Date: Fri, 14 Apr 95 16:03:31 -0400
From: michael reid <mreid@ptc.com >
~~~ Subject: more on the slice group

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.)

i also calculated for the other slice metric. in this metric,
neighbors can have the same distance from start, so a "strong"
local maximum is a position all of whose neighbors are strictly
closer to start. a "weak" local maximum is a position none of
whose neighbors are further from start.

```90 or 180 degree     number of     number of strong    number of weak
slice turns        positions       local maxima       local maxima
```
```0                  1               0                   0
1                  9               0                   0
2                 51               0                   0
3                247               0                   7
4                428               0                 212
5                 32               8                  32
```

the strict local maxima are all equivalent under symmetries of
the cube. they are the composition of pons asinorum with any
of the eight positions called "six dots".

in the same way, local maxima (within the antislice group) in the
90 degree antislice metric are local maxima in the full cube group
(quarter turn metric). perhaps mark will tell us more about this.

mike