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Richard Smith richard at ex-parrot.com Tue Sep 28 17:21:54 BST 2010
I'm going to start by cataloguing those plans that can be explained simply in terms of well-understood splices, probably in two separate emails. This will then leave the shorter list of plans that deserve further study.
This email covers all plans with one or two methods. That means there's nothing new in this email as splicing two methods (at least with a fixed treble) is well understood.
SINGLE METHOD PLANS
As we've got 75 methods (modulo lead splices and lead-end variants), 75 of the 4614 plans contain just a single method.
The fact that the plans only include a single method doesn't mean that they cannot produce extents of spliced -- for example, we can easily produce an 8-method extent of spliced using Old Oxford's lead-splices and lead-end variants. Similarly an extent of Beverley, Surfleet, Berwick and Hexham is derived from one of these single method plans.
Let's call a splice 'simple' if it can involve just two methods. So for example the three-lead splice between York and Durham is a simple splice -- sure, we can continue by combining, say, course of Beverley into the touch, but this is optional -- the touch works with just York and Durham and so the three-lead splice is 'simple'.
On the other hand, the grid splice with Ipswich, Bourne and Cambridge is not simple as all three methods are an integral part of the splice -- we cannot get an extent of just Ipswich and Bourne, for example.
For the remainder of this email, I shall refer to methods using their standard two-letter abbreviations. These are give on John Warboy's website:
It's well understood how to generate a complete list of simple splices. I'm not sure an explanation of this has ever been covered explicitly on this list, though it has been mentioned in passing. But I'm not going to break this discussion to explain how to do it -- though I might write another email on it.
Splices are usually described in terms of the minimum number of leads of the method that can be inserted. For TDMMs, this number can be 1, 2, 3, 5 or 6. In practice 2-lead splices are rare and do not occur amonst any of the standard 147. The most common form a of a 5-lead splice is the course-splice where the five leads to be replaced form a course. There are no non-course 5-lead splices using methods from the 147. Let's take these types of splices one by one.
The following lead splices exist amongst the standard 147.
(i) Using the D1, D2, D3 & D4 underworks
[Ci, Ks, Ls, Sd] / [Cf, Dk, Ny, Oc] J/M [Cw, Ns, Sl, Wr] / [Cb, Ng, Ol, Wi] K/N [Cd, Ce, Sw, Va] J
(ii) Using the S1, S2, S3 & S4 underworks
[Ba, Cs, Fg, Sk] / [Bg, Kn, Rs, Wl] J/M [Bt, Le, Md, Pv] H
(iii) Using the Westminster & Allandale underworks
[Ad, Ws] G [Co, Li] H [Ck, Wt] / [Dt, Po] K/N
(iv) Using the Beverley & Surfleet underworks
[Bk, He] / [Bv, Su] H/L [Ed, Kh] / [By, Pm] H/L [Ch, Mu] G
A note on notation. [Bk, He] / [Bv, Su] means that Bk and He are lead splices and that Bv and Su are their 2nds place lead-end variants and which also form a pair of lead splices. Whether you consider He and Bv to be lead splices is simply a matter of definition and of no great relevance here. The letters in the last column are the lead-end orders.
Because lead splices were excluded when reducing the list of methods to 75, they do not appear in the list of plans.
Unfortunately it is not possible to include both J and M variants in an extent (without also including other lead-end orders). This means that the Old Oxford group is the only one of these that can give an eight method extent. This plan is responsible for 55% of the 5.86 x 10^21 extents. This is because there are 4^30 lead splices and 2796 possible callings (allowing 2nds, 4ths and 6ths lead ends). Multiplying these together gives 3.2 x 10^21 extents.
The following is a table of all course splices using methods from the 147. This table was calculated from first principles (and is much the same as the one in Michael Foulds' books on spliced TDMM) rather than extracted from the results of the search.
Br, [Cw, Ns, Sl, Wr] / [Cb, Ng, Ol, Wi], Ma o Ab, Ro / Lf, Ne [Ci, Ks, Ls, Sd], Ox / [Cf, Dk, Ny, Oc], Ms o # Nf, Pr / Cm, Ip o [Bk, He] / [Bv, Su], Du o
[Ba, Cs, Fg, Sk], Do / [Bg, Kn, Rs, Wl], Ey o # [Ck, Wt] / [Ad, Ws], [Dt, Po] + Wh / Cl, Nb o Mo / [Ch, Mu], Nw o C1, Mp o
C3, Pn [Bt, Le, Md, Pv], Cx o Av, [Cd, Ce, Sw, Va] o Cu, Lo +
Notation. As with lead splices, a slash separates 6th place and 2nds place methods. Where a group of methods are enclosed in square brackets, they are lead splices. An o denotes that the course splice is just a half-lead variant, often with a set of lead splices. A # notes that the line contains three separate course splices, e.g. Ox, Ms and the eight lead splices are three sets of course splices. A + notes that multiple backworks are present.
It's worth calculating the number of plans that can be accounted for solely in terms of course splices. This is worthwhile because the easiest way of checking that there's nothing interesting hidden amongst the list of seemingly-ordinary plans is by checking that the search found the predicted number.
With six courses, we would expect 2^6 = 64 plans. However, our list of 4614 plans exclude rotations and reflections, and many of the 64 plans will just be rotations of each other.
If we apply the splice zero times, then we have a single method plan (already considered above). All ways of applying it once are equivalent -- we can always rotate / reflect the plan so that the splice is applied to the 123456 l.h.
What about two applications? Put succinctly, are all choices of two courses equivalent? We know that from the plain course we can reach any other course using just one bob -- therefore all pairs of (distinct) courses are related by cycling three coursing bells and are thus equivalent.
By symmetry, four, five and six applications of the splice will be the same as two, one and zero respectively. This just leaves the case of three applications of the course splice. Are all choices of three courses equivalent? No. For example, we know that a block of three bobs can join the three tenors-together courses, but the same is not true of the three split-tenors courses.
We know that any two courses must share two coursing pairs. Three distinct courses cannot all share two coursing pairs because there are only 10 pairs in total and 3*(5-2)+2 > 10. So they must either all share a single coursing pair (as the tenors together courses do, which is what allows them to be joined by a Q-set of bobs on this pair) or none (as the split tenors courses do).
How many of each type of choice of three courses are there? Once we've selected two courses, there are four remaining. Two of the unselected courses each share a (different) single coursing pair with the two courses, and therefore the other two do not share any coursing pair with both the already chosen courses. One way of looking at this is that courses A,B,C can be joined with a block of three homes, A,D,B with a block of three before, but A,B,E and A,B,F cannot be joined in any order using a block of three calls. So of the 20 ways of selecting three courses, 10 share a single coursing pair, and 10 do not. Once rotations and reflectins have been factored out, this just leaves two ways of selecting three courses.
So we have 1+1+2+1+1 = 6 plans for course splices (excluding those that none of one or other method). The table above has 18 course splices (noting that the two lines marked with a # each contain three pairs of course splices). This means that 108 = 18 * 6 out of the 4614 plans can be explained just in terms of a single course splices, perhaps applied multiple times.
The following is a table of all 6-lead splices using methods from the 147.
Do, No 2 Bl, Wk 2 [Bk, He], Pr, Wa / Bs, [Bv, Su], Cm 3  [Ed, Kh], Os, Wf / Bh, [Bt, Le, Md, Pv], Bw, [By, Pm], Cc, Mp 3  Ml / [Co, Li], Fo 3
[Ba, Cs, Fg, Sk], [Ci, Ks, Ls, Sd], Pe, Ri, Wv / [Bg, Kn, Rs, Wl], Bp, [Cf, Dk, Ny, Oc], Cn, Dn 4  Br, Lv / Ki, Ma 4 Ab, Hu / Bo, Ne 4 Km, Sh / Ti, Tr 4 Ct, Cy / Ak, Cz 4
Lu, Mo / Nw, Ww 4 [Cd, Ce, Sw, Va], Ke 4 Bc, [Cw, Ns, Sl, Wr] / Bm, [Cb, Ng, Ol, Wi] 5 Pn, So 5 Bn, Lo 5
Cx, We 5 [Ch, Mu], Cl, Gl 6 
Notation. As above, a slash separates lead end variants, and lead splices are enclosed in square brakcets. The number in the right-hand column is the fixed (pivot) bell for the splice. Where a number is given in square brackets at the end of the line, this is number of groups of mutually six-lead splicing methods on the line.
Counting the plans that these are responsible for is trivial. Because the splice uses all six rows where a given bell pivots, up to rotation, there is exactly one way of applying the splice once, one way of applying it twice, one way of applying three times, and one way of applying if four times. (Zero or five applications results in a single method extent, already considered above.)
The table above has 17 lines, but four rows list multiple six-lead splices. With n six-lead splice clusters (i.e. a row marked [n]), there are n(n-1)/2 separate pairs of six-lead splicers. This gives 17-4 + 2*(3*2/2) + 5*4/2 + 6*5/2 = 44 six-lead splices.
This means that 176 = 44 * 4 out of the 4614 plans can be explained just in terms of a single 6-lead splices, perhaps applied multiple times.
[Ad, Ws], Di 2&3 + Du, Yo 2&3 * Ca, Gl 2&3 + Cl, Cu 2&3 + Cr, [Cw, Ns, Sl, Wr] / [Cb, Ng, Ol, Wi], El 2&4 +
[Ck, Wt], Wo / [Dt, Po], Sa 2&4 + Nm, Pn 2&4 Lo, We 2&4 * Ro, St / Lf, Wm 2&5 * [Bt, Le, Md, Pv], [Co, Li] 2&5 +
[Ba, Cs, Fg, Sk], Hu / [Bg, Kn, Rs, Wl], Bo 2&6 + Km, Sn / Qu, Tr 2&6 Ct, Mo / Ak, Nw 2&6 * Br, Hm / Ma, Ta 3&5 * Hu, Lv / Bo, Ki 3&5
Cy, Lu / Cz, Ww 3&5 * C2, C3 3&5 * Ev, Wo / Sa, Te 3&6 Bn, Cx 3&6 * Di, Ms 4&5
Av, Ca 4&5 *
(Notation. As with lead splices, a slash separates 6th place and 2nds place methods. Where a group of methods are enclosed in square brackets, they are lead splices. The numbers in the right-hand column are the fixed place bells for the splice. A * notes that the splice works like London and Wells by swapping 34.16.34 for 14.36.46 at the half-lead. A + notes that multiple backworks are present.)
With 30 leads in the extent, we can apply the 3-lead splice any number of times from 0 to 10. Another way of looking at this is that there are ten ways of choosing a pair of bells from the five working bells (10 = 5*4/2). This means that there are 2^10 different plans for each extent. However, our list of 4614 plans exclude rotations and reflections, and many of the 1024 = 2^10 will just be rotations of each other which complicates things a bit.
If we apply the splice zero times, then we have a single method plan (already considered above). All ways of applying it once are equivalent -- we can always rotate / reflect the plan so that the splice is applied to the 123456 l.h. With two applications, either the two applications share a fixed bell (e.g. 2&3 and 2&4) or they do not (e.g. 2&3 and 4&5). Up to rotation and reflection, that's the only choice left. We can show these diagramatically with letters A-E indicating the five working bells and a representing each application of the splice by joining the two fixed bells.
(1.1) A --- B C D E
(2.1) A --- B --- C D E
(2.2) A --- B C --- D E
With three applications, we apparently have four possibilities.
(3.1) A --- B --- C D --- E
(3.2) A --- B --- C --- D E
(3.3) A --- B --- C E | | D
(3.4) A --- B D E \ / \ / C
However, this isn't what the search found. For example, it found five plans (up to rotation and reflection) containing 21 leads of London and 6 of Wells -- (3.2) appeared twice.
The reason is to do with parity. Because the plan only uses in-course l.h.s and l.e.s we can only rotate or reflect the plan by an even permutation. In (3.1), A and C are equivalent as are D and E. When rotating (3.1), if we find we need an odd permutation, we simply swap the labels on A and C and use an even permutation.
But with (3.2) we can't do that. Yes, A and D are equivalent as are B and C. But we cannot indepdently swap labels on one pair of these -- if we swap the labels on A and D we also need to swap the labels on B and C for the graph to remain unaltered. This means we cannot simply relabel so that an odd permutation 'rotation' converts into an even permutation. The result is that there are two versions of (3.2) which we might term a right-handed and a left-handed version.
What of four applications of the splice?
(4.1) A --- B --- C --- D --- E [has l+r versions]
(4.2) A --- B --- C --- D | | E
(4.3) A --- B --- D E \ / \ / C
(4.4) A --- B D --- E \ / \ / C
(4.5) A --- B E | | | | C --- D
(4.6) A | | B --- C --- D | | E
And finally, for five applications:
(5.1) A --- B --- C [l+r variants] \ / \ / D --- E
(5.2) A --- B --- C | | | | D --- E
(5.3) A --- B --- C --- D \ / \ / E
(5.4) A --- B --- C --- D [l+r variants] \ / \ / E
(5.5) A | | B --- C --- D \ / \ / E
(5.6) A --- B E \ / \ \ / \ C --- D
Six or more applications of the splice are, by symmetry, the same as four or fewer. This gives the total number of plans for a 3-lead splice as: 1+2+5+7+8+7+5+2+1 = 38. There are 21 3-leads splices in the table above, so that means that 3-lead splices are responsible for 798 = 21 * 38 of the 4614 plans.
Scanning through the results of the search, I find 1157 plans with one or two methods. If I add the numbers above, I get:
Single method plans . . . . . . . . . . 75 Course splices . . . . . . . . . . . . . 108 Six-lead splices . . . . . . . . . . . . 176 Three-lead splices . . . . . . . . . . . 798 --------------------------------------------- TOTAL . . . . . . . . . . . . . . . . . 1157
This isn't surprising. As I noted at the beginning of the email, the theory of spliced with just two methods is well understood and we wouldn't expect to find anything new. However, this has been a productive exercise on two counts. First, it increases my confidence that the search results are correct as it agrees with the already well-tested theory on splicing two methods. Secondly, it has allowed me to work out techniques for counting extents -- for example, identifying the potential problem with chirality (handness) of certain three-lead splices.
Of course, with 3459 plans left to study, there's still plenty to do!