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Date: Tue, 9 May 1995 08:48:23 -0400 (EDT)
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Re: more on the slice group
In-Reply-To: Message of 04/14/95 at 16:03:31 from mreid@ptc.com
On 04/14/95 at 16:03:31 mreid@ptc.com said:
>mark's post got me thinking ... i made a quick hack for the slice
>group (which is easy to represent by fixing the corners). my
>figures concur with his. i wanted to see the number of local maxima.
> 90 degree number of number of
> slice turns positions local maxima
> 0 1 0
> 1 6 0
> 2 27 0
> 3 120 0
> 4 287 0
> 5 258 24
> 6 69 69
>as i'd hoped, there are local maxima at distance 5. one such is:
> (FB') (RL') (U'D) (R2L2) =
> (R2L2) (F'B) (RL') (UD') =
> (R'L) (FB') (RL') (F'B) (U'D) =
> (U'D) (F'B) (RL') (U'D) (F'B) =
> (R'L) (UD') (F'B) (RL') (FB')
>(actually i think all are equivalent to this one under symmetries
>of the cube.)
>this is especially interesting because it is a local maximum in the
>full cube group (quarter turn metric) at distance 10q. according
>to jerry bryan's results, there are no local maxima within 9q
>of start, so this gives the closest local maximum. (there may well
>be others.)
Results for the slice group under M-conjugacy:
Level Number of Number of
Positions Local Maxima
0 1 0
1 1 0
2 2 0
3 6 0
4 16 0
5 15 1
6 9 9
Mike's conjecture that all 24 positions which are a local maxima
at level 5 are equivalent under M-conjugation is correct.
I don't yet understand why Mike's position is a local maximum in the
full cube group. But assuming it is, it is not only the shortest
local maximum, it is the first local maximum which is not
Q-transitive (i.e, we have |{m'Xm}|=24, hence we have |Symm(X)|=2,
and the size of the symmetry groups for Q-transitive positions
must be divisible by 12.).
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) (304) 293-5192
Associate Director, WVNET (304) 293-5540 fax
837 Chestnut Ridge Road BRYAN@WVNVM
Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU