Symmetries of wicks

[muzik]

The symmetries used for finite patterns (both spatial and temporal) have been classified for a while now, but to my knowledge at least there hasn't been an effort to classify the possible symmetries of patterns which are periodic in both space and time.

I'll be using pieces of DRH's notation to classify things here:

• n = asymmetry

• .c = 180-degree rotationally symmetric; center of rotation is a cell

• .e = 180-degree rotationally symmetric; center of rotation is an edge

• .k = 180-degree rotationally symmetric; center of rotation is a vertex

• -c = orthogonally mirror symmetric (1 line); line passes through cell centers

• -e = orthogonally mirror symmetric (1 line); line passes through cell vertices

• / = diagonally mirror symmetric (1 line)

• +c = orthogonally mirror symmetric (2 lines); lines intersect at a cell center

• +e = orthogonally mirror symmetric (2 lines); lines intersect at an edge

• xc = diagonally mirror symmetric (2 lines); lines intersect at a cell center

• xk = diagonally mirror symmetric (2 lines); lines intersect at a vertex

Due to how periodic unit cells can be attached together, there are often two points of symmetry, provided the original unit cell is symmetric. One corresponds to the center of the unit cell, whereas the other corresponds to where the edges connect. Depending on how the unit cell is defined, both are interchangeable. For example, this wick has both .k symmetry (red squares) and .e symmetry (green squares):

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x = 14, y = 18, rule = B3/S23:T18-6,18
13bo$10b4o$10bo$11bo$8b4o$8bo$9bo$6b4o$6bo$7bo$4b4o$4bo$5bo$2b4o$2bo$
3bo$4o$o!
[[
GRID GRIDMAJOR 0 ZOOM 16 THEME Mono
COLOR POLY #E86075 POLYSIZE 1 POLYLINE 6 -9.5 6 -9 5 -9 5 -9.5 8
COLOR POLY #E86075 POLYSIZE 1 POLYLINE 3 -7 4 -7 4 -6 3 -6 3 -7 8
COLOR POLY #E86075 POLYSIZE 1 POLYLINE 1 -4 2 -4 2 -3 1 -3 1 -4 8
COLOR POLY #E86075 POLYSIZE 1 POLYLINE -1 -1 0 -1 0 0 -1 0 -1 -1 8
COLOR POLY #E86075 POLYSIZE 1 POLYLINE -3 2 -2 2 -2 3 -3 3 -3 2 8
COLOR POLY #E86075 POLYSIZE 1 POLYLINE -5 5 -4 5 -4 6 -5 6 -5 5 8
COLOR POLY #E86075 POLYSIZE 1 POLYLINE -7 8.5 -7 8 -6 8 -6 8.5 8
COLOR POLY #0AB87B POLYSIZE 1 POLYLINE 4 -8.5 5 -8.5 5 -7.5 4 -7.5 4 -8.5 8
COLOR POLY #0AB87B POLYSIZE 1 POLYLINE 2 -5.5 3 -5.5 3 -4.5 2 -4.5 2 -5.5 8
COLOR POLY #0AB87B POLYSIZE 1 POLYLINE 0 -2.5 1 -2.5 1 -1.5 0 -1.5 0 -2.5 8
COLOR POLY #0AB87B POLYSIZE 1 POLYLINE -2 0.5 -1 0.5 -1 1.5 -2 1.5 -2 0.5 8
COLOR POLY #0AB87B POLYSIZE 1 POLYLINE -4 3.5 -3 3.5 -3 4.5 -4 4.5 -4 3.5 8
COLOR POLY #0AB87B POLYSIZE 1 POLYLINE -6 6.5 -5 6.5 -5 7.5 -6 7.5 -6 6.5 8
]]

Here's a set of possible symmetries for p1 wicks:

n

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x = 42, y = 39, rule = B3/S23:T42,42-7
40b2o$40bo3$40b2o$34b2o5bo$34bobob3o$36bobo$36b2o$34b2o$28b2o5bo$28bob
ob3o$30bobo$30b2o$28b2o$22b2o5bo$22bobob3o$24bobo$24b2o$22b2o$16b2o5bo
$16bobob3o$18bobo$18b2o$16b2o$10b2o5bo$10bobob3o$12bobo$12b2o$10b2o$4b
2o5bo$4bobob3o$6bobo$6b2o$4b2o$5bo$ob3o$obo$2o!

.c and .c

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x = 17, y = 17, rule = B3/S23:T20+12,18
10b7o$10bo5bo$11b3o$8bo5bo$8b7o2$6b7o$6bo5bo$7b3o$4bo5bo$4b7o2$2b7o$2b
o5bo$3b3o$o5bo$7o!

.c and .e

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x = 8, y = 20, rule = B3/S23:T20-5,20
bo$o$4o$3bo$2bo$bo$b4o$4bo$3bo$2bo$2b4o$5bo$4bo$3bo$3b4o$6bo$5bo$4bo$
4b4o$7bo!

.c and .k

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x = 14, y = 28, rule = B3/S23:T20+12,28
12bo$12b2o$13bo$10b3o$9bo$9b2o$10bo$9bo$9b2o$10bo$7b3o$6bo$6b2o$7bo$6b
o$6b2o$7bo$4b3o$3bo$3b2o$4bo$3bo$3b2o$4bo$b3o$o$2o$bo!

.e and .e (both lines we rotate around are parallel)

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x = 40, y = 34, rule = B3/S23:T40,40+20
b2o$obo$o4b2o$o3bobo$4bo4b2o$o2b2o3bobo$bo6bo4b2o$bob2o2b2o3bobo$bobob
o6bo4b2o$bobobob2o2b2o3bobo$2b2obobobo6bo4b2o$5bobobob2o2b2o3bobo$2b2o
2b2obobobo6bo4b2o$2bo6bobobob2o2b2o3bobo$obo3b2o2b2obobobo6bo4b2o$2o4b
o6bobobob2o2b2o3bobo$4bobo3b2o2b2obobobo6bo4b2o$4b2o4bo6bobobob2o2b2o
3bobo$8bobo3b2o2b2obobobo6bo4b2o$8b2o4bo6bobobob2o2b2o3bobo$12bobo3b2o
2b2obobobo6bo$12b2o4bo6bobobob2o2b2o$16bobo3b2o2b2obobobo$16b2o4bo6bob
obob2o2bo$20bobo3b2o2b2obobobo$20b2o4bo6bobobobo$24bobo3b2o2b2obobo$
24b2o4bo6bobo$28bobo3b2o2b2o$28b2o4bo$32bobo3b2o$32b2o4bo$36bobo$36b2o
!

.e and .e (both lines we rotate around are perpendicular - I do not know if this should be counted as a different symmetry to the above)

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x = 10, y = 27, rule = B3/S23:T20+9,27
8bo$8b2o$9bo$7bo$7b2o$8bo$6bo$6b2o$7bo$5bo$5b2o$6bo$4bo$4b2o$5bo$3bo$
3b2o$4bo$2bo$2b2o$3bo$bo$b2o$2bo$o$2o$bo!

.e and .k

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x = 14, y = 18, rule = B3/S23:T18-6,18
13bo$10b4o$10bo$11bo$8b4o$8bo$9bo$6b4o$6bo$7bo$4b4o$4bo$5bo$2b4o$2bo$
3bo$4o$o!

.k and .k

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x = 12, y = 20, rule = B3/S23:T20+10,20
bo$2o$2b2o$2bo$3bo$2b2o$4b2o$4bo$5bo$4b2o$6b2o$6bo$7bo$6b2o$8b2o$8bo$
9bo$8b2o$10b2o$10bo!

-c and -c

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x = 32, y = 11, rule = B3/S23:T32,20
3bo7bo7bo7bo$2bobo5bobo5bobo5bobo$3bo7bo7bo7bo2$b5o3b5o3b5o3b5o$o2bo2b
obo2bo2bobo2bo2bobo2bo2bo$2o3b2ob2o3b2ob2o3b2ob2o3b2o$2b3o5b3o5b3o5b3o
$2o3b5o3b5o3b5o3b3o$2bobo2bo2bobo2bo2bobo2bo2bobo2bo$b2ob2o3b2ob2o3b2o
b2o3b2ob2o!

-c and -e

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x = 35, y = 7, rule = B3/S23:T35,20
o3b2o3b2o3b2o3b2o3b2o3b2o3bo$bobo2bobo2bobo2bobo2bobo2bobo2bobo$2ob4ob
4ob4ob4ob4ob4ob2o2$2ob4ob4ob4ob4ob4ob4ob2o$bobo2bobo2bobo2bobo2bobo2bo
bo2bobo$2bo4bo4bo4bo4bo4bo4bo!

-e and -e

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x = 36, y = 7, rule = B3/S23:T36,20
2b2o4b2o4b2o4b2o4b2o4b2o$bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$b4o2b4o2b
4o2b4o2b4o2b4o2$b4o2b4o2b4o2b4o2b4o2b4o$bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo$o4b2o4b2o4b2o4b2o4b2o4bo!

/ and /

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x = 31, y = 31, rule = B3/S23:T31,31
29bo$28bobo$27bo2bo$26bob2o$25bobo$24bo2bo$23bob2o$22bobo$21bo2bo$20bo
b2o$19bobo$18bo2bo$17bob2o$16bobo$15bo2bo$14bob2o$13bobo$12bo2bo$11bob
2o$10bobo$9bo2bo$8bob2o$7bobo$6bo2bo$5bob2o$4bobo$3bo2bo$2bob2o$bobo$o
2bo$b2o!

+c and +c

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x = 7, y = 36, rule = B3/S23:T35,36
2bobo$obobobo$2obob2o$3bo$2obob2o$obobobo$2bobo$obobobo$2obob2o$3bo$2o
bob2o$obobobo$2bobo$obobobo$2obob2o$3bo$2obob2o$obobobo$2bobo$obobobo$
2obob2o$3bo$2obob2o$obobobo$2bobo$obobobo$2obob2o$3bo$2obob2o$obobobo$
2bobo$obobobo$2obob2o$3bo$2obob2o$obobobo!

+c and +e

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x = 7, y = 35, rule = B3/S23:T35,35
2obob2o$3bo$3bo$2obob2o$obobobo$2bobo$obobobo$2obob2o$3bo$3bo$2obob2o$
obobobo$2bobo$obobobo$2obob2o$3bo$3bo$2obob2o$obobobo$2bobo$obobobo$2o
bob2o$3bo$3bo$2obob2o$obobobo$2bobo$obobobo$2obob2o$3bo$3bo$2obob2o$ob
obobo$2bobo$obobobo!

+e and +e

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x = 7, y = 36, rule = B3/S23:T35,36
2bobo$b2ob2o$o5bo$o5bo$b2ob2o$2bobo$2bobo$b2ob2o$o5bo$o5bo$b2ob2o$2bob
o$2bobo$b2ob2o$o5bo$o5bo$b2ob2o$2bobo$2bobo$b2ob2o$o5bo$o5bo$b2ob2o$2b
obo$2bobo$b2ob2o$o5bo$o5bo$b2ob2o$2bobo$2bobo$b2ob2o$o5bo$o5bo$b2ob2o$
2bobo!

+e and +k

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x = 8, y = 35, rule = B3/S23:T36,35
bob2obo$2o4b2o$2b4o$2o4b2o$bob2obo$bob2obo$2o4b2o$2b4o$2o4b2o$bob2obo$
bob2obo$2o4b2o$2b4o$2o4b2o$bob2obo$bob2obo$2o4b2o$2b4o$2o4b2o$bob2obo$
bob2obo$2o4b2o$2b4o$2o4b2o$bob2obo$bob2obo$2o4b2o$2b4o$2o4b2o$bob2obo$
bob2obo$2o4b2o$2b4o$2o4b2o$bob2obo!

+k and +k

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x = 12, y = 24, rule = B3/S23:T24,24
2obob2obob2o$3bo4bo$2obo4bob2o$bobob2obobo$bobob2obobo$2obo4bob2o$3bo
4bo$2obob2obob2o$2obob2obob2o$3bo4bo$2obo4bob2o$bobob2obobo$bobob2obob
o$2obo4bob2o$3bo4bo$2obob2obob2o$2obob2obob2o$3bo4bo$2obo4bob2o$bobob
2obobo$bobob2obobo$2obo4bob2o$3bo4bo$2obob2obob2o!

xc and xc

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x = 24, y = 24, rule = B3/S23:T24,24
10b2ob2o$9bo2bo$8bobobo$8bo2bo$6b2ob2o$5bo2bo$4bobobo$4bo2bo$2b2ob2o$b
o2bo$obobo$o2bo$b2o19b2o$o20bo$o19bobo$20bo2bo$18b2ob2o$17bo2bo$16bobo
bo$16bo2bo$14b2ob2o$13bo2bo$12bobobo$12bo2bo!

xc and xk

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x = 20, y = 20, rule = B3/S23:T20,20
19bo$18bo$17bo$16bo$15bo$14bo$13bo$12bo$11bo$10bo$9bo$8bo$7bo$6bo$5bo$
4bo$3bo$2bo$bo$o!

xk and xk

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x = 24, y = 24, rule = B3/S23:T24,24
14b2obo2bo$9bo3bobobobo$8bobobobo3bo$7bo2bob2o$8b3o$11b3o$8b2obo2bo$3b
o3bobobobo$2bobobobo3bo$bo2bob2o$2b3o$5b3o$2b2obo2bo$bobobobo13bo$obo
3bo13bobo$2o17bo2bo$20b3o$2o21bo$2bo17b2obo$bo13bo3bobobo$o13bobobobo$
13bo2bob2o$14b3o$17b3o!

There are also wicks which are glide-symmetric in space, which I'm not sure how to classify. Perhaps DRH notation could somehow be amended with a symbol such as ~ to indicate this?

Here's one with n symmetry that glide-reflects across an "e" type line:

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x = 24, y = 6, rule = B3/S23:T24,24
4b2o4b2o4b2o4b2o$bo2bo2bo2bo2bo2bo2bo2bo$2o3b3o3b3o3b3o3bo$2b3o3b3o3b
3o3b3o$bo2bo2bo2bo2bo2bo2bo2bo$b2o4b2o4b2o4b2o!

n symmetric across a "c" type line:

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x = 24, y = 13, rule = B3/S23:T24,24
6b2o6b2o6b2o$7bo7bo7bo$6bo7bo7bo$6b2o6b2o6b2o$o2bo4bo2bo4bo2bo$o2b6o2b
6o2b5o2$5o2b6o2b6o2bo$4bo2bo4bo2bo4bo2bo$2b2o6b2o6b2o$2bo7bo7bo$3bo7bo
7bo$2b2o6b2o6b2o!

-e symmetric across an "e" type line:

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x = 24, y = 4, rule = B3/S23:T24,8
2b2o4b2o4b2o4b2o$bo2bo2bo2bo2bo2bo2bo2bo$bo2bo2bo2bo2bo2bo2bo2bo$o4b2o
4b2o4b2o4bo!

-e symmetric across a "c" type line:

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x = 24, y = 7, rule = B3/S23:T24,12
2b2o4b2o4b2o4b2o$bo2bo2bo2bo2bo2bo2bo2bo$bo2bo2bo2bo2bo2bo2bo2bo$ob2ob
2ob2ob2ob2ob2ob2obo$bo2bo2bo2bo2bo2bo2bo2bo$bo2bo2bo2bo2bo2bo2bo2bo$o
4b2o4b2o4b2o4bo!

-c symmetric across an "e" type line:

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x = 24, y = 6, rule = B3/S23:T24,12
2ob2ob2ob2ob2ob2ob2ob2o$bobo3bobo3bobo3bobo$o3b3o3b3o3b3o3b2o$b3o3b3o
3b3o3b3o$o3bobo3bobo3bobo3bo$2ob2ob2ob2ob2ob2ob2ob2o!

-c symmetric across a "c" type line:

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x = 24, y = 9, rule = B3/S23:T24,12
2bo7bo7bo$5o3b5o3b5o$5bobo5bobo5bobo$6ob7ob7obo2$2ob7ob7ob5o$bobo5bobo
5bobo$o3b5o3b5o3b4o$6bo7bo7bo!

/ symmetric across a "/" type line:

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x = 24, y = 24, rule = B3/S23:T24,24
23bo$23bo$21b2o$20bo$20bo$18b2o$17bo$17bo$15b2o$14bo$14bo$12b2o$11bo$
11bo$9b2o$8bo$8bo$6b2o$5bo$5bo$3b2o$2bo$2bo$2o!

Also / symmetric across a "/" type line, but seemingly differently - perhaps n/ and n/e should be considered distinct after all:

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x = 23, y = 24, rule = B3/S23:T24,24
20b3o2$18b3o$15bo2bo2bo$14bobobobo$13bo2bo2bo$14b3o2$12b3o$9bo2bo2bo$
8bobobobo$7bo2bo2bo$8b3o2$6b3o$3bo2bo2bo$2bobobobo$bo2bo2bo$2b3o2$3o$o
2bo17bo$obo17bobo$bo17bo2bo!

There's almost certainly n symmetric cases that glide-reflect across the / line type, and probably other symmetries I've missed (if so, please list these below).

For example, this is glide-reflective and has diagonal mirror symmetry, but I think it also has .e symmetry:

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x = 22, y = 22, rule = B3/S23:T22,22
19b2o$18bo2bo$17bo2bo$16bob2o$15bobo$14bo2bo$14bobo$12b2obo$11bo2bo$
10bo2bo$9bob2o$8bobo$7bo2bo$7bobo$5b2obo$4bo2bo$3bo2bo$2bob2o$bobo$o2b
o$obo$bo!

Any further insight into this topic? I doubt much new ground is being treaded here, and someone has already done a full breakdown of this before, but on the off chance that this is something nobody has put serious thought into before with respect to CA then some assistance from others would be appreciated.

[muzik]

Here's some "asymmetric" glide-symmetric diagonal wicks:

Type /

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x = 67, y = 64, rule = B3/S23
65b2o$62bo3bo$62b4o$60b2o$56b2obobo2b2o$56b2obo4b2o$59bo$59b2o$57b2o$
54bo3bo$54b4o$52b2o$48b2obobo2b2o$48b2obo4b2o$51bo$51b2o$49b2o$46bo3bo
$46b4o$44b2o$40b2obobo2b2o$40b2obo4b2o$43bo$43b2o$41b2o$38bo3bo$38b4o$
36b2o$32b2obobo2b2o$32b2obo4b2o$35bo$35b2o$33b2o$30bo3bo$30b4o$28b2o$
24b2obobo2b2o$24b2obo4b2o$27bo$27b2o$25b2o$22bo3bo$22b4o$20b2o$16b2obo
bo2b2o$16b2obo4b2o$19bo$19b2o$17b2o$14bo3bo$14b4o$12b2o$8b2obobo2b2o$
8b2obo4b2o$11bo$11b2o$9b2o$6bo3bo$6b4o$4b2o$2obobo2b2o$2obo4b2o$3bo$3b
2o!

Type /e

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x = 62, y = 56, rule = B3/S23
58b2o$49b2o5bo2bo$49bobo4b3o$51bo2b2o$50b2obobo2b3o$53bo4bo2bo$53b2o5b
2o$51b2o$42b2o5bo2bo$42bobo4b3o$44bo2b2o$43b2obobo2b3o$46bo4bo2bo$46b
2o5b2o$44b2o$35b2o5bo2bo$35bobo4b3o$37bo2b2o$36b2obobo2b3o$39bo4bo2bo$
39b2o5b2o$37b2o$28b2o5bo2bo$28bobo4b3o$30bo2b2o$29b2obobo2b3o$32bo4bo
2bo$32b2o5b2o$30b2o$21b2o5bo2bo$21bobo4b3o$23bo2b2o$22b2obobo2b3o$25bo
4bo2bo$25b2o5b2o$23b2o$14b2o5bo2bo$14bobo4b3o$16bo2b2o$15b2obobo2b3o$
18bo4bo2bo$18b2o5b2o$16b2o$7b2o5bo2bo$7bobo4b3o$9bo2b2o$8b2obobo2b3o$
11bo4bo2bo$11b2o5b2o$9b2o$2o5bo2bo$obo4b3o$2bo2b2o$b2obobo2b3o$4bo4bo
2bo$4b2o5b2o!

I believe this means we have a complete set as far as p1 wicks are concerned - do correct me if counterexamples exist.

[confocaloid]

This two-state isotropic CA with range-1 Moore neighbourhood on the two-dimensional square tiling supports mod-1 wicks of all periods:

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x = 60, y = 56, rule = B1e2i3r5i6a/S2ai3i4et:T60,80
2b2o4b2o2b2o$2bo5bo3bo$2b2o4b2o2b2o$2bo5bo$2b8o2b2o$2bo5bo3bo$2b2o4b2o
2b2o$2bo5bo3bo$2b2o4b2o2b2o3$60o2$11bo2$60o3$ob2ob2ob2ob2ob2ob2ob2ob2o
b2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2obo$2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo$60o$2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b
o2bo2bo2bo2bo2bo2bo2bo$ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob2ob
2ob2ob2ob2obo3$2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b
2o2b2o$2bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo$60o$2bo3bo3bo3bo
3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo$2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o
2b2o2b2o2b2o2b2o2b2o2b2o$2bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo3bo
$2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o3$2b2o3b
2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o$2bo4bo4bo4bo4bo4bo4bo4bo4bo
4bo4bo4bo$2b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o$2bo4bo4bo4b
o4bo4bo4bo4bo4bo4bo4bo4bo$60o$2bo4bo4bo4bo4bo4bo4bo4bo4bo4bo4bo4bo$2b
2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o$2bo4bo4bo4bo4bo4bo4bo4b
o4bo4bo4bo4bo$2b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o3$2b2o4b
2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o$2bo5bo5bo5bo5bo5bo5bo5bo5bo5bo$2b2o
4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o$2bo5bo5bo5bo5bo5bo5bo5bo5bo5bo$
60o$2bo5bo5bo5bo5bo5bo5bo5bo5bo5bo$2b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o4b
2o4b2o$2bo5bo5bo5bo5bo5bo5bo5bo5bo5bo$2b2o4b2o4b2o4b2o4b2o4b2o4b2o4b2o
4b2o4b2o$2bo5bo5bo5bo5bo5bo5bo5bo5bo5bo$2b2o4b2o4b2o4b2o4b2o4b2o4b2o4b
2o4b2o4b2o!