Help with symmetries

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Saka
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Help with symmetries

Post by Saka » November 25th, 2015, 12:55 am

I need help with soup symmetries, I know that C1 means no symmetry, but what do the other symmetries mean? (e.g. D4_x4, C2_2, D8 x4)

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biggiemac
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Re: Help with symmetries

Post by biggiemac » November 25th, 2015, 2:01 am

So the names are tied to group theory, and refer to the set of things you can do that make the soup look like it started. C1 is the trivial group, the only thing you can do is nothing. C2 and C4 are groups with 2 and 4 steps, rotations by a half or quarter circle (C means cyclic). So a C2 soup looks the same when rotated 180 degrees. A C4 soup looks the same when rotated 90 degrees. But because we are on a grid, there are three different alignments of the 180 degree rotated piece. The extra number tells that: for C2_2 a cell and its rotated image are in corners of a odd by even rectangle, C2_4 an even by even, and C2_1 an odd by odd.

The D is for the dihedral group. It's like the cyclic group but with reflections too. D2 is like C1 with a mirror down the middle, and again there is a choice of even or odd. There is an additional choice though, of whether the mirror is vertical/horizontal or diagonal. If the mirror is vertical/horizontal, it is +, for diagonal it is x. So D2 +2 refers to a soup that can be made to look like it started either by doing nothing or by reflecting along a horizontal line with an even distance between similar cells. Similarly, D4 resembles C2, but with the reflections too, and now there are 2 independent parities, one for the spin and one for the flip. Thus with D4, 1 means odd odd, 2 odd even, and 4 even even. So D4 soups look the same if you do nothing, if you spin 180, if you mirror appropriately or if you mirror and spin (4 possible operations, so the group is D4, which is the method behind identifying the number immediately following the letter).

D8 has the most options, it refers to soups that are identical if rotated by any multiple of 90 degrees or if flipped appropriately about any axis. A flip about a horizontal axis followed by a 90 degree rotation is the same as a flip about a diagonal axis so there's no + vs x in D8, only the even or odd.

You can tell the symmetry of an object with the same group theory. If some ash object can be mirrored about a line with an odd distance between neighbors (e.g., MWSS on MWSS 1), then it is probably more common from soups with D2_1 symmetry (which includes D4_1, D4_2, and D8_1 but not D4_4). 28p7.2, for another example, has C2_2 symmetry, so our best bet for finding a p7 from a soup would be to modify apgnano to support C2_2 -nudge nudge-.

Hope this helps! I started by writing from memory but I went and checked with catagolue and found I had a number of things wrong with the parity numbers, so I edited that all up and now it's fixed. Hopefully nobody saw it while it was still wrong..
Physics: sophistication from simplicity.

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Saka
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Re: Help with symmetries

Post by Saka » November 25th, 2015, 3:35 am

I think I got it, but I'm not sure, I prefer pictures

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SuperSupermario24
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Re: Help with symmetries

Post by SuperSupermario24 » November 25th, 2015, 2:11 pm

Speaking of symmetries, I have a couple questions about rule table symmetries:
-What does the "permute" symmetry mean?
-What is "rotate8"? How would that work with a Moore neighborhood?

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bobo2b3o2b2o2bo3bobo$obobobo3bo2bobo3bobo$obobob2o2bo2bobo3bobo$o3bobo3bo2bobobobo$o3bob3o2b2o3bobo2bo!

danieldb
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Re: Help with symmetries

Post by danieldb » November 25th, 2015, 2:44 pm

SuperSupermario24 wrote:Speaking of symmetries, I have a couple questions about rule table symmetries:
-What does the "permute" symmetry mean?
-What is "rotate8"? How would that work with a Moore neighborhood?
Permute means that for

Code: Select all

1,1,1,1,0,0,0,0,0,1
All S3 coditions are set

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SuperSupermario24
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Re: Help with symmetries

Post by SuperSupermario24 » November 25th, 2015, 2:51 pm

danieldb wrote:Permute means that for

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1,1,1,1,0,0,0,0,0,1
All S3 coditions are set
Oh, that's useful. Thanks.

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bobo2b3o2b2o2bo3bobo$obobobo3bo2bobo3bobo$obobob2o2bo2bobo3bobo$o3bobo3bo2bobobobo$o3bob3o2b2o3bobo2bo!

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Alexey_Nigin
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Re: Help with symmetries

Post by Alexey_Nigin » November 27th, 2015, 5:19 pm

Saka wrote:I think I got it, but I'm not sure, I prefer pictures
OK, here they are:

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#C [[ VIEWONLY TITLE "C1" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 16, y = 16, rule = B3/S23
16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b
2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o
!

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#C [[ VIEWONLY TITLE "C2_1" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 31, y = 31, rule = B3/S23
16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b
2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$31o
$15bo14bo$15bo14bo$15bo9b2o3bo$15bo8b4o2bo$15bo8b4o2bo$15bo2b10o2bo$
15bo14bo$15bo14bo$15bo3b2o4b2o3bo$15bo2bo2bo2bo2bo2bo$15bo2bo2bo2bo2bo
2bo$15bo3b2o4b2o3bo$15bo14bo$15bo14bo$15b16o!

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#C [[ VIEWONLY TITLE "C2_2" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 31, y = 32, rule = B3/S23
16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b
2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o
$15b16o$15bo14bo$15bo14bo$15bo9b2o3bo$15bo8b4o2bo$15bo8b4o2bo$15bo2b
10o2bo$15bo14bo$15bo14bo$15bo3b2o4b2o3bo$15bo2bo2bo2bo2bo2bo$15bo2bo2b
o2bo2bo2bo$15bo3b2o4b2o3bo$15bo14bo$15bo14bo$15b16o!

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#C [[ VIEWONLY TITLE "C2_4" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 32, y = 32, rule = B3/S23
16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b
2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o
$16b16o$16bo14bo$16bo14bo$16bo9b2o3bo$16bo8b4o2bo$16bo8b4o2bo$16bo2b
10o2bo$16bo14bo$16bo14bo$16bo3b2o4b2o3bo$16bo2bo2bo2bo2bo2bo$16bo2bo2b
o2bo2bo2bo$16bo3b2o4b2o3bo$16bo14bo$16bo14bo$16b16o!

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#C [[ VIEWONLY TITLE "C4_1" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 31, y = 31, rule = B3/S23
31o$o14bo14bo$o14bo14bo$o3b2o4b2o3bo3b3o3b2o3bo$o2bo2bo2bo2bo2bo2b4o2b
o2bo2bo$o2bo2bo2bo2bo2bo2b4o2bo2bo2bo$o3b2o4b2o3bo3b3o3b2o3bo$o14bo5bo
8bo$o14bo5bo8bo$o2b10o2bo5bo3b2o3bo$o2b4o8bo5bo2bo2bo2bo$o2b4o8bo5bo2b
o2bo2bo$o3b2o9bo5bo3b2o3bo$o14bo14bo$o14bo14bo$31o$o14bo14bo$o14bo14bo
$o3b2o3bo5bo9b2o3bo$o2bo2bo2bo5bo8b4o2bo$o2bo2bo2bo5bo8b4o2bo$o3b2o3bo
5bo2b10o2bo$o8bo5bo14bo$o8bo5bo14bo$o3b2o3b3o3bo3b2o4b2o3bo$o2bo2bo2b
4o2bo2bo2bo2bo2bo2bo$o2bo2bo2b4o2bo2bo2bo2bo2bo2bo$o3b2o3b3o3bo3b2o4b
2o3bo$o14bo14bo$o14bo14bo$31o!

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#C [[ VIEWONLY TITLE "C4_4" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 32, y = 32, rule = B3/S23
32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b3o3b2o3bo$o2bo2bo2bo2bo2b2o2b
4o2bo2bo2bo$o2bo2bo2bo2bo2b2o2b4o2bo2bo2bo$o3b2o4b2o3b2o3b3o3b2o3bo$o
14b2o5bo8bo$o14b2o5bo8bo$o2b10o2b2o5bo3b2o3bo$o2b4o8b2o5bo2bo2bo2bo$o
2b4o8b2o5bo2bo2bo2bo$o3b2o9b2o5bo3b2o3bo$o14b2o14bo$o14b2o14bo$32o$32o
$o14b2o14bo$o14b2o14bo$o3b2o3bo5b2o9b2o3bo$o2bo2bo2bo5b2o8b4o2bo$o2bo
2bo2bo5b2o8b4o2bo$o3b2o3bo5b2o2b10o2bo$o8bo5b2o14bo$o8bo5b2o14bo$o3b2o
3b3o3b2o3b2o4b2o3bo$o2bo2bo2b4o2b2o2bo2bo2bo2bo2bo$o2bo2bo2b4o2b2o2bo
2bo2bo2bo2bo$o3b2o3b3o3b2o3b2o4b2o3bo$o14b2o14bo$o14b2o14bo$32o!

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#C [[ VIEWONLY TITLE "D2_+1" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 16, y = 31, rule = B3/S23
16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b
2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o
$o14bo$o14bo$o3b2o9bo$o2b4o8bo$o2b4o8bo$o2b10o2bo$o14bo$o14bo$o3b2o4b
2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b2o3bo$o14bo$o14bo$16o!

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#C [[ VIEWONLY TITLE "D2_+2" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 16, y = 32, rule = B3/S23
16o$o14bo$o14bo$o3b2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b
2o3bo$o14bo$o14bo$o2b10o2bo$o2b4o8bo$o2b4o8bo$o3b2o9bo$o14bo$o14bo$16o
$16o$o14bo$o14bo$o3b2o9bo$o2b4o8bo$o2b4o8bo$o2b10o2bo$o14bo$o14bo$o3b
2o4b2o3bo$o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2bo$o3b2o4b2o3bo$o14bo$o14bo$
16o!

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#C [[ VIEWONLY TITLE "D2_x" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 16, y = 16, rule = B3/S23
16o$o14bo$o14bo$o7bo6bo$o6bobo5bo$o7bo3bo2bo$o11bo2bo$o3bo7bo2bo$o2bob
o6bo2bo$o3bo6bo3bo$o9bo4bo$o8bo5bo$o4b4o6bo$o14bo$o14bo$16o!

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#C [[ VIEWONLY TITLE "D4_+1" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 31, y = 31, rule = B3/S23
31o$o14bo14bo$o14bo14bo$o3b2o4b2o3bo3b2o4b2o3bo$o2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo$o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$o3b2o4b2o3bo3b2o4b2o3bo$o14b
o14bo$o14bo14bo$o2b10o2bo2b10o2bo$o2b4o8bo8b4o2bo$o2b4o8bo8b4o2bo$o3b
2o9bo9b2o3bo$o14bo14bo$o14bo14bo$31o$o14bo14bo$o14bo14bo$o3b2o9bo9b2o
3bo$o2b4o8bo8b4o2bo$o2b4o8bo8b4o2bo$o2b10o2bo2b10o2bo$o14bo14bo$o14bo
14bo$o3b2o4b2o3bo3b2o4b2o3bo$o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$o2bo2bo2b
o2bo2bo2bo2bo2bo2bo2bo$o3b2o4b2o3bo3b2o4b2o3bo$o14bo14bo$o14bo14bo$31o
!

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#C [[ VIEWONLY TITLE "D4_+2" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 32, y = 31, rule = B3/S23
32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b2o2b
o2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b2o3b
o$o14b2o14bo$o14b2o14bo$o2b10o2b2o2b10o2bo$o2b4o8b2o8b4o2bo$o2b4o8b2o
8b4o2bo$o3b2o9b2o9b2o3bo$o14b2o14bo$o14b2o14bo$32o$o14b2o14bo$o14b2o
14bo$o3b2o9b2o9b2o3bo$o2b4o8b2o8b4o2bo$o2b4o8b2o8b4o2bo$o2b10o2b2o2b
10o2bo$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b
2o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b
2o3bo$o14b2o14bo$o14b2o14bo$32o!

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#C [[ VIEWONLY TITLE "D4_+4" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 32, y = 32, rule = B3/S23
32o$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b2o2b
o2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b2o3b
o$o14b2o14bo$o14b2o14bo$o2b10o2b2o2b10o2bo$o2b4o8b2o8b4o2bo$o2b4o8b2o
8b4o2bo$o3b2o9b2o9b2o3bo$o14b2o14bo$o14b2o14bo$32o$32o$o14b2o14bo$o14b
2o14bo$o3b2o9b2o9b2o3bo$o2b4o8b2o8b4o2bo$o2b4o8b2o8b4o2bo$o2b10o2b2o2b
10o2bo$o14b2o14bo$o14b2o14bo$o3b2o4b2o3b2o3b2o4b2o3bo$o2bo2bo2bo2bo2b
2o2bo2bo2bo2bo2bo$o2bo2bo2bo2bo2b2o2bo2bo2bo2bo2bo$o3b2o4b2o3b2o3b2o4b
2o3bo$o14b2o14bo$o14b2o14bo$32o!

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#C [[ VIEWONLY TITLE "D4_x1" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 31, y = 31, rule = B3/S23
31o$o14bo14bo$o14bo14bo$o7bo6bo5b3o6bo$o6bobo5bo5bobo6bo$o7bo3bo2bo2bo
2b3o6bo$o11bo2bo2b2o10bo$o3bo7bo2bo2b2o5b3o2bo$o2bobo6bo2bo2b3o4bobo2b
o$o3bo6bo3bo3b2o4b3o2bo$o9bo4bo3b4o7bo$o8bo5bo4b5o5bo$o4b4o6bo6b4o4bo$
o14bo14bo$o14bo14bo$31o$o14bo14bo$o14bo14bo$o4b4o6bo6b4o4bo$o5b5o4bo5b
o8bo$o7b4o3bo4bo9bo$o2b3o4b2o3bo3bo6bo3bo$o2bobo4b3o2bo2bo6bobo2bo$o2b
3o5b2o2bo2bo7bo3bo$o10b2o2bo2bo11bo$o6b3o2bo2bo2bo3bo7bo$o6bobo5bo5bob
o6bo$o6b3o5bo6bo7bo$o14bo14bo$o14bo14bo$31o!

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#C [[ VIEWONLY TITLE "D4_x4" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 32, y = 32, rule = B3/S23
32o$o14b2o14bo$o14b2o14bo$o7bo6b2o5b3o6bo$o6bobo5b2o5bobo6bo$o7bo3bo2b
2o2bo2b3o6bo$o11bo2b2o2b2o10bo$o3bo7bo2b2o2b2o5b3o2bo$o2bobo6bo2b2o2b
3o4bobo2bo$o3bo6bo3b2o3b2o4b3o2bo$o9bo4b2o3b4o7bo$o8bo5b2o4b5o5bo$o4b
4o6b2o6b4o4bo$o14b2o14bo$o14b2o14bo$32o$32o$o14b2o14bo$o14b2o14bo$o4b
4o6b2o6b4o4bo$o5b5o4b2o5bo8bo$o7b4o3b2o4bo9bo$o2b3o4b2o3b2o3bo6bo3bo$o
2bobo4b3o2b2o2bo6bobo2bo$o2b3o5b2o2b2o2bo7bo3bo$o10b2o2b2o2bo11bo$o6b
3o2bo2b2o2bo3bo7bo$o6bobo5b2o5bobo6bo$o6b3o5b2o6bo7bo$o14b2o14bo$o14b
2o14bo$32o!

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#C [[ VIEWONLY TITLE "D8_1" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 31, y = 31, rule = B3/S23
31o$o14bo14bo$o14bo14bo$o7bo6bo6bo7bo$o6bobo5bo5bobo6bo$o7bo3bo2bo2bo
3bo7bo$o11bo2bo2bo11bo$o3bo7bo2bo2bo7bo3bo$o2bobo6bo2bo2bo6bobo2bo$o3b
o6bo3bo3bo6bo3bo$o9bo4bo4bo9bo$o8bo5bo5bo8bo$o4b4o6bo6b4o4bo$o14bo14bo
$o14bo14bo$31o$o14bo14bo$o14bo14bo$o4b4o6bo6b4o4bo$o8bo5bo5bo8bo$o9bo
4bo4bo9bo$o3bo6bo3bo3bo6bo3bo$o2bobo6bo2bo2bo6bobo2bo$o3bo7bo2bo2bo7bo
3bo$o11bo2bo2bo11bo$o7bo3bo2bo2bo3bo7bo$o6bobo5bo5bobo6bo$o7bo6bo6bo7b
o$o14bo14bo$o14bo14bo$31o!

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#C [[ VIEWONLY TITLE "D8_4" GRID ]]
#C [[ COLOR BACKGROUND 48 48 48 COLOR ALIVE 255 255 255 ]]
x = 32, y = 32, rule = B3/S23
32o$o14b2o14bo$o14b2o14bo$o7bo6b2o6bo7bo$o6bobo5b2o5bobo6bo$o7bo3bo2b
2o2bo3bo7bo$o11bo2b2o2bo11bo$o3bo7bo2b2o2bo7bo3bo$o2bobo6bo2b2o2bo6bob
o2bo$o3bo6bo3b2o3bo6bo3bo$o9bo4b2o4bo9bo$o8bo5b2o5bo8bo$o4b4o6b2o6b4o
4bo$o14b2o14bo$o14b2o14bo$32o$32o$o14b2o14bo$o14b2o14bo$o4b4o6b2o6b4o
4bo$o8bo5b2o5bo8bo$o9bo4b2o4bo9bo$o3bo6bo3b2o3bo6bo3bo$o2bobo6bo2b2o2b
o6bobo2bo$o3bo7bo2b2o2bo7bo3bo$o11bo2b2o2bo11bo$o7bo3bo2b2o2bo3bo7bo$o
6bobo5b2o5bobo6bo$o7bo6b2o6bo7bo$o14b2o14bo$o14b2o14bo$32o!
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muzik
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Re: Help with symmetries

Post by muzik » December 4th, 2018, 3:41 pm

Now that apgsearch has the ability to search hexagonal rules, and that corresponding symmetries for the hexagonal grid will likely eventually be supported, can examples of such symmetries be provided? It'd be helpful to have these documented on the wiki.

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Re: Help with symmetries

Post by GUYTU6J » November 2nd, 2021, 11:08 am

muzik wrote:
December 4th, 2018, 3:41 pm
Now that apgsearch has the ability to search hexagonal rules, and that corresponding symmetries for the hexagonal grid will likely eventually be supported, can examples of such symmetries be provided? It'd be helpful to have these documented on the wiki.
Now that I have written a special template designed for the demos, let's begin to consider hexagonal symmetries.
...Wait, there is a subtle problem. On the square grid we have:
C2_1: Rotation around the center of a cell. ...
C2_2: Rotation around the midpoint of a side of a cell. ...
C2_4: Rotation around a corner of a cell. ...
But on the hexagonal grid, a corner of a cell is no longer a center of C2 rotational symmetry, and so there shouldn't be an equivalent of C2_4, yet it has been supported by apgsearch. A random soup (below, left) reveals that its center of symmetry is actually on the midpoint of a side of a cell.

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x = 92, y = 32, rule = B2/S34H
16b2obo2b3o5b2o28b2o5b3o2bob2o$16b3obo5b3o34b3o5bob3o$19b3o2b2o3b3o28b
3o3b2o2b3o$16bobo2b4o3b3o30b3o3b4o2bobo$16b2ob3obo5bobo28bobo5bob3ob2o
$17bo2b2obob2o3b2o28b2o3b2obob2o2bo$21b2ob4o2bo30bo2b4ob2o$18bobo3b2ob
3obo28bob3ob2o3bobo$19bob2o2b2o2b3o28b3o2b2o2b2obo$18bob5o6bo28bo6b5ob
o$16b2obob3o4bob2o28b2obo4b3obob2o$16bo5b2o2bo2b3o28b3o2bo2b2o5bo$17b
3ob2ob3obob2o28b2obob3ob2ob3o$19bo3bo2bo4bo28bo4bo2bo3bo$16bob4o2bobo
2bobo28bobo2bobo2b4obo$16b3o5bobobo34bobobo5b3o$3bobobo5b3o60b3o5bobob
o$obo2bobo2b4obo60bob4o2bobo2bobo$o4bo2bo3bo66bo3bo2bo4bo$2obob3ob2ob
3o62b3ob2ob3obob2o$3o2bo2b2o5bo60bo5b2o2bo2b3o$2obo4b3obob2o60b2obob3o
4bob2o$o6b5obo64bob5o6bo$3o2b2o2b2obo66bob2o2b2o2b3o$ob3ob2o3bobo64bob
o3b2ob3obo$bo2b4ob2o70b2ob4o2bo$2o3b2obob2o2bo62bo2b2obob2o3b2o$obo5bo
b3ob2o60b2ob3obo5bobo$b3o3b4o2bobo60bobo2b4o3b3o$3o3b2o2b3o66b3o2b2o3b
3o$3b3o5bob3o60b3obo5b3o$2o5b3o2bob2o60b2obo2b3o5b2o!
However, we cannot call it simply by C2_2, because there are actually two of these. By flipping with respect to x-axis in Golly's square-grid mode, another soup (above, right) is obtained that has the same type of symmetry but shows different behaviour. So how to tell them apart?

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muzik
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Re: Help with symmetries

Post by muzik » November 3rd, 2021, 1:16 pm

GUYTU6J wrote:
November 2nd, 2021, 11:08 am
...Wait, there is a subtle problem. On the square grid we have:
C2_1: Rotation around the center of a cell. ...
C2_2: Rotation around the midpoint of a side of a cell. ...
C2_4: Rotation around a corner of a cell. ...
But on the hexagonal grid, a corner of a cell is no longer a center of C2 rotational symmetry, and so there shouldn't be an equivalent of C2_4, yet it has been supported by apgsearch.
...
However, we cannot call it simply by C2_2, because there are actually two of these. By flipping with respect to x-axis in Golly's square-grid mode, another soup (above, right) is obtained that has the same type of symmetry but shows different behaviour. So how to tell them apart?
Interesting. It seems we may have a new potential entry for apgsearch's currently "missing hexagonal symmetries", assuming we haven't missed something and it is, in fact, already included (nontrivially as opposed to as part of a higher symmetry):
calcyman wrote:
December 20th, 2018, 10:09 am
As of version 4.71, lifelib and apgsearch support the following hexagonal symmetries:

["1x256", "2x128", "4x64", "8x32", "C1", "C2_4", "C2_1", "C3_1", "C6", "D2_x", "D2_xo", "D4_x4", "D4_x1", "D6_1", "D6_1o", "D12"]

The missing hexagonal symmetries are C3_3 and D6_3.

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Re: Help with symmetries

Post by muzik » December 10th, 2021, 1:04 pm

On a closely related note, how many different mod/time symmetry types are known on a square grid?

Here's a list of known oscillator cases I've compiled: [SUPERSEDED - see my next post in this thread]

[1/1] Time-asymmetric
Affects any spatial symmetry

Code: Select all

x = 11, y = 8, rule = B3/S23
5b3o$3bo$3bo3bo$3bobo$5bo2b2o$7bo!
[1/2] Flips horizontally
Affects any spatial symmetry excluding D4_+n and D8_n (needs confirmation)

Code: Select all

x = 9, y = 9, rule = B3/S23
$3bo$3bobo$bo4bo$b6o2$3b2o$3b2o!
[1/2] {Rotates 180 degrees} xor {flips horizontally and vertically}
Affects any spatial symmetry excluding C2_n, C4_n, D4_xn and D8_n (needs confirmation)

Code: Select all

x = 12, y = 8, rule = B3/S23
$7b2o$2b3obobo2$3bobo2bo$2bo4bo$2b2o3bo!
[1/2] Flips diagonally
Affected symmetries unknown, known to affect C1 and D2_x (needs confirmation)

Code: Select all

x = 11, y = 10, rule = B3/S23
$b2o$bo3b2o$2bobobo2$2b2o2bo$4bo2bo$6b2o!
[1/2] Flips horizontally xor vertically xor diagonally
Can only affect C2_n or C4_n (needs confirmation - especially unsure about C2_2 as I don't know if that would count as diagonally)

Code: Select all

x = 8, y = 8, rule = B3/S23
3bo$3bobo$bo$6b2o$2o$6bo$2bobo$4bo!
[1/2] {Flips horizontally xor vertically} xor {rotates 90 degrees CW xor CCW}
May just be the C2_n version of the case three entries above this one - more info needed

Code: Select all

x = 7, y = 7, rule = B3/S23
b2ob2o$ob2o2bo$2o4bo$bo3bo$o4b2o$o2b2obo$b2ob2o!
[1/2] Flips diagonally xor {rotates 90 degrees CW xor CCW}
Affected symmetries unknown - more info needed

Code: Select all

x = 1, y = 3, rule = B3/S23
o$o$o!
[1/4] Rotates 90 degrees CW or CCW

Code: Select all

x = 12, y = 12, rule = B3/S23
6b2o$6b2o2$4b4o$2obobo2bo$2obo2bobo$3b2o3bob2o$3bo4bob2o$4b4o2$4b2o$4b
2o!
Last edited by muzik on December 13th, 2021, 7:37 pm, edited 6 times in total.

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Re: Help with symmetries

Post by dvgrn » December 10th, 2021, 1:55 pm

muzik wrote:
December 10th, 2021, 1:04 pm
[1/2] Flips horizontally xor vertically
Can only affect C2_n or C4_n where n is 1 or 4 (needs confirmation)

Code: Select all

x = 8, y = 8, rule = B3/S23
3bo$3bobo$bo$6b2o$2o$6bo$2bobo$4bo!
That example oscillator could also be considered to flip diagonally. I'm not sure if that's always true for this category, but it's true for the other examples that I can think of.

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Re: Help with symmetries

Post by muzik » December 13th, 2021, 7:33 pm

So I was raking through known oscillators to look for examples for each known symmetry case as well as to find any I hadn't accounted for, and while looking through DRH-oscillators, I ended up finding out that Dean Hickerson ended up beating me to the punch before I knew how to kick:
Dean Hickerson wrote:
September 16th, 2000, 7:30 am
The 'period' of an oscillator (or spaceship) is the smallest positive integer P for which generation P of the object is congruent to and in the same orientation as generation 0. The 'mod' of an oscillator (or spaceship) is the smallest positive integer M for which generation M of the object is congruent to generation 0, but not necessarily in the same orientation. The quotient q=P/M is always either 1, 2, or 4. To specify both P and M, we often write "period P.M" or "period P/q".

There are 43 types of symmetry that an oscillator can have, taking into account both the symmetry of a single generation and the change of orientation (if any) M generations later. There are 16 types of symmetry that a pattern can have in a single generation. Each of these is given a one or two character name, as follows:
  • n no symmetry

    -c mirror symmetry across a horizontal axis through cell centers
    -e mirror symmetry across a horizontal axis through cell edges

    / mirror symmetry across one diagonal

    .c 180 degree rotational symmetry about a cell center
    .e 180 degree rotational symmetry about a cell edge
    .k 180 degree rotational symmetry about a cell corner

    +c mirror symmetry across horizontal and vertical axes meeting
    at a cell center
    +e mirror symmetry across horizontal and vertical axes meeting
    at a cell edge
    +k mirror symmetry across horizontal and vertical axes meeting
    at a cell corner

    xc mirror symmetry across 2 diagonals meeting at a cell center
    xk mirror symmetry across 2 diagonals meeting at a cell corner

    rc 90 degree rotational symmetry about a cell center
    rk 90 degree rotational symmetry about a cell corner

    *c 8-fold symmetry about a cell center
    *k 8-fold symmetry about a cell corner
For a period P/1 object, specifying the symmetry of generation 0 tells us all there is to know about the oscillator's symmetry. For a period P/2 or P/4 object, we also need to know how gen M is related to gen 0. For the P/2 case, gen M can be either a mirror image of gen 0, a 180 degree rotation of it, or a 90 degree rotation of it if the pattern has 180 degree rotational symmetry. For the P/4 case gen M must be a 90 degree rotation of gen 0. In any case, if we merge all gens which are multiples of M, the resulting pattern will have more symmetry than the original oscillator. We describe the complete symmetry class of the oscillator by appending the one or two character description of the union's symmetry to that of gen 0's symmetry. For example, if gen 0 has 180 degree rotational symmetry about a cell center, and gen M is obtained by reflecting gen 0 across a diagonal, then the union of gens 0 and M is symmetric across both diagonals, so its symmetry class is denoted ".cxc".

The 43 possible symmetry types are:
  • period/mod = 1: nn -c-c -e-e // .c.c .e.e .k.k +c+c
    +e+e +k+k xcxc xkxk rcrc rkrk *c*c *k*k

    period/mod = 2: n-c n-e n/ n.c n.e n.k
    -c+c -c+e -e+e -e+k
    /xc /xk
    .c+c .cxc .crc .e+e .k+k .kxk .krk
    +c*c +k*k xc*c xk*k rc*c rk*k

    period/mod = 4: nrc nrk
Conveniently, this means that we now know all 43 possible cases for space and time symmetry for stationary periodic patterns on square grids, as well as having examples for each of them. So since we know each case and how they behave, maybe it'd be a good idea to document all of these on the Symmetry page on the wiki to describe this notation as well as describe each symmetry type and give clear examples...

...except that the Symmetry page is already considerably large in its current state and it seems unwise to keep on adding more to it. Not to mention that since this notation also covers period-1 patterns, it technically would also duplicate all of the existing non-gutter non-D8_2 non-hexagonal examples on there anyway.

So I think it might be a good idea to split up the Symmetry page into at least two subpages: one for "general symmetries", which deals with symmetries as they appear in soup searching, spaceship searching, and the analysis of patterns and rules in general, and another for "pattern symmetries", which concerns specifically periodic patterns such as the oscillator cases discussed here, as well as spaceships and other things such as puffers.

The "general" and "pattern" naming scheme I'm using here might not be clear and there's probably a better way to make such a distinction. (I'm not exactly in the right mood to be making good name ideas - if it's any indication of my current state of mind, earlier I had to stop my hand from trying to turn off my desktop monitor by pulling said monitor down as if it were a laptop lid.) "General symmetries" can be used to describe what can be proven to be true of any pattern in a given rule, whereas for periodic patterns such behaviour may only arise coincidentally, even if said pattern is in a rule which would preserve that attribute anyway. (Example: a bi-block in HighLife technically has gutter symmetry, but this is coincidental as B6 disallows permanent gutter formation, therefore gutters are a "general symmetry" due to being rule-specific and existing permanently for all patterns containing containing it in rules that support it, but not a "pattern symmetry" since despite periodic patterns that exhibit gutters being able to exist in many rules, not all of said rules preserve gutters. Is there a less convoluted way to say this?)

If anyone has anything else to add on this subject, please do - splitting this page probably isn't a trivial feat and likely won't be without controversy. In addition, preparation of visual examples for the 43 "pattern symmetry" cases, as well as investigation into what exists for spaceships as well as stuff for hexagonal and triangular grids, is more than welcome. I may post known symmetry cases below later.

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Re: Help with symmetries

Post by muzik » December 13th, 2021, 10:24 pm

Follow-up to the above post: would I be correct in saying that these are all the symmetry types for spaceships which a square grid can possess? Not sure if I'm abusing notation here, as the union of a spaceship before and after its mod wouldn't necessarily be of a higher symmetry for these cases (although this may be a possibility if alignment to the grid is not required).

n

Code: Select all

x = 9, y = 9, rule = LifeHistory
.A$A.A$A2.A$.2A2$A5.A$.A3.A.A$2A3.A2.A$A3.2A2.A!
[[ AUTOSTART ZOOM 16 GPS 20 TRACKLOOP 7 0.0 -1/7 HISTORYSTATES 0 AGESTATES 0 GRID ]]
-c

Code: Select all

x = 15, y = 82, rule = LifeHistory
7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$
7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$
7.C$6.ADA$5.A.D.A$6.ACA$7.D$4.2A.D.2A$2.A3.ADA3.A$.2A3.ADA3.2A$A5.ADA
5.A$.A.2A.ADA.2A.A$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.
D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D
$7.D$7.D$7.D$7.D$7.D$7.D$7.D$7.D!
[[ AUTOSTART ZOOM 16 GPS 20 TRACKLOOP 3 0.0 -1/3 HISTORYSTATES 0 AGESTATES 0 GRID ]]
-e

Code: Select all

x = 8, y = 105, rule = LifeHistory
6$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.
2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$
3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2C$3.2C$2.A
2DA$2.A2CA$.A.2C.A$2.A2DA$3.2D$.2A2D2A$A2.2D2.A$3.2D$.A.2D.A$2.A2DA$
3.2C$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D
$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.
2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$3.2D$
3.2D$3.2D$3.2D!
[[ AUTOSTART ZOOM 16 GPS 20 TRACKLOOP 10 0.0 -1/10 HISTORYSTATES 0 AGESTATES 0 GRID ]]
/

Code: Select all

x = 128, y = 128, rule = LifeHistory
127.D$126.D$125.D$124.D$123.D$122.D$121.D$120.D$119.D$118.D$117.D$
116.D$115.D$114.D$113.D$112.D$111.D$110.D$109.D$108.D$107.D$106.D$
105.D$104.D$103.D$102.D$101.D$100.D$99.D$98.D$97.D$96.D$95.D$94.D$93.
D$92.D$91.D$90.D$89.D$88.D$87.D$86.D$85.D$84.D$83.D$82.D$81.D$80.D$
79.D$78.D$77.D$76.D$55.4A16.D$53.2A4.2A13.D$53.2A5.A12.D$55.2A.A.A11.
D$60.A10.D$56.A3.A9.D$56.A4.A7.D$58.3A3.A3.D$58.2A4.A2.D$64.2AD$65.DA
$64.D.3A$63.D$62.D$61.D7.A$60.D7.A.5A$59.D7.2A5.A$58.D8.2A3.A2.A$57.D
17.A$56.D12.2A.A2.A$55.D16.A2.A$54.D18.2A$53.D19.2A$52.D$51.D$50.D$
49.D$48.D$47.D$46.D$45.D$44.D$43.D$42.D$41.D$40.D$39.D$38.D$37.D$36.D
$35.D$34.D$33.D$32.D$31.D$30.D$29.D$28.D$27.D$26.D$25.D$24.D$23.D$22.
D$21.D$20.D$19.D$18.D$17.D$16.D$15.D$14.D$13.D$12.D$11.D$10.D$9.D$8.D
$7.D$6.D$5.D$4.D$3.D$2.D$.D$D!
[[ AUTOSTART ZOOM 16 GPS 5 TRACKLOOP 5 -1/5 1/5 HISTORYSTATES 0 AGESTATES 0 GRID ]]
n-c

Code: Select all

x = 4, y = 52, rule = LifeHistory
2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$
2.D$2.D$2.D$2.D$2.D$2.D$.ACA$A.DA$2.DA$2.DA$A.C$2.D$2.D$2.D$2.D$2.D$
2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$2.D$
2.D$2.D!
[[ AUTOSTART ZOOM 32 GPS 5 TRACKLOOP 2 0.0 -1/2 HISTORYSTATES 0 AGESTATES 0 GRID ]]
n-e

Code: Select all

x = 34, y = 107, rule = LifeHistory
16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.
2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$
16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.
2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$
16.2D$16.2D$16.2D$16.2D$16.2D$4.2A10.2C10.2A$4.2A10.2C10.2A$3.A2.A8.A
2DA8.A2.A$3.4A8.A2CA8.4A$2.A.2A.A6.A.2C.A6.A.2A.A$3.A2.A8.A2DA8.A2.A$
16.2D$2.2A2.2A6.2A2D2A6.2A2.2A$.A6.A4.A2.2D2.A4.A6.A$16.2D$2.A4.A6.A.
2D.A6.A4.A$3.A2.A8.A2DA8.A2.A$4.2A10.2C10.2A$16.2D$.2A4.2A4.2A.2D.2A
4.2A4.2A$A2.A2.A2.A2.A2.A2DA2.A2.A2.A2.A2.A$.2A4.2A4.2A.2D.2A4.2A4.2A
$2.A4.A6.A.2D.A6.A4.A$16.2D$14.A.2D$16.2D$16.DC2A$16.DC$16.2D$16.2D$
16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.
2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$
16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D$16.2D!
[[ AUTOSTART ZOOM 8 TRACKLOOP 20 0.0 -1/10 HISTORYSTATES 0 AGESTATES 0 GRID ]]
n/

Code: Select all

x = 43, y = 44, rule = LifeHistory
42.D$41.2D$40.2D$39.2D$38.2D$37.2D$36.2D$35.2D$34.2D$33.2D$32.2D$31.
2D$30.2D$29.2D$28.2D$27.2D$26.2D$25.2D$24.2D$22.A2D$22.CDA$21.DCA$20.
2D$19.2D$18.2D$17.2D$16.2D$15.2D$14.2D$13.2D$12.2D$11.2D$10.2D$9.2D$
8.2D$7.2D$6.2D$5.2D$4.2D$3.2D$2.2D$.2D$2D$D!
[[ AUTOSTART ZOOM 32 GPS 5 TRACKLOOP 4 -1/4 1/4 HISTORYSTATES 0 AGESTATES 0 GRID ]]
Last edited by muzik on December 13th, 2021, 10:43 pm, edited 3 times in total.

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Re: Help with symmetries

Post by GUYTU6J » December 13th, 2021, 10:27 pm

muzik wrote:
December 13th, 2021, 7:33 pm
...
The "general" and "pattern" naming scheme I'm using here might not be clear and there's probably a better way to make such a distinction. ...
The terminology could be "static symmetry" and "kinetic symmetry", distinguished by whether evolution through more than one generation is relevant.

Previous discussion on time symmetry (or temporal symmetry in some other posts) can be found here:
viewtopic.php?f=7&t=1777

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Re: Help with symmetries

Post by muzik » December 16th, 2021, 7:08 pm

Is the list of square grid gutter symmetries on the wiki's Symmetry page complete? Only 1D gutters count (0D gutters, such as a single-cell gutter implied by the lack of B0, B4c, B4e and B8, are considered trivial and ignored, as are gutters which cannot be preserved by non-trivial rules inside the range-1 Moore INT rulespace).

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Re: Help with symmetries

Post by muzik » December 20th, 2021, 11:11 pm

GUYTU6J wrote:
December 13th, 2021, 10:27 pm
muzik wrote:
December 13th, 2021, 7:33 pm
...
The "general" and "pattern" naming scheme I'm using here might not be clear and there's probably a better way to make such a distinction. ...
The terminology could be "static symmetry" and "kinetic symmetry", distinguished by whether evolution through more than one generation is relevant.

Previous discussion on time symmetry (or temporal symmetry in some other posts) can be found here:
viewtopic.php?f=7&t=1777
I've now split the page into three: Kinetic symmetry for periodic object space/time symmetries, Static symmetry for the usual space symmetry, and Minor static symmetries for stuff like gutters.

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Re: Help with symmetries

Post by erictom333 » December 20th, 2021, 11:44 pm

muzik wrote:
December 13th, 2021, 10:24 pm
n/

Code: Select all

x = 43, y = 44, rule = LifeHistory
42.D$41.2D$40.2D$39.2D$38.2D$37.2D$36.2D$35.2D$34.2D$33.2D$32.2D$31.
2D$30.2D$29.2D$28.2D$27.2D$26.2D$25.2D$24.2D$22.A2D$22.CDA$21.DCA$20.
2D$19.2D$18.2D$17.2D$16.2D$15.2D$14.2D$13.2D$12.2D$11.2D$10.2D$9.2D$
8.2D$7.2D$6.2D$5.2D$4.2D$3.2D$2.2D$.2D$2D$D!
[[ AUTOSTART ZOOM 32 GPS 5 TRACKLOOP 4 -1/4 1/4 HISTORYSTATES 0 AGESTATES 0 GRID ]]
That'd be n/e: even diagonal symmetry is possible with glide-reflecting spaceships.

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muzik
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Re: Help with symmetries

Post by muzik » March 2nd, 2023, 7:56 am

erictom333 wrote:
December 20th, 2021, 11:44 pm
That'd be n/e: even diagonal symmetry is possible with glide-reflecting spaceships.
This is actually a valid point. Two possible types of diagonal reflections possible for diagonal spaceships: one across a line that passes through cell centered and vertices (example below, we'll call this just n/) and one that goes through edges (we'll call this one n/e).

Code: Select all

x = 76, y = 76, rule = LifeHistory
D$.D$2.D$3.D$4.D$5.D$6.D$7.D$8.D$9.D$10.D$11.D$12.D$13.D$14.D$15.D$16.
D$17.D$18.D$19.D$20.D$21.D$22.D$23.D$24.D$25.D$26.D$27.D$28.D$29.D$30.
D$31.D$32.D$33.D$34.D$35.D$36.D$37.D$38.D$39.D4.2A.A7.3A$40.D3.A3.A6.
A$41.D2.A4.A7.A7.2A$42.D3.2A7.A.A4.2A.A$43.D4.A.A4.3A.A2.2A$39.3A2.D9.
A3.2A2.2A$39.A5.D4.2A.4A.2A$42.A3.D4.7A$39.A2.A4.D$40.A2.A4.D$41.A7.D
$43.A.A4.D$45.2A4.D$46.A5.D3A$45.2A6.C2.A$44.3A5.A.D.A$39.2A.2A.2A7.A
D$39.A3.A.2A9.CA$39.A.3A2.A5.A4.D.A$44.2A8.3A.D.A$43.3A13.DA$58.A.D$59.
A.D$42.3A17.D$42.3A18.D$64.D$41.2A22.D$41.A24.D$67.D$68.D$69.D$70.D$71.
D$72.D$73.D$74.D$75.D!
[[ ZOOM 12 LOOP 60 X 8 Y 8 ]]
I assume this is a spaceship-exclusive kinetic symmetry? I don't know of any oscillators that follow it (and I don't think skewed diagonal gutters would count).

erictom333
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Re: Help with symmetries

Post by erictom333 » March 2nd, 2023, 5:45 pm

muzik wrote:
March 2nd, 2023, 7:56 am
erictom333 wrote:
December 20th, 2021, 11:44 pm
That'd be n/e: even diagonal symmetry is possible with glide-reflecting spaceships.
This is actually a valid point. Two possible types of diagonal reflections possible for diagonal spaceships: one across a line that passes through cell centered and vertices (example below, we'll call this just n/) and one that goes through edges (we'll call this one n/e).

Code: Select all

x = 76, y = 76, rule = LifeHistory
D$.D$2.D$3.D$4.D$5.D$6.D$7.D$8.D$9.D$10.D$11.D$12.D$13.D$14.D$15.D$16.
D$17.D$18.D$19.D$20.D$21.D$22.D$23.D$24.D$25.D$26.D$27.D$28.D$29.D$30.
D$31.D$32.D$33.D$34.D$35.D$36.D$37.D$38.D$39.D4.2A.A7.3A$40.D3.A3.A6.
A$41.D2.A4.A7.A7.2A$42.D3.2A7.A.A4.2A.A$43.D4.A.A4.3A.A2.2A$39.3A2.D9.
A3.2A2.2A$39.A5.D4.2A.4A.2A$42.A3.D4.7A$39.A2.A4.D$40.A2.A4.D$41.A7.D
$43.A.A4.D$45.2A4.D$46.A5.D3A$45.2A6.C2.A$44.3A5.A.D.A$39.2A.2A.2A7.A
D$39.A3.A.2A9.CA$39.A.3A2.A5.A4.D.A$44.2A8.3A.D.A$43.3A13.DA$58.A.D$59.
A.D$42.3A17.D$42.3A18.D$64.D$41.2A22.D$41.A24.D$67.D$68.D$69.D$70.D$71.
D$72.D$73.D$74.D$75.D!
[[ ZOOM 12 LOOP 60 X 8 Y 8 ]]
I assume this is a spaceship-exclusive kinetic symmetry? I don't know of any oscillators that follow it (and I don't think skewed diagonal gutters would count).
Yes, even diagonal symmetry is not possible with non-moving objects, as the even-diagonal reflection of a pattern would not be aligned with the grid; even-diagonal reflection is only possible as a glide reflection.

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muzik
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Re: Help with symmetries

Post by muzik » March 5th, 2023, 6:19 pm

erictom333 wrote:
March 2nd, 2023, 5:45 pm
Yes, even diagonal symmetry is not possible with non-moving objects, as the even-diagonal reflection of a pattern would not be aligned with the grid; even-diagonal reflection is only possible as a glide reflection.
The other thing I want to know: is it possible for two spaceships to have the same diagonal speed and period, but one has n/ symmetry and the other has n/e symmetry? From what I've seen so far, glide-reflective c/2, c/4, c/6, c/8, ... spaceships all appear to be n/e, whereas 2c/4, 2c/8, ... diagonal spaceships appear to follow n/.

Can we be sure that n/e isn't just n/ in disguise, a symmetry that only arises for some speeds and not others (unlike n-f and n-e, which can apply to a spaceship regardless of its speed, as long as the period and displacement are right)?

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Re: Help with symmetries

Post by toroidalet » March 6th, 2023, 4:22 am

muzik wrote:
March 5th, 2023, 6:19 pm
The other thing I want to know: is it possible for two spaceships to have the same diagonal speed and period, but one has n/ symmetry and the other has n/e symmetry? From what I've seen so far, glide-reflective c/2, c/4, c/6, c/8, ... spaceships all appear to be n/e, whereas 2c/4, 2c/8, ... diagonal spaceships appear to follow n/.

Can we be sure that n/e isn't just n/ in disguise, a symmetry that only arises for some speeds and not others (unlike n-f and n-e, which can apply to a spaceship regardless of its speed, as long as the period and displacement are right)?
This is a good point. Actually, it turns out that there is only one glide symmetry for each diagonal speed.
Consider a spaceship with speed (n,n)c/2m. Consider the spaceship after m generations (half a period). If we fix an arbitrary cell, it will be advanced by some vector (a,b) with a+b=n {a}. Now, if we move the marked cell one cell down or to the left, it is now advanced by (a-1,b+1). Similarly, moving it one cell up or to the right changes it to (a+1,b-1).
Next, consider the difference a-b. Moving the marked cell will change the difference to (a-1)-(b+1)=(a-b)-2 or (a+1)-(b-1)=(a-b)+2. If n is even, then a-b will also be even, and so by moving the marked cell, you can find a spot where the difference is 0. This means the marked cell moves by (n/2,n/2) and is then reflected, leaving it in the same spot, so the axis of reflection will pass through that cell.
On the other hand, if n is odd, then there will be a diagonal line of cells where a-b=1 and a line next to it where a-b=-1. The axis of reflection cannot be on one side of both lines, so it must be between them.
Here are some sample even and odd-displacement ships:

Code: Select all

x = 62, y = 35, rule = LifeHistory
51.A$50.CEA$50.A3$19.CA$12.D5.AE21.D12.CB$13.D5.A22.D10.BE$.A12.D10.B
17.D10.B$CEA4.9D8.CEB6.12D15.B$A13.D9.B18.D15.CEB$13.D28.D16.B$12.D
28.D9$50.A$49.CEA$49.A3$19.CA$18.AE34.CB$12.D6.A21.D11.BE$13.D28.D11.
B$.A12.D10.B17.D$CEA4.9D8.CEB6.12D15.B$A13.D9.B18.D15.CEB$13.D28.D16.
B$12.D28.D!
As you can see, moving the marked cell one space over changes the difference by 2, and that when the displacement is even, there is a spot where the difference is 0, while an odd displacement has two neighboring lines where the difference is ±1.

{a}: In the next m generations, it will be advanced by (b,a) for a total displacement of (a+b,b+a), which we know is (n,n).
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Re: Help with symmetries

Post by Haycat2009 » August 29th, 2023, 4:18 am

Since D8_2 symmetry exists ( Which is unusual in the sense that it is not symmetical per se ), What birth and survival conditions sets are required to maintain C4_2 and D4_x2 symmetry?
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