Red Ed wrote:udosuk, I'd be interested to know whether the isomorph below is, to "trained eyes", better or worse than the one you gave - and why.
- Code: Select all
+-------+-------+-------+
| 2 3 1 | 4 8 6 | 5 9 7 |
| 7 5 9 | 1 2 3 | 4 8 6 |
| 8 6 4 | 7 5 9 | 2 3 1 |
+-------+-------+-------+
| 1 2 3 | 8 6 4 | 7 5 9 |
| 9 7 5 | 2 3 1 | 6 4 8 |
| 4 8 6 | 5 9 7 | 1 2 3 |
+-------+-------+-------+
| 3 1 2 | 6 4 8 | 9 7 5 |
| 5 9 7 | 3 1 2 | 8 6 4 |
| 6 4 8 | 9 7 5 | 3 1 2 |
+-------+-------+-------+
The automorphism group is <C,RxSx,C2B>.
Almost the same, but I think
eleven and I are more used to see Column Sticks than Row Sticks, so a transpose of the grid could be more comfortable to the eyes.
These are the 5 basic symmetry groups I can directly "see" from it:
RS: (1)(2)(3)(47)(58)(69)
MR: (123)(486)(597)
JC @ c456, GC/ @ c123789: (123)(486)(597)
GC\ @ c456, JC @ c123789: (132)(468)(579)
GC/ @ c456, GC\ @ c123789: (1)(2)(3)(4)(5)(6)(7)(8)(9)
Red Ed wrote:udosuk, upon re-reading some of the posts above, it seems that your criteria for a nice morph differ from mine: you want to see as many symmetries as possible in their natural form; I want to specify the aut group in as concise a form as possible.
Just do what you like for the next report. Because in most cases I think we can't display all symmetries in one isomorph (e.g. \M+/M & QT) anyway. If we can't do the best for this criteria, then might as well set it to suit another criteria better.
Red Ed wrote:If I understood that correctly, your requirements are much easier to fulfil than mine. I would suggest doing so as follows. You give me a list of every symmetry that you would like to see in the morphed grid, and a score for each. E.g. "D" might be worth 10 points, "D2" just 5. Probably the best score is 1 point for every symmetry in its natural form - whatever you want that to mean - and 0 for everything else. Make sure the symmetries are described precisely! Then it's a simple job to write a program to find a morph with maximum score.
I'm not sure that's easy - anyway, here is a try:
(Just a note - I'm more comfortable using my own 2-letter codes for symmetries then your/
eleven's "D", "RxSx" etc, so pardon for not using them.)
Each of HT (order-2) & the order-9 groups (FR/FC, WR/WC, BR/BC, FD) gets 30 pts, because they're the hardest to see from a shuffled grid. Note the order-9 groups implicitly imply HT anyway, so both should be displayed.
The mixed order-9 groups (e.g. 1FR2WR, 1WR2FR) gets 24 pts, 80% of the pure ones.
If we have DDS (\M + /M), it's worth 20 pts, as I think it gives us a better view than QT (in case it also exists) for other groups.
Also same for DSS (Double Sticks Symmetries) (CS+RS), 20 pts.
Each of individual \M, /M, CS, RS gets 10 pts.
QT gets 8 pts.
Each of the order-3 groups (MR/MC, MD, JR/JC, JD, GR/GC in any orientation) gets 5 pts (they're the easiest to see from a shuffled grid).
Each of the mixed order-3 groups (e.g. 1MR2MD, 1JR2GR) gets 4 pts, again 80% of the pure forms.
(Note: in case of double orientations (e.g. JR+JC), make sure both are displayed and scores counted.)
(Note 2: for the mixed groups (1AA2BB) make sure the 1-group (AA) is always displayed in the middle band/stack and the 2-group (BB) on the sides.)
That's about it, let's see if it works well.