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3D cubic grid symmetries and naming
Posted: June 29th, 2025, 2:50 pm
by muzik
Asked about this on Polytope Discord around a year ago. The response was this, I'm not sure how comprehensive it is:
- IMG_9758.jpeg (1.38 MiB) Viewed 1712 times
Code: Select all
symmetries with singular central/fixed points:
cubic symmetry centered on a cube or vertex
chiral cubic symmetry centered on a cube or vertex
pyritic symmetry centered on a cube or vertex
tetrahedral symmetry centered on a cube or vertex
chiral tetrahedral symmetry centered on a cube or vertex
triangular antiprismatic symmetry centered on a cube or vertex
triangular proantiprismatic symmetry centered on a cube or vertex
chiral triangular prismatic symmetry centered on a cube or vertex
square prismatic symmetry centered on a cube, face, edge, or vertex
square proprismatic symmetry centered on a cube, face, edge, or vertex
chiral square prismatic symmetry centered on a cube, face, edge, or vertex
brick symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
digonal antiprismatic symmetry centered on a cube, face, edge, or vertex
espic symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
chiral brick symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
prodappic symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
central inversion symmetry centered on a cube, face, edge, or vertex
symmetries with central/fixed lines:
triangular pyramidal symmetry (line goes through vertices and cubes)
chiral triangular pyramidal symmetry (line goes through vertices and cubes)
square pyramidal symmetry (line goes through cubes and faces)
square pyramidal symmetry (line goes through edges and vertices)
chiral square pyramidal symmetry (line goes through cubes and faces)
chiral square pyramidal symmetry (line goes through edges and vertices)
digonal pyramidal symmetry (line goes through cubes and faces)
digonal pyramidal symmetry (line goes through edges and vertices)
digonal pyramidal symmetry (line goes through cubes and edges)
digonal pyramidal symmetry (line goes through faces and vertices)
chiral digonal pyramidal symmetry (line goes through cubes and faces)
chiral digonal pyramidal symmetry (line goes through edges and vertices)
chiral digonal pyramidal symmetry (line goes through cubes and edges)
chiral digonal pyramidal symmetry (line goes through faces and vertices)
symmetries with central/fixed planes (these are just reflections):
plane goes through cubes, edges, and faces
plane goes through faces, edges, and vertices
plane goes through cubes, faces, edges, and vertices
symmetries with central/fixed everything:
no symmetry
which is like around 90
Assuming someone hasn't went ahead and done so already, do we have a naming system for symmetries on the cubic grid like we do on the square grid (specifically the Catagolue style system)? If not, how would we go about devising one? The highest level of symmetry would be D48_1 or D48_8 (cubic symmetry centered on a cube or vertex), and the lowest C1. I don't know how we'd sort all of the rest of these, however.
This list is currently present on the cubic honeycomb page on the wiki as a commented out block of text and it'd be great if we can finally get it set up in a readable way (alongside a bunch of other symmetry-related things I've been putting off for longer than I should have).
Re: 3D cubic grid symmetries and naming
Posted: June 30th, 2025, 2:51 pm
by muzik
Simple example cases:
Much like the tub is a simple example of something with D8_1 symmetry in 2D:
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x = 3, y = 3, rule = B3/S23
bo$obo$bo!
this arrangement of 6 cells, effectively a 3D generalization of the tub, has D48_1 symmetry (maximal symmetry, cell-centered)
Code: Select all
3D version=1 size=50 pos=23,23,24
x=3 y=3 z=3 rule=3D4..7,9/4,5,7
$bo/bo$obo$bo/$bo!
- 3dtub.png (4.21 KiB) Viewed 1680 times
Likewise, whereas the pond is a simple D8_4 symmetric object in 2D:
Code: Select all
x = 4, y = 4, rule = B3/S23
b2o$o2bo$o2bo$b2o!
this 24-cell 3D object, effectively a 3D pond generalization, has D48_8 symmetry (maximal symmetry, vertex-centered):
Code: Select all
3D version=1 size=50 pos=22,22,24
x=4 y=4 z=4 rule=3D4..7,9/4,5,7
$boo$boo/boo$obbo$obbo$boo/boo$obbo$obbo$boo/$boo$boo!
- 3dpond.png (7.51 KiB) Viewed 1680 times
I plan to supply visual examples of objects bound by many of these symmetries since we don't have an in-browser means of viewing 3D patterns so far.
Re: 3D cubic grid symmetries and naming
Posted: July 1st, 2025, 7:11 am
by muzik
Here are objects with C24_1 and C24_8 symmetry:
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3D version=1 size=50 pos=22,22,22
x=7 y=7 z=7 rule=3D4..7,9/4,5,7
7o$4oboo$ob5o$3ob3o$5obo$oob4o$7o/oob4o$o5bo$o$o5bo$6bo$o5bo$4oboo/
5obo$6bo$o5bo$o5bo$o5bo$o$ob5o/3ob3o$o5bo$o5bo$$o5bo$o5bo$3ob3o/ob5o
$o$o5bo$o5bo$o5bo$6bo$5obo/4oboo$o5bo$6bo$o5bo$o$o5bo$oob4o/7o$oob4o
$5obo$3ob3o$ob5o$4oboo$7o!
- 3dc24_1.png (59.5 KiB) Viewed 1645 times
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3D version=1 size=50 pos=22,22,22
x=6 y=6 z=6 rule=3D4..7,9/4,5,7
6o$3oboo$o3boo$oo3bo$oob3o$6o/oob3o$o4bo$o$5bo$o4bo$3oboo/oo3bo$5bo
3$o$o3boo/o3boo$o3$5bo$oo3bo/3oboo$o4bo$5bo$o$o4bo$oob3o/6o$oob3o$oo
3bo$o3boo$3oboo$6o!
- 3dc24_8.png (37.62 KiB) Viewed 1645 times
Screenshots are largely unuseful here as it's hard to tell what cubes are in the foreground and which are in the background.
The 2D equivalent objects are as follows - the 3D objects are cubes with these on each of the six faces, such that any 90 degree rotation around one of the three axes, 120 degree rotations around a vertex or 180 degree rotations around an edge will result in it looking identical, but mirror symmetries are not permitted:
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x = 7, y = 7, rule = B3/S23
7o$4ob2o$ob5o$3ob3o$5obo$2ob4o$7o!
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x = 6, y = 6, rule = B3/S23
6o$3ob2o$o3b2o$2o3bo$2ob3o$6o!
Re: 3D cubic grid symmetries and naming
Posted: July 2nd, 2025, 6:12 am
by muzik
The following are objects with pyritohedral symmetry - the faces have D4_+ symmetries, but the vertices are chiral:
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3D version=1 size=50 pos=23,23,23
x=5 y=5 z=5 rule=3D4..7,9/4,5,7
5o$ooboo$ooboo$ooboo$5o/5o$o3bo$$o3bo$5o/o3bo$o3bo$$o3bo$o3bo/5o$o3b
o$$o3bo$5o/5o$ooboo$ooboo$ooboo$5o!
- 3dpyrite1.png (29.61 KiB) Viewed 1606 times
Code: Select all
3D version=1 size=50 pos=22,22,22
x=6 y=6 z=6 rule=3D4..7,9/4,5,7
6o$oobboo$oobboo$oobboo$oobboo$6o/6o$o4bo3$o4bo$6o/o4bo$o4bo3$o4bo$
o4bo/o4bo$o4bo3$o4bo$o4bo/6o$o4bo3$o4bo$6o/6o$oobboo$oobboo$oobboo$
oobboo$6o!
- 3dpyrite8.png (34.76 KiB) Viewed 1606 times
I'm not sure what these would be called in a naming system like what Catagolue uses. D24_+1 and D24_+8
could work as they're dihedrally symmetric and otherwise have all the symmetries C24_1 and C24_8 have, but I'm not sure if they're ambiguous and if other symmetries could fit this description as well.
Re: 3D cubic grid symmetries and naming
Posted: July 3rd, 2025, 12:26 pm
by muzik
A regular tetrahedron can be made quite simply through an alternation process on a cube, with either an odd or even edge length. Other objects can also have this symmetry. I also don't know how we'd name this systematically, though.
Odd edge length example:
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3D version=1 size=50 pos=22,22,22
x=7 y=7 z=7 rule=3D4..7,9/4,5,7
o$bo$bbo$3bo$4bo$5bo$6bo/bo$3o$b3o$bb3o$3b3o$4b3o$5bo/bbo$b3o$5o$b5o
$bb5o$3b3o$4bo/3bo$bb3o$b5o$7o$b5o$bb3o$3bo/4bo$3b3o$bb5o$b5o$5o$b3o
$bbo/5bo$4b3o$3b3o$bb3o$b3o$3o$bo/6bo$5bo$4bo$3bo$bbo$bo$o!
- 3dtet_1.png (49.66 KiB) Viewed 1565 times
Even edge length:
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3D version=1 size=50 pos=21,21,21
x=8 y=8 z=8 rule=3D4..7,9/4,5,7
o$bo$bbo$3bo$4bo$5bo$6bo$7bo/bo$3o$b3o$bb3o$3b3o$4b3o$5b3o$6bo/bbo$
b3o$5o$b5o$bb5o$3b5o$4b3o$5bo/3bo$bb3o$b5o$7o$b7o$bb5o$3b3o$4bo/4bo
$3b3o$bb5o$b7o$7o$b5o$bb3o$3bo/5bo$4b3o$3b5o$bb5o$b5o$5o$b3o$bbo/6b
o$5b3o$4b3o$3b3o$bb3o$b3o$3o$bo/7bo$6bo$5bo$4bo$3bo$bbo$bo$o!
- 3dtet_8.png (34.66 KiB) Viewed 1565 times
Re: 3D cubic grid symmetries and naming
Posted: July 4th, 2025, 6:24 pm
by muzik
Here's some chiral tetrahedra - the details that make symmetry obvious are getting increasingly more difficult to represent in individual screenshots.
Cell-centered:
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3D version=1 size=50 pos=22,22,22
x=7 y=7 z=7 rule=3D4..7,9/4,5,7
o$bo$bbo$3bo$4bo$5bo$6bo/bo$o$3bo$$3bo$6bo$5bo/bbo$$o3bo$bo3bo$bbo3b
o$$4bo/3bo$bbobo$$o5bo$$bbobo$3bo/4bo$$bbo3bo$bo3bo$o3bo$$bbo/5bo$6b
o$3bo$$3bo$o$bo/6bo$5bo$4bo$3bo$bbo$bo$o!
- 3dchitet_1.png (36.93 KiB) Viewed 1504 times
Vertex-centered:
Code: Select all
3D version=1 size=50 pos=21,21,21
x=8 y=8 z=8 rule=3D4..7,9/4,5,7
o$bo$bbo$3bo$4bo$5bo$6bo$7bo/bo$o$3bo3$4bo$7bo$6bo/bbo$$o3bo$bo3bo$
bbo3bo$3bo3bo$$5bo/3bo$bbo$5bo$o$7bo$bbo$5bo$4bo/4bo$5bo$bbo$7bo$o$
5bo$bbo$3bo/5bo$$3bo3bo$bbo3bo$bo3bo$o3bo$$bbo/6bo$7bo$4bo3$3bo$o$b
o/7bo$6bo$5bo$4bo$3bo$bbo$bo$o!
- 3dchitet_8.png (51.35 KiB) Viewed 1504 times
Re: 3D cubic grid symmetries and naming
Posted: July 5th, 2025, 6:44 pm
by muzik
Here's octahedra with opposite faces deleted, which I assume is the triangular antiprismatic symmetry mentioned:
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3D version=1 size=50 pos=22,22,22
x=7 y=7 z=7 rule=3D4..7,9/4,5,7
o$bo$bbo$3bo$4bo$5bo$6bo/bo$o$3bo$$3bo$6bo$5bo/bbo$$o3bo$bo3bo$bbo3b
o$$4bo/3bo$bbobo$$o5bo$$bbobo$3bo/4bo$$bbo3bo$bo3bo$o3bo$$bbo/5bo$6b
o$3bo$$3bo$o$bo/6bo$5bo$4bo$3bo$bbo$bo$o!
- 3d3anti1.png (28.73 KiB) Viewed 1478 times
Code: Select all
3D version=1 size=50 pos=21,21,21
x=8 y=8 z=8 rule=3D4..7,9/4,5,7
o$bo$bbo$3bo$4bo$5bo$6bo$7bo/bo$o$3bo3$4bo$7bo$6bo/bbo$$o3bo$bo3bo$
bbo3bo$3bo3bo$$5bo/3bo$bbo$5bo$o$7bo$bbo$5bo$4bo/4bo$5bo$bbo$7bo$o$
5bo$bbo$3bo/5bo$$3bo3bo$bbo3bo$bo3bo$o3bo$$bbo/6bo$7bo$4bo3$3bo$o$b
o/7bo$6bo$5bo$4bo$3bo$bbo$bo$o!
- 3d3anti8.png (39.28 KiB) Viewed 1478 times
Re: 3D cubic grid symmetries and naming
Posted: July 8th, 2025, 8:28 pm
by CARuler
like like describe symmetries as operations on a vector
here's one symmetry: [x,y,z]->[-y,x,-z]
example:
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3D version=1 size=50 pos=22,22,22
x=3, y=3, z=4, rule=3D4..7,9/4,5,7
2o2$b2o/bo2$bo/b$obo/2bo$obo$o!
(i don't know how to put pictures on the forums sorry)
Re: 3D cubic grid symmetries and naming
Posted: July 29th, 2025, 3:10 am
by gzaytman
How about basing names on
existing notation for 3d point groups. Maybe
Schoenflies notation possibly a modified version that uses the group order as the subscript to be consistant with the Catalogue notation in 2d.
Re: 3D cubic grid symmetries and naming
Posted: August 5th, 2025, 8:44 am
by breaker's glider gun
Re: 3D cubic grid symmetries and naming
Posted: August 7th, 2025, 8:03 am
by gzaytman
No, that's Conway's notation for polyhedra. That's different form his group notation described in your first link.
Re: 3D cubic grid symmetries and naming
Posted: August 7th, 2025, 8:19 am
by breaker's glider gun
gzaytman wrote: August 7th, 2025, 8:03 am
No, that's Conway's notation for polyhedra. That's different form his group notation described in your first link.
Ah, my bad. Conway was just too awesome, making multiple types of notation. Could someone find a more technical description of the one we want?
Re: 3D cubic grid symmetries and naming
Posted: August 28th, 2025, 9:35 pm
by gzaytman
Ok, I've created symbols based on Conway notation for the 92 cubic grid symmetries
here, that's
hopefully free from mistakes.
I admit that in a few cases my subscripts aren't completely intuitive.
I'll probably add the generating transformations later.
Re: 3D cubic grid symmetries and naming
Posted: August 29th, 2025, 5:07 am
by muzik
Excellent (though should visual examples or pattern files be provided alongside each symmetry like is done on the 2D symmetries page)?
Re: 3D cubic grid symmetries and naming
Posted: August 29th, 2025, 11:23 pm
by gzaytman
Ok, I've added a list of generators to the page. I figured putting them all at the bottom in a machine-readable format would be the most helpful. The names
should match the descriptions on the text.
muzik wrote: August 29th, 2025, 5:07 am
Excellent (though should visual examples or pattern files be provided alongside each symmetry like is done on the 2D symmetries page)?
If someone who is more familiar with the pattern editing tools than me wants to use the generators to create pattern files, feel free to.
Edit: Also does anyone have suggestions for a better format for the generators?
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 9:58 am
by muzik
The following, I assume, would be "±C3_k"?
Code: Select all
3D version=1 size=50 pos=23,23,23
x=4 y=4 z=4 rule=3D0..6/F
$bo/$b3o$bbo$bbo/bo$bo$3o/$$bbo!
- 3d-twelve-protrappic.png (12.56 KiB) Viewed 1132 times
I assume we're certain that our enumeration of 92 symmetries is complete?
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 12:01 pm
by gzaytman
muzik wrote: August 30th, 2025, 9:58 am
The following, I assume, would be "±C3_k"?
Code: Select all
3D version=1 size=50 pos=23,23,23
x=4 y=4 z=4 rule=3D0..6/F
$bo/$b3o$bbo$bbo/bo$bo$3o/$$bbo!
3d-twelve-protrappic.png
That appears to be correct.
muzik wrote: August 30th, 2025, 9:58 am
I assume we're certain that our enumeration of 92 symmetries is complete?
The groups as given by the generators are correct. Assigning them to names and descriptions was done by hand, so there might still be errors.
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 1:54 pm
by gzaytman
In addition to the static symmetries there are:
220 kinetic symmetries with period/mod=2,
7 with period/mod=3,
14 with period/mod=4, and
4 with period/mod=6
for a total of 245 kinetic symmetries with period different from the mod and 337 total kinetic symmetries.
I've added them to the
page.
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 2:37 pm
by CARuler
there's also [x,y,z]->[y,z,x]
Code: Select all
3D version=1 size=50 pos=2,2,2
o/2o$o!
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 3:31 pm
by muzik
gzaytman wrote: August 30th, 2025, 1:54 pm
In addition to the static symmetries there are:
220 kinetic symmetries with period/mod=2,
7 with period/mod=3,
14 with period/mod=4, and
4 with period/mod=6
for a total of 245 kinetic symmetries with period different from the mod and 312 total kinetic symmetries.
I've added them to the
page.
Was just going to ask about kinetic symmetries. There haven't been that many oscillators found in 3D rules, especially ones with interesting symmetries. Here's a short table that acts as a blinker - would I be correct in saying the static symmetry is CD4_fe and the kinetic symmetry is ±D4_fe or am I incorrect on that?
Code: Select all
3D version=1 size=50 pos=24,24,25 gen=1
x=3 y=2 z=1 rule=3D7,8,18,19/5,8,13,14
obo$3o!
Probably a bit early to be creating a Hickerson-style notation for kinetic symmetries, since we don't even have one for the hexagonal/triangular grids yet (something I definitely plan on attempting).
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 3:48 pm
by gzaytman
muzik wrote: August 30th, 2025, 3:31 pm
Here's a short table that acts as a blinker - would I be correct in saying the static symmetry is CD4_fe and the kinetic symmetry is ±D4_fe or am I incorrect on that?
Code: Select all
3D version=1 size=50 pos=24,24,25 gen=1
x=3 y=2 z=1 rule=3D7,8,18,19/5,8,13,14
obo$3o!
That is incorrect, the static symmetry of your blinker is CD4_cfo, and c̶o̶m̶p̶o̶s̶i̶t̶e̶ ̶s̶y̶m̶m̶e̶t̶r̶y̶ ̶i̶s̶ ̶i̶n̶d̶e̶e̶d̶ ̶±̶D̶4̶_̶f̶e̶.
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 4:05 pm
by gzaytman
gzaytman wrote: August 30th, 2025, 3:48 pm
muzik wrote: August 30th, 2025, 3:31 pm
Here's a short table that acts as a blinker - would I be correct in saying the static symmetry is CD4_fe and the kinetic symmetry is ±D4_fe or am I incorrect on that?
Code: Select all
3D version=1 size=50 pos=24,24,25 gen=1
x=3 y=2 z=1 rule=3D7,8,18,19/5,8,13,14
obo$3o!
That is incorrect, the static symmetry of your blinker is CD4_cfo, and c̶o̶m̶p̶o̶s̶i̶t̶e̶ ̶s̶y̶m̶m̶e̶t̶r̶y̶ ̶i̶s̶ ̶i̶n̶d̶e̶e̶d̶ ̶±̶D̶4̶_̶f̶e̶.
No, scratch that, the composite symmetry is actually DD8_fk.
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 5:16 pm
by gzaytman
gzaytman wrote: August 30th, 2025, 4:05 pm
gzaytman wrote: August 30th, 2025, 3:48 pm
muzik wrote: August 30th, 2025, 3:31 pm
Here's a short table that acts as a blinker - would I be correct in saying the static symmetry is CD4_fe and the kinetic symmetry is ±D4_fe or am I incorrect on that?
Code: Select all
3D version=1 size=50 pos=24,24,25 gen=1
x=3 y=2 z=1 rule=3D7,8,18,19/5,8,13,14
obo$3o!
That is incorrect, the static symmetry of your blinker is CD4_cfo, and c̶o̶m̶p̶o̶s̶i̶t̶e̶ ̶s̶y̶m̶m̶e̶t̶r̶y̶ ̶i̶s̶ ̶i̶n̶d̶e̶e̶d̶ ̶±̶D̶4̶_̶f̶e̶.
No, scratch that, the composite symmetry is actually DD8_fk.
The group CD4_cfo is called that because the rotation axis goes through the centers of a cell and a face and the two reflection planes are orthogonal.
As for DD8_fk, I admit my notation for the DD8's is probably the most confusing. I ultimately decided to call it DD8_fk because its chiral subgroup is D4_fk.
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 5:19 pm
by muzik
For spaceships: is there at least one symmetry corresponding to each possible 2D oscillator kinetic symmetry?
Re: 3D cubic grid symmetries and naming
Posted: August 30th, 2025, 5:23 pm
by gzaytman
muzik wrote: August 30th, 2025, 5:19 pm
For spaceships: is there at least one symmetry corresponding to each possible 2D oscillator kinetic symmetry?
For orthogonal spaceships, yes.