 4-part palindromes fit the theory which was developed for this paper.

The first way is to single a pair of bobs in alternate parts to swap working bells 4 and 6. This may be done either at a1 or at a2. The following is a breakdown of the part-ends and apical transfigures:

[4.06] order 4m Apical Transfigures 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ The set of 8 transfigures 3 2 4 5 6 7 8 s 3 2 4 6 5 8 7 * does not form a group! 2 3 6 5 4 7 8 s 2 3 5 4 6 8 7 $ 3 2 6 5 4 7 8 3 2 5 4 6 8 7 *

The second way (which is the reverse of the first) is to swap working bells 4 and 5, by singles for bobs at b1 or b2 in alternate parts. The breakdown is:

[4.06] order 4m Apical Transfigures 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ Again, the set of 8 3 2 4 5 6 7 8 s 3 2 4 6 5 8 7 * transfigures does not 2 3 5 4 6 7 8 s 2 3 6 5 4 8 7 $ form a group! 3 2 5 4 6 7 8 3 2 6 5 4 8 7 *

In the two cases, the first two rows of the columns are the same and conform to the 2-part system [2.01]-[6.36].

Further, the 4-part peal may be extended by singling-in an extra whole course in alternate parts; doing so in every part produces falseness. The result is the peal of 7,646 by A.J. Cox. Unlike Example 3 above, the partheads of the 4-part do form a group, but the extended set including the image leadends do not form an outer group.

Having the space available, but at the risk of being thought a crank, I cannot resist airing a philosophical matter which puzzles and intrigues me. When one is composing peals, there is a definite impression that peals exist as mathematical entities awaiting discovery. Is this so? Is composition truly creative? Or did all possible peals exist just after the Big Bang?

One can argue that such existence is too extravagant. For we may find more and more complex methods on higher numbers of bells, all having their problems of the composing of an extent. To paraphrase a lecturer I remember: "Now I say, take a number n, however large, and I can find a number of bells m, so that the number of possible extents on m bells is larger than n...".

Do peals really exist before they have been composed? Bishop Berkeley would have said "No!". George Berkeley (1685-1763) was latterly Bishop of Cloyne, near Cork in southern Ireland. He "maintained that the world we see and touch is not an abstract independent substance, of which conscious mind may be an effect, but is the very world which is presented to our senses, and which depends for its actuality on being perceived" [Chambers dictionary].

This matter of creativity was touched on in a Times supplement article of 1995, about Colin Wyld's discovery of a way to obtain an extent of Stedman Triples with bobs only. Was it the fact of his discovery that made it possible for the attempts of others to do the same thing? This case is particularly interesting, because ringers had been trying to achieve Wyld's result, and also trying to prove it unattainable, for over a century before the matter was resolved.

Strange things have been shown to happen, even in the material field. Scientists have for some time been devising experiments to test the logic of the behaviour of matter and have obtained startling results; for instance, electrons behave in different fashions according to whether they are being observed or not.

When we send an expedition to the moon, or a probe to photograph the moons of Jupiter, always more problems are posed than solved. Can the physics of a moon of Jupiter be affected by the fact that it is being observed? Can the behaviour of sets of permutations be governed by the creative acts of composers? Does the universe get progressively more complex as we strive to understand it?

This category of composition lends itself to palindromic methods. Systems listed above which can keep the tenors "together" are marked (TT). For a system to qualify, the partend group can have no more than 5 working bells (implemented as bells 2 to 6) and the overall group must have extra swaps available to enable 7x8 to swap at all apices. It is useful to examine the positions of the tenors in the 12 classes of major methods. The classification below is that used by Corrigan in his *370 Surprise Major Methods* (1948) though it may have had earlier origins. Each lead is given by its leadhead and leadend. The possible positions of apices of a palindrome are where the tenors cross over at leadend or mid-lead, and are marked by asterisks.

It will be seen that each category has one mid-lead apex and three possible kinds of apices at Plains or Bobs, four in all of transfigure kind (2,2,2,1,1); a fifth kind with a Single at Home, of kind (2,2,1,1,1,1) is also useful. In the mx category, two consecutive bobs at home, or none at all, flank a mid-lead apex, whereas an isolated bob at home is itself a possible apex. Each category has two places in the course where no call can be made or taken, so that the leads on either side may be combined as an element in the search, should this be feasible and convenient. These places are marked with & signs. The nature of the single in the mx category determines its position as an apex in the course.

In the cases of Bobs Before and Bobs at Home which shunt, the shunts are marked on the left, "back" or "forward".

For each category, the arrangement of the calls in the course is symmetrical about the apices, bobs at Middle and Wrong being images of one another.

2nds Place Methods (4ths place bobs assumed) Category a Category b --------------- Home --------------- Home 1 . . . . . 7 8 1 . . . . . 7 8 1 . . . . 7 . 8 1 . . 7 . 8 . . --------------- Wrong &-------------- 1 . . . 7 . 8 . 1 . 7 . 8 . . . 1 . . 7 . 8 . . 1 8 7 . . . . . &-------------- | --------------- 1 . 7 . 8 . . . | 1 8 . 7 . . . . 1 7 . 8 . . . . | 1 . . . 8 . 7 . --------------- | --------------- Middle *Bob | 1 7 8 . . . . . *Mid- *Bob | 1 . . . . 8 . 7 *Mid- Before| 1 8 7 . . . . . lead Before| 1 . . . . 7 . 8 lead (back) --------------- (for- | --------------- Wrong 1 8 . 7 . . . . ward)| 1 . . . 7 . 8 . 1 . 8 . 7 . . . | 1 7 . 8 . . . . &-------------- | --------------- 1 . . 8 . 7 . . 1 7 8 . . . . . 1 . . . 8 . 7 . 1 . 8 . 7 . . . --------------- Middle &-------------- 1 . . . . 8 . 7 1 . . 8 . 7 . . 1 . . . . . 8 7 1 . . . . . 8 7 --------------- Home --------------- Home 1 . . . . . 7 8 *P *B *S 1 . . . . . 7 8 *P *B *S Cambridge, Pudsey, Yorkshire, Superlative, Lincolnshire. Category c Category d --------------- Home --------------- Home 1 . . . . . 7 8 1 . . . . . 7 8 1 7 . 8 . . . . 1 8 7 . . . . . --------------- | --------------- | 1 7 8 . . . . . | 1 8 . 7 . . . . | 1 . . . 8 . 7 . | 1 . . . . 7 . 8 | --------------- Middle | --------------- Wrong | 1 . . . . 8 . 7 | 1 . . . 7 . 8 . | 1 . . 7 . 8 . . | 1 . 8 . 7 . . . | &-------------- | &-------------- *Bob | 1 . 7 . 8 . . . *Mid- *Bob | 1 . . 8 . 7 . . *Mid- Before| 1 . 8 . 7 . . . lead Before| 1 . . 7 . 8 . . lead (back)| &-------------- (for- | &-------------- | 1 . . 8 . 7 . . ward)| 1 . 7 . 8 . . . | 1 . . . . 7 . 8 | 1 . . . 8 . 7 . | --------------- Wrong | --------------- Middle | 1 . . . 7 . 8 . | 1 . . . . 8 . 7 | 1 8 7 . . . . . | 1 7 . 8 . . . . --------------- | --------------- 1 8 . 7 . . . . 1 7 8 . . . . . 1 . . . . . 8 7 1 . . . . . 8 7 --------------- Home --------------- Home 1 . . . . . 7 8 *P *B *S 1 . . . . . 7 8 *P *B *S Category e Category f --------------- Home --------------- Home 1 . . . . . 7 8 1 . . . . . 7 8 1 . 8 . 7 . . . 1 . . . 8 . 7 . &-------------- --------------- Middle 1 . . 8 . 7 . . 1 . . . . 8 . 7 1 7 . 8 . . . . 1 . 8 . 7 . . . --------------- &-------------- | 1 7 8 . . . . . 1 . . 8 . 7 . . | 1 . . . . 7 . 8 1 8 7 . . . . . | --------------- Wrong | --------------- *Bob | 1 . . . 7 . 8 . *Mid- *Bob | 1 8 . 7 . . . . *Mid- Before| 1 . . . 8 . 7 . lead Before| 1 7 . 8 . . . . lead (back)| --------------- Middle (for- | --------------- | 1 . . . . 8 . 7 ward) 1 7 8 . . . . . | 1 8 7 . . . . . 1 . . 7 . 8 . . --------------- &-------------- 1 8 . 7 . . . . 1 . 7 . 8 . . . 1 . . 7 . 8 . . 1 . . . . 7 . 8 &-------------- --------------- Wrong 1 . 7 . 8 . . . 1 . . . 7 . 8 . 1 . . . . . 8 7 1 . . . . . 8 7 --------------- Home --------------- Home 1 . . . . . 7 8 *P *B *S 1 . . . . . 7 8 *P *B *S London, Rutland. 8ths Place Methods (6ths place bobs assumed, except in group mx) Category g Category h --------------- --------------- Home 1 . . . . . 7 8 1 . . . . . 7 8 1 . . 7 . 8 . . 1 7 . 8 . . . . &-------------- --------------- 5ths 1 . . . 7 . 8 . 1 . 7 . 8 . . . *Mid- 1 7 . 8 . . . . 1 . 8 . 7 . . . lead --------------- 5ths --------------- In 1 . 7 . 8 . . . 1 8 . 7 . . . . 1 8 7 . . . . . 1 . . . . . 8 7 --------------- Before | --------------- 1 7 8 . . . . . *P *B *S | 1 . . . . 8 . 7 1 . 8 . 7 . . . | 1 . . 7 . 8 . . --------------- In | &-------------- 1 8 . 7 . . . . Home | 1 . . . 7 . 8 . 1 . . . 8 . 7 . *B | 1 8 7 . . . . . &-------------- (for- | --------------- Before 1 . . 8 . 7 . . wards)| 1 7 8 . . . . . *P *B *S 1 . . . . . 8 7 | 1 . . . 8 . 7 . Home | --------------- | &-------------- *B | 1 . . . . 8 . 7 *Mid- | 1 . . 8 . 7 . . (for- | 1 . . . . 7 . 8 lead | 1 . . . . 7 . 8 wards)| --------------- | --------------- 1 . . . . . 7 8 1 . . . . . 7 8 Category j Category k --------------- --------------- 1 . . . . . 7 8 1 . . . . . 7 8 1 8 7 . . . . . 1 . 8 . 7 . . . --------------- Before --------------- In 1 7 8 . . . . . *P *B *S 1 8 . 7 . . . . 1 . . . . . 8 7 1 . . 7 . 8 . . | --------------- &-------------- | 1 . . . . 8 . 7 1 . . . 7 . 8 . *Mid- | 1 7 . 8 . . . . 1 . . . 8 . 7 . lead | --------------- 5ths &-------------- | 1 . 7 . 8 . . . 1 . . 8 . 7 . . | 1 . . . 8 . 7 . 1 7 . 8 . . . . Home | &-------------- --------------- 5ths *B | 1 . . 8 . 7 . . *Mid- 1 . 7 . 8 . . . (for- | 1 . . 7 . 8 . . lead 1 . . . . . 8 7 ward)| &-------------- | --------------- | 1 . . . 7 . 8 . Home | 1 . . . . 8 . 7 | 1 . 8 . 7 . . . *B | 1 8 7 . . . . . | --------------- In (for- | --------------- Before | 1 8 . 7 . . . . ward)| 1 7 8 . . . . . *P *B | 1 . . . . 7 . 8 | 1 . . . . 7 . 8 *S | --------------- | --------------- 1 . . . . . 7 8 1 . . . . . 7 8 Category l Category mx (4ths place bobs) --------------- *B (*S) --------------- 1 . . . . . 7 8 Home | 1 . . . . . 7 8 *Mid- 1 . . . 8 . 7 . (back) | 1 . . . . . 8 7 lead &-------------- --------------- 1 . . 8 . 7 . . Middle | 1 . . . . 8 . 7 1 8 7 . . . . . (back) | 1 . . . 8 . 7 . --------------- Before --------------- 1 7 8 . . . . . *P *B *S 1 . . 8 . 7 . . 1 . . 7 . 8 . . 1 . 8 . 7 . . . &-------------- &-------------- 1 . . . 7 . 8 . 1 8 . 7 . . . . 1 . . . . . 8 7 1 8 7 . . . . . | --------------- --------------- Before | 1 . . . . 8 . 7 1 7 8 . . . . . *P *B | 1 . 8 . 7 . . . 1 7 . 8 . . . . (*S) Home | --------------- In &-------------- *B | 1 8 . 7 . . . . *Mid- 1 . 7 . 8 . . . (for- | 1 7 . 8 . . . . lead 1 . . 7 . 8 . . ward)| --------------- 5ths --------------- | 1 . 7 . 8 . . . Wrong | 1 . . . 7 . 8 . | 1 . . . . 7 . 8 (back) | 1 . . . . 7 . 8 | --------------- --------------- 1 . . . . . 7 8 1 . . . . . 7 8 Bristol, Belfast.

During an exhaustive search for part-peals of Cambridge Major with the tenors together, it was found that many of the peals were palindromes, and some could be broked down into round blocks with palindromic symmetry, and reconstituted in many ways (see Price 1989). Although a general one-part search was in 1988 (and maybe still is?) beyond the bounds of possibility even with a computer, it was found possible to carry out a complete search for one-part palindromes on a mainframe computer; a search which is now straightforward with a PC. Of particular interest were peals (pp 13, 15, 18 etc) which broke down into round blocks, all symmetrical about the same apical transfigure.

A similar structure was discovered when the author was working on long lengths of London Surprise (tenors parted). It was found that the falsity of London lends itself to a whole-course method of composition, and a random technique was devised to discover mutually true courses, utilising various part plans corresponding to available groups of courseheads.

Using a 5-part plan (of five working bells rotating) this random technique quickly produced sets of 65 true courses, one of which is the basis of Peter Border's 14,336 in which a whole course is omitted for the sake of linkage. It was a straightforward matter to find out how to link them all into a 14,560. But the fascinating thing about these 65 courses, folded by the 5-part plan into 13 course-types, is that they have a symmetrical palindromic structure. 5 of the course-types are on the axis of symmetry (this axis being not just a simple transfigure of kind [2, 3x4, 5x6, 7x8] but also its rotations on a 5-part plan), and the other 8 courses are in four image pairs. The 5 course-types on the axis each have two apices on the axis, and there are two apices on the axis at Home Bob linkages. Thus the assembly has 12 possible apices. The only number of palindromes which will include all courses is five. Using a Bob Home apex cuts out a Plain Home apex on the other side of a Q-set, so there must be 10 apices used. It is impossible to find a single palindromic linkage.

How to get one extra course? 66 is divisible by 3, and attempts were made to find 66 true courses by using the two different 3-part plans, [3.02] and [6.33]. The simpler one of just 3 bells rotating met with success. And again, the 22 course-types were disposed with palindromic symmetry! This time, there were 4 course-types on the axis and 9 image pairs, giving 8 apices to be used; again, a single palindrome was impossible.

There were in fact four distinct solutions to 66 true courses, only one of which had linkages by two different calls only. The other solutions were not checked for symmetry, but as they differed only by a few courses, symmetry is probable.

How to get yet another true course? 67 is a prime number. Extensive efforts were made by classifying the 360 positive courses on a one-part palindromic axis, and using the same random technique, without success.

Part-peals and one-part palindromes of Cambridge were (hopefully) exhausted in 1989, but what about one-part palindromic structures with more than two apices? The evidence is that these are likely to exist. This suggests a tree search which, having discovered a palindrome of length at least equal to a set parameter, assembles the remaining true leads and tries the apices left as the start of a further palindrome, and so on. This process might not be so astronomical as appears at first thought, since there are only a limited number of apices available. The basic assumption will be that a resulting set of palindromes must have the same axis of symmetry, an assumption justified by observation. Of course, the problem of linkage will remain, but the fact that linkages exist does not imply that a one-part palindrome could result (and hence the block would have been discovered previously); as has happened elsewhere, several apparently different solutions might give the clue to linkages, asymmetric treatment of which might give a peal-length block. Progress might be made in this direction, but attempts to implement these ideas have so far ended in disappointment.

However, part-success was achieved with Stedman Triples using the 40-part group [7.11], in which there are six apical six-types, all of which must be used, requiring three different palindromic blocks. Such blocks were discovered, but linkages for a peal were inadequate.

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