[r-t] Fw: Avon Double Cyclic Doubles

[Philip Saddleton]

If you write out the rows of a multi-part extent in a table, with a 
column for each part, the rows of the table (hereafter called lines to 
avoid confusion) correspond to the cosets of the group given by the part 
ends. The transposition between rows in adjacent columns of the same 
line is the same for each adjacent pair in a line, and different for 
each line (the conjugates of the part end transposition). Thus for a 
5-part extent of doubles, starting at each row of the part will give a 
different part end, and there are 4! possibilities. Four of these will 
be cyclic.

Incidentally, v2 is simply v1 backwards, hence the part ends occur in 
the reverse order.

PABS

On 24/07/2015 12:23, Philip Earis wrote:
> John Bissell has sent this reply to my recent email, which I am 
> forwarding with his permission.  Please see his questions below...
>
> -----Original Message----- From: J. J. Bissell
> Sent: Friday, July 24, 2015 3:38 PM
> To: pje24 at cantab.net
> Subject: Re: [r-t] Avon Double Cyclic Doubles
>
> Dear Philip,
>
> Yes that’s correct. Someone mentioned to me that the extent of Banana 
> Doubles has a call every 12 rows, and that made me wonder whether it’s 
> related (derived from?) a 24 rows per lead principle. So I generated 
> the extent, and translated it to give it both cyclic lead heads and 
> the double half-lead symmetry property. Of course, the translation 
> makes it wrong hunting on the front (a method we’ve nicknamed Nova), 
> so I finished with a rotation.
>
> This prompts a further question: is it always possible to translate a 
> plain course extent principle to give it cyclic lead heads? If so, why?
>
> For example, noting that Avon Doubles has a parity change every row, I 
> thought it would be interesting to see if there exists a plain course 
> extent principle with half-lead double symmetry such that the row 
> parity changes every six rows. This is what I came up with:
>
> Parity Doubles: 3.1.3.5.3.125.3.5.3.1.3.145.3.5.3.1.3.145.3.1.3.5.3.125
>
> Now, if instead one adopts
>
> 1.3.5.3.125.3.5.3.1.3.145.3.5.3.1.3.145.3.1.3.5.3.125.3
>
> (i.e., starts from the second row), then one recovers the cyclic lead 
> heads, half-leads, and quarter leads; though personally I prefer the 
> right-hunting on the front rotation:
>
> Parity Doubles v2: 
> 3.125.3.5.3.1.3.145.3.1.3.5.3.145.3.1.3.5.3.125.3.5.3.1
>
> I’m sure there must be a very simple reason for this, but it is not 
> obvious to me.
>
> Thanks,
>
>
> John
>
>
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