Date: Mon, 26 Sep 94 14:29:31 -0400 (EDT)
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ ~~~ Subject: Re: < U, R> Group (W-conjugate results)

I have completed the God's Algorithm calculations for the <U,R> group
in terms of W-conjugate classes (or really, in terms of representative
elements of W-conjugate classes), with the results below. In general,
the use of W-conjugates reduces the size of the problem by about
four times. However, I was surprised to see that for levels 1, 3, 5,
7, 9, and 11 the number of cubes was exactly four times larger than the
number of W-conjugate classes. My interpretation is that all cubes
in <U,R> at these levels are completely "asymmetric" with respect to
W. (They are somewhat symmetric with respect to M, of course.) However,
when level 13 turned out not to have a ratio of exactly 4 between cubes
and W-conjugate classes, I was rescued from the task of explaining why
all cubes at an odd distance from Start were asymmetric.

```Level   W-Conjugate  Branching      Total   Branching   Ratio
Classes     Factor       Cubes     Factor      of
Cubes to
Classes
```
``` 0             1                        1              1
1             1      1                 4     4        4
2             3      3                10     2.5      3.333
3             6      2                24     2.4      4
4            15      2.5              58     2.416    3.866
5            35      2.333           140     2.413    4
6            85      2.429           338     2.414    3.976
7           204      2.4             816     2.414    4
8           493      2.417         1,970     2.414    3.996
9         1,189      2.412         4,756     2.414    4
10         2,863      2.408        11,448     2.407    3.999
11         6,862      2.397        27,448     2.398    4
12        16,324      2.379        65,260     2.378    3.998
13        38,550      2.362       154,192     2.363    3.9997
14        90,192      2.340       360,692     2.339    3.9992
15       206,898      2.294       827,540     2.294    3.9997
16       462,893      2.237     1,851,345     2.237    3.9996
17       992,268      2.144     3,968,840     2.144    3.9998
18     1,973,209      1.989     7,891,990     1.988    3.9996
19     3,415,314      1.731    13,659,821     1.755    3.9996
20     4,618,491      1.352    18,471,682     1.352    3.9995
21     4,147,448      0.898    16,586,822     0.898    3.9993
22     2,010,449      0.485     8,039,455     0.485    3.9988
23       378,110      0.118     1,511,110     0.188    3.9965
24        11,894      0.031        47,351     0.031    3.9811
25            27      0.002            87     0.002    3.222
```
```Total  18,373,824               73,483,200              3.9993

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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
```

If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?