[r-t] Extents of doubles

[Alexander Holroyd]

Well done Richard.

Can we see a list of the 101 2-blow ones?
And maybe the legal glide-symmetrics (if there are any)?

On Tue, 22 Aug 2006, Richard Smith wrote:

> Richard Smith wrote:
>
>>> Can anyone generate an exhaustive list of all 24 change-per-lead doubles
>>> principles whose plain course generates the extent?
>>
>> It ought to be feasible.
>
> Well, somewhat later than anticipated, I have a list of all
> 24 change-per-lead doubles principles.  Once you've factored
> out reflections, rotations, translations and "Kent / Oxford
> variants", there are 52,227,975 doubles principles
> whose plain course generates the extent.
>
> By "Kent / Oxford variant", I mean the same as Ander in his
> list of minimus extents -- if a block of four rows can be
> rung in both the orders (a,b,c,d) and (a,c,b,d) such that
> both are joined by valid changes, then they are K/O
> variants.  For example, 5.125.5 and 345.125.345 are a pair
> of K/O variants.  All pairs are characterised by swapping a
> dodge for places.  This particular example is precisely the
> difference between Plain Bob and Reverse Canterbury.  (The
> term "Kent / Oxford variant" is a bit of a misnomer as Kent
> and Oxford do *not* differ by a K/O variant.)  On five
> bells, there are 17 pairs of K/O variants.
>
> These fifty-odd million principles can be categorised by
> symmetry as follows:
>
>  52,214,650    Asymmetric
>      12,953    Palindromic
>         372    Glide-symmetric
>  ----------
>  52,227,975
>
> It would appear (unless there's a bug in my code) that there
> are no rotationally-symmetric principles.
>
> They can also be classified in terms of maximum consecutive
> blows in one place:
>
>         101     2 blows  \
>     506,059     3 blows   | "Legal"
>   8,703,933     4 blows  /
>  13,055,579     5 blows
>  11,590,519     6 blows
>   7,733,797     7 blows
>   4,551,747     8 blows
>   2,599,963     9 blows
>   1,496,379    10 blows
>     855,875    11 blows
>     489,505    12 blows
>     271,352    13 blows
>     154,848    14 blows
>      89,658    15 blows
>      54,516    16 blows
>      31,830    17 blows
>      17,940    18 blows
>      10,434    19 blows
>       5,818    20 blows
>       3,832    21 blows
>       1,576    22 blows
>       2,714    24 blows
>  ----------
>  52,227,975
>
> Note that, unsurprisingly, no methods have 23 consecutive
> blows in one place.  Unfortunately, this analysis is
> somewhat spoilt by the fact that K/O variants do not
> necessarily contain the same number blows in one place and
> the choice of which K/O variant to discard is, to some
> extent, arbitrary.
>
> As the list runs to several gigabytes, I can't readily make
> it available.  However, if anyone wants me to do any further
> analysis on this data, let me know and I'll try.
>
> RAS
>
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