Let us start an examination of the theory by studying the one-part palindrome. Consider the lead of treble-bob (or surprise) below:

--------------- 1 5 7 2 8 4 6 3 leadhead - - - - - - - - 2 6 8 5 3 4 7 1 6 2 5 8 4 3 7 1 - - - - - - - - 1 8 7 6 5 3 2 4 leadend ---------------

It has been telescoped by writing only the first and last rows and the two centre rows when the treble lies behind. It could be the second lead of Cambridge, in the course 42356.

At the half-lead, pairs 2x6, 5x8, 3x4 swap and 7 lies still. Because of the symmetry of the transpositions in the lead, this transfigure is the same as that between the leadhead and leadend. The four rows illustrate the fundamental relationship between transpositions and transfigures:

1 5 7 2 8 4 6 3 -------> 2 6 8 5 3 4 7 1 x | | | y | y | | V V x 1 8 7 6 5 3 2 4 -------> 6 2 5 8 4 3 7 1

where x is the overall transposition of the first halflead (the second halflead is its inverse transposition) and y is the transfigure determined by the places made while the treble lies behind.

Thus the relation between leadhead and leadend (or between any pair of rows of the lead symmetrically disposed) is a 2-part transfigure (of 3 swaps and 2 bells fixed). This relation between leadhead and leadend can also be regarded as a transposition, and for a particular method this transposition is the same for every lead, as it is the result of a fixed sequence of transpositions according to the places made in the method. Furthermore, this transposition must be a 2-part one as pairs of bells swapping at half-lead also swap their positions between leadhead and leadend. We can further telescope the lead as:

| V --------------- leadhead 1 5 7 2 8 4 6 3 | 2-part leadend 1 8 7 6 5 3 2 4 | transposition --------------- | V

Since the transposition is 2-part, it is its own inverse. This will prove important later.

Consider this lead as part of a one-part palindrome. What will be its image lead on the other side of the palindrome?

The one-part palindrome will have two apices, each of them either at the treble's lead, or at mid-lead when the treble lies behind. Usually, the treble plus one other bell will make places at each apex, the work of these two bells being exact images in the palindrome; and three pairs of bells will swap at each apex, the same three pairs at each. For each pair, the work of one bell on one side of the palindrome is the image of the work of the other bell on the other side. However, exceptionally it has been shown that one-part palindromes exist with singles at the apices ( Price 1989) and hence the apical transfigure will have three working bells making a place and only two pairs swapping. Such an apex cannot be at mid-lead, nor at a Before if the tenors are to remain unparted (because it would reverse the tenors); it must be at a Home single.

Although a course without calls has 7 leads, in order to leave the tenors unparted calls may only be made at four positions in a course. These positions are symmetrically disposed in relation to the work of the tenors, and it is necessary to have the tenors swapping at the apices, so that possible call positions pair off as images in a palindromic structure. All known palindromes with the tenors unparted conform to this. Let us therefore assume that the pair of bells 78 swap at the apices. Of the working bells 23456, usually one bell makes a place (at plain or single, or at mid-lead) and two pairs swap; but it is also possible at a single that three of the bells make a place, and one pair swap. In either case, a 2-part transfigure is defined by the apices. Consider as an example that an apex at a leadend defines the transfigure (2, 3x4, 5x6) which is labelled x below:

+-------------+ +-------------+ | p | | q | +-------------+ +-------------+ ^ ^ transposition y | | | APEX | 1 2 5 6 3 4 7 8 <------------ 1 2 6 5 4 3 8 7 row b transfigure x row a

(N.B. The "uparrow" symbol ^ and letter V are used here inadequately as arrows in these diagrams)

The rows a and b are consecutive rows, the leadend and leadhead of an apex with transfigure x. p is any row on the side of the palindrome following the apex quoted, and q is its image on the other side. From row 12563478 to row p is a sequence of transpositions, equivalent overall to transposition y, while row q to row 12654387 is transposition z. From the nature of a palindrome, y is the inverse transposition to z. Hence we have:

p <--------- q | | transposition z | | transposition z V V 1 2 5 6 3 4 7 8 <--------- 1 2 6 5 4 3 8 7 transfigure x

It follows from the fundamental relation that p and q are related by the transfigure x. Thus corresponding rows of the palindrome are related by the same transfigure x.

Return now to the sample lead we discussed previously.

^ | | V --------------- --------------- Lead 1 6 8 2 7 3 5 4 1 5 7 2 8 4 6 3 Lead A* - - - - - - - - - - - - - - - - A 1 7 8 5 6 4 2 3 1 8 7 6 5 3 2 4 --------------- --------------- ^ | | V | Apex (2, 3x4, 5x6, 7x8) | ------<------------------<------

Lead A is the sample lead, and Lead A* is its image lead. The four rows (leadhead and leadend of each lead) are related thus:

1 6 8 2 7 3 5 4 <-----------> 1 5 7 2 8 4 6 3 ^ x ^ | w w | V x V 1 7 8 5 6 4 2 3 <-----------> 1 8 7 6 5 3 2 4 Transfigurex swaps positions offigures3x4, 5x6, 7x8 Transpositionw swaps figures inpositions2x5, 4x7, 6x8

This is a special form of the fundamental relation in which both the transposition and transfigure involved are 2-part and hence self-inverse. x is the transfigure at the apices, w is the internal transposition of a lead. An important deduction follows: every possible lead (whether the tenors are parted or not) has a unique image lead. For example, Lead A has leadhead 15728463. To this row, apply transfigure x and transposition w (in either order) to obtain 17856423, the leadhead of Lead A*. Since the arrows are 2-way (the operations being self-inverse), 17856423 by the same process gives 15728463. This would not be true, were the operations not self-inverse.

If the tenors are to remain unparted, the uniqueness applies also, provided that the apical transfigures have 7x8 swapping at each. Refer to the analysis of treble-bob major method categories with tenors unparted, towards the end of this paper.

Thus, we may take all possible leads and classify them into pairs. But there is an exception to this process; a lead might turn out to be its own image. This happens when the centre-line of the lead has the same transfigure as a palindrome apex, i.e. it is a possible centre-lead apex. In this case, applying the operations x and w successively to the leadhead will produce no overall change. Thus, in the process of segregating leads (or lead types, for composition in more than one part) into image-pairs, a certain number may be isolated in this way as being self-images. The circumstances in which this does not happen are in systems where the apical transfigures are all of kind (2,2,1,1,1); in these, mid-lead apices are not possible and all apices will be at leadhead Singles. But if mid-lead calls are being used, the situation is clearly different.

If we are including singles in the composition process (and composition without singles is very restricted) with tenors unparted, there are 5! = 120 possible courses with 7 leads in each, a total of 840 leads to classify into image pairs and possible mid-lead apices. How many mid-lead apices will there be? There are 120 mid-leads, of which one-fifth will have the correct place bell, and of these the remaining 4 bells must pair off with a probability of one-third of being correct. Thus there should be just 8 possible mid-lead apices. We can therefore predict that there will be 8 possible mid-lead apices, and the remaining 832 leads will fall into 416 discrete pairs.

What now of calls? Treating a plain lead as a call, there will be three possible calls to link leads (some of which will be invalid as parting the tenors); but in each case, the call as a transposition linking leadend to leadhead is a 2-part one and hence self-inverse. Thus the same logic applies to calls as to internal lead transpositions. The crucial point is this:

Lead B Lead C ^ | transposition z | | transposition z | V Lead A* Lead A

Leads A and A* are image leads in the palindrome. A call is made from the leadend of Lead A*, with transposition z, producing the leadhead of Lead B. On the other side of the palindrome, Lead C is the lead which produces Lead A by the same call transposition. Are Leads B and C also images? They must be, by the same argument as above, if we substitute the call transposition for the internal lead transposition.

^ | | V --------------- --------------- Lead 1 6 8 2 7 3 5 4 1 5 7 2 8 4 6 3 Lead A* - - - - - - - - - - - - - - - - A 1 7 8 5 6 4 2 3 1 8 7 6 5 3 2 4 BOB --------------- BOB --------------- 1 8 7 5 4 6 3 2 1 7 8 6 3 5 4 2 Lead - - - - - - - - - - - - - - - - Lead D* 1 4 7 3 8 2 5 6 1 3 8 4 7 2 6 5 D --------------- --------------- ^ | | V | Apex (2, 3x4, 5x6, 7x8) | -----<----------------------<----

Lead A gives Lead D by a bob Before, while lead D* gives Lead A* by a bob. The rows on either side of the bobs are related thus:

1 7 8 5 6 4 2 3 <-----------> 1 8 7 6 5 3 2 4 ^ x ^ | BOB BOB | V x V 1 8 7 5 4 6 3 2 <-----------> 1 7 8 6 3 5 4 2 Transfigure x swaps positions of figures 3x4, 5x6, 7x8 Transposition Bob swaps figures in positions 2x3, 5x6, 7x8

Now suppose that Leads A and A*, mutual images, are classed as image pair type P, and Leads D and D* as type Q. Lead A by a bob gives Lead D, but Lead A* does not give Lead D* by a bob but rather the other way round. Hence we cannot generalise and say that type P gives type Q by a bob. The image-pair types cannot be used for implementing calls, but on the other hand they can be used for the purposes of proof, because we have established a 1:1 correspondence between leads.

This suggests a technique of proof which will avoid a tree search having to construct and test for proof the two sides of a palindrome separately. We may give each lead (or lead-type) a "proof number", a number associated (arbitrarily) with the two leads involved - the lead and its image. In the case of leads containing a possible mid-lead apex, they have no image lead so their proof number is unique; Such leads may only appear at an apex. Then, starting at a known apex, only one side (the "forward" side) of the palindrome will be constructed by a tree search using trial and error; as each lead is added to the chain, its proof number is checked against those of the leads already assembled, that there is no repetition.

Thus in the case cited above, the proof array will be dimensioned 424 by x, x being the maximum number of false image-pairs per image-pair, whereas the call array will be 840 x 3.

What of the possible apices which are not at mid-lead? They are at Befores or Homes, with a call or a plain lead. They can be detected during the calculation of the call array, because the leads on either side of a possible apex will have the same proof number.

^ | | V --------------- --------------- Lead 1 5 8 3 7 4 2 6 1 6 7 4 8 3 2 5 A* - - - - - - - - - - - - - - - - A 1 7 8 2 5 6 3 4 1 8 7 2 6 5 4 3 --------------- --------------- | BOB V | Apex (2, 3x4, 5x6, 7x8) | ------<----------------------<-----

In the example above, Lead A with leadhead 16748325 gives, with a Bob Before, Lead A* with leadhead 17825634. These two leads are mutual images, and joined by a Bob Before they could form an apex of a palindrome with the required transfigure (however, they can occur on the two sides of a palindrome without being at an apex). If the arrangement above is to be a starting apex, then the tree search will start with Lead A*; whereas if it occurs as the final apex of a tree search, Lead A will be the final element in the tree.

Thus, during the compilation of the call array (a process which does not involve the image-pair types directly) the cases must be marked where the call in question links leads which are mutual images, and hence is a possible apex. A system can be evolved in which the earlier of such a lead-pair, and the later one, are labelled in order to identify the possible apices of a palindrome.

A peal which is a one-part palindrome may be regarded as a two-part peal in structure, the group of partends being rounds together with the apical transfigure; but one of the parts is rung in reverse. However, in terms of leads this is not quite correct, as either or both apices may be at centre-lead; but in terms of rows the statement is correct.

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