3D cubic grid symmetries and naming

[muzik]

For spaceships: is there at least one symmetry corresponding to each possible 2D oscillator kinetic symmetry?

For orthogonal spaceships, yes.

If I'm analyzing this correctly, orthogonal spaceships would have square-grid kinetic symmetries, diagonal spaceships would have some sort of rectangular symmetries forbidding diagonal reflections and 90-degree rotations but preserving orthogonal reflections and 180-degree rotations, and paragonal spaceships would presumably have hexagonal-grid-like symmetries?

[gzaytman]

If I'm analyzing this correctly, orthogonal spaceships would have square-grid kinetic symmetries, diagonal spaceships would have some sort of rectangular symmetries forbidding diagonal reflections and 90-degree rotations but preserving orthogonal reflections and 180-degree rotations, and paragonal spaceships would presumably have hexagonal-grid-like symmetries?

Correct. The paradiagonal spaceships are only allowed the hexagonal symmetries that preserve "vertex coloring". Color all the vertices on a hexagonal grid one of two colors such that adjacent vertices are different colors. The symmetries available to paradiagonal spaceships are the ones that preserve this coloring.

[gzaytman]

Ok, since I had the code already written I decided to try counting tesseract symmetries. The results, assuming I didn't make a mistake, are:
686 static symmetries,
2535 with period/mod=2,
38 with period/mod=3,
235 with period/mod=4,
46 with period/mod=6, and
2 with period/mod=8.
I can get generators for these symmetries, but naming them seems messy.

Edit: found a typo in my program and fixed the numbers.

[muzik]

Ok, since I had the code already written I decided to try counting tesseract symmetries.
I can get generators for these symmetries, but naming them seems messy.

Would probably be a good idea to have these noted down anyway for future reference. I assume it's possible to enumerate symmetries on {3,4,3,3} and {3,3,4,3} as well, for 4D completeness's sake?

[gzaytman]

Ok, I've added the 65 spaceship symmetries of the 3D cubic grid here.

If there are no objections, I'll be transferring the page to the mainspace shortly.

[muzik]

Seems good to me, ideally we'd have graphics alongside these but I have no idea how to make anything that works good.

Code: Select all

3D version=1 size=96 pos=40,40,40
x=16 y=16 z=16 rule=3D7,8,15,17,21..26/7..9,13,17..26
16o$o14bo$o14bo$o9boo3bo$o8b4obbo$o8b4obbo$obb10obbo$o14bo$o14bo$o3b
oo4boo3bo$obbobbobbobbobbo$obbobbobbobbobbo$o3boo4boo3bo$o14bo$o14b
o$16o/o14bo15$o14bo/o14bo15$o14bo/o3boo4boo3bo4$o$o$o14bo4$o14bo$o14b
o4$o3boo4boo3bo/obbobbobbobbobbo3$o$o$o$o14bo3$o14bo3$o14bo3$obbobb
obbobbobbo/obbobbobbobbobbo3$o$o$o$o14bo3$o14bo3$o14bo3$obbobbobbobb
obbo/o3boo4boo3bo4$o$o$o14bo4$o14bo$o14bo4$o3boo4boo3bo/o14bo6$o14b
o9$o14bo/o14bo6$o14bo9$o14bo/obb10obbo4$15bo$15bo$o14bo4$o14bo$o14b
o4$obb10obbo/o8b4obbo3$15bo$15bo$15bo$o14bo3$o14bo3$o14bo3$obb4o8bo
/o8b4obbo3$15bo$15bo$15bo$o14bo3$o14bo3$o14bo3$obb4o8bo/o9boo3bo4$15b
o$15bo$o14bo4$o14bo$o14bo4$o3boo9bo/o14bo15$o14bo/o14bo15$o14bo/16o
$o14bo$o14bo$o3boo9bo$obb4o8bo$obb4o8bo$obb10obbo$o14bo$o14bo$o3boo
4boo3bo$obbobbobbobbobbo$obbobbobbobbobbo$o3boo4boo3bo$o14bo$o14bo$
16o!

threedimensionalc1.png (41.98 KiB) Viewed 1412 times

Are there any plans to make a user page for the tesseractic symmetry generators, or to enumerate the 16-cell/24-cell grid symmetries? Certainly I want to know how many tetrahedral symetries there are - we'd have the alternation of cube facets, for one, as well as the tesseract's vertices. Would the facets of the 4-demicube be a third, distinct tetrahedral symmetry, or would it duplicate one of the aforementioned two?

[muzik]

A question related to kinetic symmetries: the vertex of {4,3,3,3} (the cell of the standard 5D grid {4,3,3,3,4} used in Visions of Chaos) is {3,3,3}, the regular 5-cell. Is it therefore possible for an oscillator in a five-dimensional rule to have a mod a fifth of its period? In general, will the n–dimensional hypercubic grid be able to support oscillators of mod n (e.g. mods of 7 on a hepteractic grid, 9 on an enneractic grid, and so on)?

[gzaytman]

Is it therefore possible for an oscillator in a five-dimensional rule to have a mod a fifth of its period? In general, will the n–dimensional hypercubic grid be able to support oscillators of mod n (e.g. mods of 7 on a hepteractic grid, 9 on an enneractic grid, and so on)?

Yes. In fact if you use the An lattice instead of the standard lattice, it is possible to have period/mod=n+1 oscillators in n dimensions.

For example, in two dimensions the A2 lattice, i.e., the hexagonal tiling, can have period/mod=3.

[gzaytman]

Ok, since I had the code already written I decided to try counting tesseract symmetries. The results, assuming I didn't make a mistake, are:
686 static symmetries,
2535 with period/mod=2,
38 with period/mod=3,
235 with period/mod=4,
46 with period/mod=6, and
2 with period/mod=8.
I can get generators for these symmetries, but naming them seems messy.

Edit: found a typo in my program and fixed the numbers.

Ok, I've given the tesseractic symmetries names based on the names for 4D symmetry groups from chapter 4 of Conway and Smith's On Quaternions and Octonions. I have posted the list of groups and their generators as well as the kinetic symmetries here. Aside from the names, which only make sense if you understand how quaternions work, I haven't given them human-friendly descriptions.

[muzik]

Ok, I've given the tesseractic symmetries names based on the names for 4D symmetry groups from chapter 4 of Conway and Smith's On Quaternions and Octonions. I have posted the list of groups and their generators as well as the kinetic symmetries here. Aside from the names, which only make sense if you understand how quaternions work, I haven't given them human-friendly descriptions.

I assume the kinetic symmetries list only applies to oscillators (i.e. the spaceship symmetries are yet to be computed so far)?

[gzaytman]

I assume the kinetic symmetries list only applies to oscillators (i.e. the spaceship symmetries are yet to be computed so far)?

Correct.

[gzaytman]

Ok, I've computed and posted the 24-cell honeycomb symmetries here. These also apply to its dual honeycomb, the 16-cell honeycomb. There are
501 static symmetries,
1569 with period/mod=2,
77 with period/mod=3,
143 with period/mod=4,
70 with period/mod=6,
4 with period/mod=8, and
1 with period/mod=12.

A new phenomenon that shows up is that two of the period/mod=8 symmetries have the same static symmetry and composite symmetry and are in fact conjugate under time inversion, but not under any hyperspace transformation.

[muzik]

I've added a table here that lists the known symmetry information for notable grids so far. Also there is now gutteroid information listed here for the square grid and here for the hexagonal grid. I assume that in 3D, gutteroids can take the form of central cells, orthogonal, diagonal and paragonal lines, and orthogonal, diagonal and paragonal planes?

Another question I have: is the fact that the tesseractic honeycomb has kinetic symmetries of mod p/8 and the 16-cell and 24-cell honeycombs have kinetic symmetries of mod p/12 linked to the fact that the tesseract and 24-cell have octagonal and dodecagonal Petrie polygons? If so, does this mean that the penteractic grid has a p/10 mod symmetry (and the n-dimensional cubic honeycomb has a p/2n mod symmetry)? In addition, would a p/5 symmetry arise from this if you rotate by twice the angle, and if so, is this a distinct p/5 time symmetry from the one that arises from a penteract's vertex?

[gzaytman]

I've added a table here that lists the known symmetry information for notable grids so far. Also there is now gutteroid information listed here for the square grid and here for the hexagonal grid. I assume that in 3D, gutteroids can take the form of central cells, orthogonal, diagonal and paragonal lines, and orthogonal, diagonal and paragonal planes?

What's a gutteroid?

Another question I have: is the fact that the tesseractic honeycomb has kinetic symmetries of mod p/8 and the 16-cell and 24-cell honeycombs have kinetic symmetries of mod p/12 linked to the fact that the tesseract and 24-cell have octagonal and dodecagonal Petrie polygons? If so, does this mean that the penteractic grid has a p/10 mod symmetry (and the n-dimensional cubic honeycomb has a p/2n mod symmetry)?

Yes, and yes.

In addition, would a p/5 symmetry arise from this if you rotate by twice the angle, and if so, is this a distinct p/5 time symmetry from the one that arises from a penteract's vertex?

The p/5 symmetry would involve rotating by 2pi/5 in one plane, and twice that in a different plane. In high dimensions there's enough room to rotate by different angles in different planes. You still might get distinct symmetries depending on how the static and composite symmetry groups interact.

[muzik]

I've added a table here that lists the known symmetry information for notable grids so far. Also there is now gutteroid information listed here for the square grid and here for the hexagonal grid. I assume that in 3D, gutteroids can take the form of central cells, orthogonal, diagonal and paragonal lines, and orthogonal, diagonal and paragonal planes?

What's a gutteroid?

Any area of an oscillator that oscillates at a lower period specifically due to symmetry constraints, for example the line of such cells in this -c+c:

Code: Select all

x = 3, y = 2, rule = B2-ac3aeiq4krtwz5r6i7e8/S1c2ae3eijy4airt5k6-k7e
obo$bo!
[[ AUTOIDENTIFY ]]

and the central cell of this nrc:

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x = 2, y = 3, rule = B2ce3iny4aeinqw5acr6-ci7c/S12e3knr4acjrt5ai6e
2o2$bo!
[[ AUTOIDENTIFY ]]

[gzaytman]

I assume that in 3D, gutteroids can take the form of central cells, orthogonal, diagonal and paragonal lines, and orthogonal, diagonal and paragonal planes?

Not paradiagonal planes, but otherwise, yes.

[gzaytman]

I posted a list of (I believe) minimal objects preserved by the 3d symmetry groups here. Currently, they're given as lists of points, I could probably also give them in rle3 format. Unfortunately, the wiki does not support 3d graphics and I can't post sage notebooks to the wiki.
If someone is interested in a particular format or wants to get sent the sage notebook, let me know.

[gzaytman]

Ok, here are all the rle3's for 3d-symmetries. I've also added them to the wiki page.

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3D version=1 size=13 pos=3,3,3
#C O24_c: chiral cubic symmetry centered on a cell
x=7 y=7 z=7 rule=3D0..6/F
$4bo$bo2$5bo$2bo/2bo2$3bo2bo$2bobo$o2bo2$4bo/5bo$o2bo2$bo3bo2$3bo2bo$bo/$2bobo$bo3bo2$bo3bo$2bobo/bo$3bo2bo2$
bo3bo2$o2bo$5bo/4bo2$o2bo$2bobo$3bo2bo2$2bo/$2bo$5bo2$bo$4bo!

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3D version=1 size=11 pos=3,3,3
#C T12_c: chiral tetrahedral symmetry centered on a cell
x=5 y=5 z=5 rule=3D0..6/F
$2bo2$2bo/$3bo$o3bo$bo/bobo2$2bo2$bobo/$bo$o3bo$3bo/$2bo2$2bo!

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3D version=1 size=11 pos=3,3,4
#C D8_c: chiral square prismatic symmetry centered on a cell
x=5 y=5 z=3 rule=3D0..6/F
bo$4bo2$o$3bo/$2bo$bobo$2bo/3bo$o2$4bo$bo!

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3D version=1 size=11 pos=3,3,4
#C C4_cf: chiral square pyramidal symmetry, a.k.a., 90 degree rotation (line goes through cells and faces)
x=5 y=5 z=2 rule=3D0..6/F
$2bo$bobo$2bo/3bo$o2$4bo$bo!

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3D version=1 size=11 pos=4,3,4
#C D4_cf: chiral brick symmetry centered on a cell, orthogonal orientation
x=3 y=5 z=3 rule=3D0..6/F
$2bo2$o/bo2$obo2$bo/$o2$2bo!

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3D version=1 size=11 pos=3,3,4
#C D4_ce: chiral brick symmetry centered on a cell, diagonal orientation
x=5 y=5 z=3 rule=3D0..6/F
bo4$3bo/$bo$2bo$3bo/$o2$4bo!

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3D version=1 size=9 pos=3,3,3
#C D6_ck or simply D6_c: chiral triangular prismatic symmetry centered on a cell
x=3 y=3 z=3 rule=3D0..6/F
2$bo/2bo$bo/$o!

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3D version=1 size=9 pos=3,3,3
#C C3_ck or simply C3: chiral triangular pyramidal symmetry, a.k.a., 120 degree rotation (line goes through vertices and cells)
x=3 y=3 z=3 rule=3D0..6/F
2$bo/2bo$2bo$bo/$2o!

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3D version=1 size=9 pos=3,3,3
#C C2_cf: chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through cells and faces)
x=3 y=2 z=3 rule=3D0..6/F
2o/$bo/b2o!

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3D version=1 size=9 pos=3,3,3
#C C2_ce: chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through cells and edges)
x=2 y=2 z=3 rule=3D0..6/F
o/$o/$bo!

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3D version=1 size=9 pos=3,3,3
#C C1: no symmetry
x=2 y=3 z=3 rule=3D0..6/F
o/$o/$bo$bo!

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3D version=1 size=7 pos=3,3,3
#C ±O24_c: cubic symmetry centered on a cell
x=1 y=1 z=1 rule=3D0..6/F
o!

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3D version=1 size=11 pos=3,3,3
#C ±T12_c: pyritic symmetry centered on a cell
x=5 y=5 z=5 rule=3D0..6/F
2$bobo/2bo2$2bo2$2bo/$obobo$bobo$obobo/2bo2$2bo2$2bo/2$bobo!

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3D version=1 size=9 pos=4,4,3
#C ±D8_c: square prismatic symmetry centered on a cell
x=1 y=1 z=3 rule=3D0..6/F
o/o/o!

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3D version=1 size=11 pos=3,3,5
#C ±C4_cf: square proprismatic symmetry centered on a cell
x=5 y=5 z=1 rule=3D0..6/F
3bo$obo$bobo$2bobo$bo!

Code: Select all

3D version=1 size=9 pos=3,3,4
#C ±D4_cf: brick symmetry centered on a cell, orthogonal orientation
x=3 y=3 z=1 rule=3D0..6/F
obo$3o$obo!

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3D version=1 size=9 pos=3,3,4
#C ±D4_ce: brick symmetry centered on a cell, diagonal orientation
x=3 y=3 z=1 rule=3D0..6/F
o$bo$2bo!

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3D version=1 size=9 pos=3,3,3
#C ±D6_ck or simply ±D6_c: triangular antiprismatic symmetry centered on a cell
x=3 y=3 z=3 rule=3D0..6/F
o/$bo/2$2bo!

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3D version=1 size=11 pos=3,3,3
#C ±C3_ck or simply ±C3_c: triangular proantiprismatic symmetry centered on a cell
x=5 y=5 z=5 rule=3D0..6/F
$2bo/$bo$o/bo2$2bo2$3bo/2$4bo$3bo/3$2bo!

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3D version=1 size=9 pos=3,4,3
#C ±C2_cf: espic symmetry centered on a cell, axis goes through face
x=3 y=1 z=3 rule=3D0..6/F
o/3o/2bo!

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3D version=1 size=9 pos=3,3,3
#C ±C2_ce: espic symmetry centered on a cell, axis goes through edge
x=3 y=3 z=3 rule=3D0..6/F
o/o$bo$2bo/2$2bo!

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3D version=1 size=11 pos=4,4,3
#C ±C1_c: central inversion symmetry centered on a cell
x=3 y=3 z=5 rule=3D0..6/F
o/$o/$bo/$2bo/2$2bo!

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3D version=1 size=9 pos=3,3,3
#C TO24_c: tetrahedral symmetry centered on a cell
x=3 y=3 z=3 rule=3D0..6/F
2bo2$o/$bo/o2$2bo!

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3D version=1 size=8 pos=3,3,3
#C CD6_ck or simply CD6: triangular pyramidal symmetry (line goes through vertices and cells)
x=2 y=2 z=2 rule=3D0..6/F
2o$o/o!

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3D version=1 size=9 pos=3,3,3
#C CD8_cf: square pyramidal symmetry (line goes through cells and faces)
x=3 y=3 z=2 rule=3D0..6/F
$bo/bo$obo$bo!

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3D version=1 size=9 pos=3,3,3
#C DD8_cf: digonal antiprismatic symmetry centered on a cell, digon is diagonal
x=3 y=3 z=3 rule=3D0..6/F
2bo$bo$o/$bo/o$bo$2bo!

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3D version=1 size=9 pos=3,3,3
#C DD8_ce: digonal antiprismatic symmetry centered on a cell, digon is orthogonal
x=3 y=3 z=3 rule=3D0..6/F
bo2$bo/$bo/$obo!

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3D version=1 size=11 pos=3,3,4
#C CC4_cf: prodappic symmetry centered on a cell
x=5 y=5 z=3 rule=3D0..6/F
3bo4$bo/$2bo$bobo$2bo/$o2$4bo!

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3D version=1 size=9 pos=3,3,4
#C CD4_cfo: digonal pyramidal symmetry (line goes through cells and faces), orthogonal orientation
x=3 y=2 z=1 rule=3D0..6/F
bo$obo!

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3D version=1 size=9 pos=3,3,3
#C CD4_cfd: digonal pyramidal symmetry (line goes through cells and faces), diagonal orientation
x=3 y=3 z=2 rule=3D0..6/F
$bo/o2$2bo!

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3D version=1 size=8 pos=3,3,3
#C CD4_ce: digonal pyramidal symmetry (line goes through cells and edges)
x=2 y=2 z=1 rule=3D0..6/F
2o$o!

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3D version=1 size=9 pos=4,3,3
#C CC2_c: reflection (plane goes through cells, edges, and faces)
x=1 y=3 z=2 rule=3D0..6/F
o$o/2$o!

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3D version=1 size=9 pos=3,3,3
#C CC2_/: reflection (plane goes through cells, faces, edges, and vertices)
x=3 y=3 z=2 rule=3D0..6/F
o/$bo$2bo!

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3D version=1 size=11 pos=3,3,4
#C D8_f: chiral square prismatic symmetry centered on a face
x=5 y=5 z=2 rule=3D0..6/F
bo$2bobo$bobo$obo$3bo/3bo$obo$bobo$2bobo$bo!

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3D version=1 size=9 pos=3,3,3
#C D4_fe: chiral brick symmetry centered on a face, orthogonal orientation
x=3 y=3 z=2 rule=3D0..6/F
2bo$3o$o/o$3o$2bo!

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3D version=1 size=11 pos=3,3,4
#C D4_fk: chiral brick symmetry centered on a face, diagonal orientation
x=5 y=5 z=2 rule=3D0..6/F
$o$bobo$4bo/bo$2bo2$2bo$3bo!

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3D version=1 size=9 pos=3,3,3
#C C2_fe: chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and edges)
x=3 y=2 z=2 rule=3D0..6/F
bo$2bo/bo$o!

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3D version=1 size=8 pos=3,3,3
#C C2_fk: chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and vertices)
x=2 y=2 z=2 rule=3D0..6/F
2o/o$o!

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3D version=1 size=8 pos=3,3,3
#C ±D8_f: square prismatic symmetry centered on a face
x=1 y=1 z=2 rule=3D0..6/F
o/o!

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3D version=1 size=9 pos=3,4,3
#C ±D4_fe: brick symmetry centered on a face, orthogonal orientation
x=3 y=1 z=2 rule=3D0..6/F
3o/3o!

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3D version=1 size=9 pos=3,3,3
#C ±D4_fk: brick symmetry centered on a face, diagonal orientation
x=3 y=3 z=2 rule=3D0..6/F
o$bo$2bo/o$bo$2bo!

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3D version=1 size=11 pos=3,3,4
#C ±C4_fc: square proprismatic symmetry centered on a face
x=5 y=5 z=2 rule=3D0..6/F
3bo$obo$bobo$2bobo$bo/3bo$obo$bobo$2bobo$bo!

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3D version=1 size=9 pos=3,3,3
#C ±C2_fc: espic symmetry centered on a face, axis goes through cell
x=3 y=3 z=2 rule=3D0..6/F
o$3o$2bo/o$3o$2bo!

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3D version=1 size=9 pos=4,3,3
#C ±C2_fe: espic symmetry centered on a face, axis goes through edge
x=1 y=3 z=2 rule=3D0..6/F
o$o/$o$o!

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3D version=1 size=9 pos=3,3,3
#C ±C2_fk: espic symmetry centered on a face, axis goes through vertex
x=3 y=3 z=2 rule=3D0..6/F
$bo$2bo/o$bo!

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3D version=1 size=9 pos=3,3,3
#C ±C1_f: central inversion symmetry centered on a face
x=3 y=3 z=2 rule=3D0..6/F
$b2o$2bo/o$2o!

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3D version=1 size=9 pos=3,3,3
#C DD8_fe: digonal antiprismatic symmetry centered on a face, digon is diagonal
x=3 y=3 z=2 rule=3D0..6/F
o$bo$2bo/2bo$bo$o!

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3D version=1 size=9 pos=3,3,3
#C DD8_fk: digonal antiprismatic symmetry centered on a face, digon is orthogonal
x=3 y=3 z=2 rule=3D0..6/F
$obo/bo2$bo!

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3D version=1 size=10 pos=3,3,3
#C CC4_fc: prodappic symmetry centered on a face
x=3 y=3 z=4 rule=3D0..6/F
2bo2$o/$obo/bo2$bo/o2$2bo!

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3D version=1 size=10 pos=4,4,3
#C CD4_fe: digonal pyramidal symmetry (line goes through faces and edges)
x=1 y=2 z=4 rule=3D0..6/F
$o/o/o/$o!

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3D version=1 size=10 pos=4,4,3
#C CD4_fk: digonal pyramidal symmetry (line goes through faces and vertices)
x=2 y=2 z=4 rule=3D0..6/F
$bo/o/o/$bo!

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3D version=1 size=9 pos=3,3,3
#C CC2_f: reflection (plane goes through faces, edges, and vertices)
x=3 y=2 z=2 rule=3D0..6/F
2o$2bo/2o$2bo!

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3D version=1 size=10 pos=3,3,3
#C D8_e: chiral square prismatic symmetry centered on an edge
x=3 y=4 z=4 rule=3D0..6/F
$o$2bo/2bo$bo$bo$o/o$bo$bo$2bo/$2bo$o!

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3D version=1 size=10 pos=3,3,4
#C D4_ef: chiral brick symmetry centered on an edge, orthogonal orientation
x=3 y=4 z=2 rule=3D0..6/F
2bo$bo$bo$o/o$bo$bo$2bo!

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3D version=1 size=10 pos=3,3,3
#C D4_ec: chiral brick symmetry centered on an edge, diagonal orientation
x=3 y=4 z=4 rule=3D0..6/F
$o/2bo$bo/2$bo$2bo/2$o!

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3D version=1 size=10 pos=4,3,3
#C C4_ek: chiral square pyramidal symmetry, a.k.a., 90 degree rotation (line goes through edges and vertices)
x=2 y=4 z=4 rule=3D0..6/F
2$bo/bo$o$o/$o$o$bo/$bo!

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3D version=1 size=10 pos=4,4,3
#C C2_ek: chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through edges and vertices)
x=2 y=2 z=4 rule=3D0..6/F
o/bo/$bo/$o!

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3D version=1 size=8 pos=3,3,3
#C ±D8_e: square prismatic symmetry centered on an edge
x=1 y=2 z=2 rule=3D0..6/F
o$o/o$o!

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3D version=1 size=10 pos=4,3,4
#C ±D4_ef: brick symmetry centered on an edge, orthogonal orientation
x=1 y=4 z=2 rule=3D0..6/F
o$o$o$o/o$o$o$o!

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3D version=1 size=8 pos=3,3,3
#C ±D4_ec: brick symmetry centered on an edge, diagonal orientation
x=1 y=2 z=2 rule=3D0..6/F
o/$o!

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3D version=1 size=10 pos=4,3,3
#C ±C4_ek: square proprismatic symmetry centered on an edge
x=1 y=4 z=4 rule=3D0..6/F
2$o/o$o$o/$o$o$o/$o!

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3D version=1 size=10 pos=4,3,4
#C ±C2_ek: espic symmetry centered on an edge, axis goes through vertex
x=1 y=4 z=2 rule=3D0..6/F
o$o/2$o$o!

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3D version=1 size=9 pos=3,3,3
#C ±C2_ef: espic symmetry centered on an edge, axis goes through face
x=3 y=2 z=2 rule=3D0..6/F
b2o$2o/b2o$2o!

Code: Select all

3D version=1 size=9 pos=3,3,3
#C ±C2_ec: espic symmetry centered on an edge, axis goes through cell
x=3 y=2 z=2 rule=3D0..6/F
b2o/$2o!

Code: Select all

3D version=1 size=10 pos=3,3,4
#C ±C1_e: central inversion symmetry centered on an edge
x=3 y=4 z=2 rule=3D0..6/F
2bo$bo/2$bo$o!

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3D version=1 size=9 pos=3,3,3
#C DD8_ef: digonal antiprismatic symmetry centered on an edge, digon is diagonal
x=3 y=2 z=2 rule=3D0..6/F
b2o$2o/2o$b2o!

Code: Select all

3D version=1 size=10 pos=3,3,3
#C DD8_ec: digonal antiprismatic symmetry centered on an edge, digon is orthogonal
x=3 y=4 z=4 rule=3D0..6/F
$o$o/2bo$bo$bo$2bo/2bo$bo$bo$2bo/$o$o!

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3D version=1 size=10 pos=4,3,3
#C CD8_ek: square pyramidal symmetry (line goes through edges and vertices)
x=2 y=4 z=4 rule=3D0..6/F
$bo$bo/bo$o$o$bo/bo$o$o$bo/$bo$bo!

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3D version=1 size=10 pos=3,3,3
#C CC4_ek: prodappic symmetry centered on an edge
x=3 y=4 z=4 rule=3D0..6/F
2$o/2bo$bo$bo/$bo$bo$2bo/$o!

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3D version=1 size=10 pos=4,3,4
#C CD4_eko: digonal pyramidal symmetry (line goes through edges and vertices), orthogonal orientation
x=2 y=4 z=2 rule=3D0..6/F
bo$o$o$bo/bo$o$o$bo!

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3D version=1 size=10 pos=4,3,3
#C CD4_ekd: digonal pyramidal symmetry (line goes through edges and vertices), diagonal orientation
x=2 y=4 z=4 rule=3D0..6/F
bo/$o/2$o/3$bo!

Code: Select all

3D version=1 size=12 pos=3,3,3
#C O24_k: chiral cubic symmetry centered on a vertex
x=6 y=6 z=6 rule=3D0..6/F
$2bo$4bo$bo$3bo/3bo$bo2bo$o$5bo$bo2bo$2bo/bo$5bo3$o$4bo/4bo$o3$5bo$bo/2bo$bo2bo$5bo$o$bo2bo$3bo/$3bo$bo$4bo$2bo!

Code: Select all

3D version=1 size=12 pos=3,3,3
#C T12_k: chiral tetrahedral symmetry centered on a vertex
x=6 y=6 z=6 rule=3D0..6/F
$3bo3$2bo/2$3bobo$obo/4bo$3bo$4bo$bo$2bo$bo/bo$2bo$bo$4bo$3bo$4bo/2$obo$3bobo/$2bo3$3bo!

Code: Select all

3D version=1 size=10 pos=3,3,3
#C D8_k: chiral square prismatic symmetry centered on a vertex
x=4 y=4 z=4 rule=3D0..6/F
2bo$o$3bo$bo/$b2o$b2o/$b2o$b2o/bo$3bo$o$2bo!

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3D version=1 size=12 pos=4,3,5
#C D4_ke: chiral brick symmetry centered on a vertex, orthogonal orientation
x=4 y=6 z=2 rule=3D0..6/F
bo$o$o$3bo$3bo$2bo/2bo$3bo$3bo$o$o$bo!

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3D version=1 size=10 pos=3,3,4
#C D4_kf: chiral brick symmetry centered on a vertex, diagonal orientation
x=4 y=4 z=2 rule=3D0..6/F
$2o$2b2o/bo$bo$2bo$2bo!

Code: Select all

3D version=1 size=10 pos=3,3,3
#C D6_kc or simply D6_k: chiral triangular prismatic symmetry centered on a vertex
x=4 y=4 z=4 rule=3D0..6/F
2$bo/2bo$bo2$2bo/$o2bo$2bo/2$bo!

Code: Select all

3D version=1 size=8 pos=3,3,3
#C ±O24_k: cubic symmetry centered on a vertex
x=2 y=2 z=2 rule=3D0..6/F
2o$2o/2o$2o!

Code: Select all

3D version=1 size=12 pos=3,3,3
#C ±T12_k: pyritic symmetry centered on a vertex
x=6 y=6 z=6 rule=3D0..6/F
2$bo2bo$bo2bo/2b2o$bo2bo3$bo2bo$2b2o/$o4bo3$o4bo/$o4bo3$o4bo/2b2o$bo2bo3$bo2bo$2b2o/2$bo2bo$bo2bo!

Code: Select all

3D version=1 size=10 pos=4,4,3
#C ±D8_k: square prismatic symmetry centered on a vertex
x=2 y=2 z=4 rule=3D0..6/F
2o$2o/2o$2o/2o$2o/2o$2o!

Code: Select all

3D version=1 size=10 pos=3,3,4
#C ±C4_ke: square proprismatic symmetry centered on a vertex
x=4 y=4 z=2 rule=3D0..6/F
2bo$3o$b3o$bo/2bo$3o$b3o$bo!

Code: Select all

3D version=1 size=10 pos=3,3,4
#C ±D4_ke: brick symmetry centered on a vertex, orthogonal orientation
x=4 y=4 z=2 rule=3D0..6/F
o2bo$4o$4o$o2bo/o2bo$4o$4o$o2bo!

Code: Select all

3D version=1 size=8 pos=3,3,3
#C ±D4_kf: brick symmetry centered on a vertex, diagonal orientation
x=2 y=2 z=2 rule=3D0..6/F
o$bo/o$bo!

Code: Select all

3D version=1 size=8 pos=3,3,3
#C ±D6_kc or simply ±D6_k: triangular antiprismatic symmetry centered on a vertex
x=2 y=2 z=2 rule=3D0..6/F
o/$bo!

Code: Select all

3D version=1 size=10 pos=3,3,3
#C ±C3_kc or simply ±C3_k: triangular proantiprismatic symmetry centered on a vertex
x=4 y=4 z=4 rule=3D0..6/F
2$bo/2bo$bo$3bo/$o$2bo$bo/$2bo!

Code: Select all

3D version=1 size=10 pos=3,4,4
#C ±C2_ke: espic symmetry centered on a vertex, axis goes through edge
x=4 y=2 z=2 rule=3D0..6/F
2o$2o/2b2o$2b2o!

Code: Select all

3D version=1 size=10 pos=4,4,3
#C ±C2_kf: espic symmetry centered on a vertex, axis goes through face
x=2 y=2 z=4 rule=3D0..6/F
o/o/$bo/$bo!

Code: Select all

3D version=1 size=10 pos=3,4,4
#C ±C1_k: central inversion symmetry centered on a vertex
x=4 y=2 z=2 rule=3D0..6/F
o$bo/2bo$3bo!

Code: Select all

3D version=1 size=8 pos=3,3,3
#C TO24_k: tetrahedral symmetry centered on a vertex
x=2 y=2 z=2 rule=3D0..6/F
o$bo/bo$o!

Code: Select all

3D version=1 size=10 pos=4,4,3
#C DD8_ke: digonal antiprismatic symmetry centered on a vertex, digon is diagonal
x=2 y=2 z=4 rule=3D0..6/F
o$bo/o$bo/bo$o/bo$o!

Code: Select all

3D version=1 size=10 pos=3,3,4
#C DD8_kf: digonal antiprismatic symmetry centered on a vertex, digon is orthogonal
x=4 y=4 z=2 rule=3D0..6/F
$o2bo$o2bo/b2o3$b2o!

Code: Select all

3D version=1 size=10 pos=3,3,3
#C CC4_ke: prodappic symmetry centered on a vertex
x=4 y=4 z=4 rule=3D0..6/F
$o$3bo/$bo$2bo/$2bo$bo/2bo3$bo!

[muzik]

To my knowledge, linking from a mainspace article to a userspace article is usually frowned upon; is there anything that needs to be done to the two 4D symmetry enumeration pages before they can be moved to the mainspace like was done for the 3D symmetries page?

[gzaytman]

To my knowledge, linking from a mainspace article to a userspace article is usually frowned upon;

I don't understand the purpose of that rule. It would strike me that linking to the user page from mainspace would help people find who else is interested in a niche topic.

is there anything that needs to be done to the two 4D symmetry enumeration pages before they can be moved to the mainspace like was done for the 3D symmetries page?

Not particularly. I'm not sure what qualifies as mainspace worthy.

[muzik]

is there anything that needs to be done to the two 4D symmetry enumeration pages before they can be moved to the mainspace like was done for the 3D symmetries page?

Not particularly. I'm not sure what qualifies as mainspace worthy.

They both seem fine to me, I was wondering if you were keeping them in the userspace on the basis that they were incomplete (perhaps you planned to enumerate spaceship symmetries as well and didn't want to move to the mainspace until those were all listed).

[muzik]

It's a huge struggle with the mess the wiki is in nowadays but I've tried arranging cubic symmetries into a table to improve readability.

Regarding spaceship symmetries: does "any in orthogonal/diagonal plane" mean any direction including knightwise inside the plane, or only symmetric directions inside of the plane (orthogonal, diagonal)? Time-symmetric knightships seem like a rather strange concept I hadn't yet considered.

[gzaytman]

Regarding spaceship symmetries: does "any in orthogonal/diagonal plane" mean any direction including knightwise inside the plane, or only symmetric directions inside of the plane (orthogonal, diagonal)?

Any direction.

Time-symmetric knightships seem like a rather strange concept I hadn't yet considered.

They're knightships in a plane and time-symmetric in the direction orthogonal to that plane.

[muzik]

Counting symmetries is easy enough, but is it possible to compute the number and types of unique "transformations", i.e. the things LifeViewer provides when identifying something?

They're not one-to-one with kinetic symmetries, e.g. n.c, n.e and n.k 3both map to Rot180 (since they all involve 180-degree rotation), whereas a n/ object could be either Flip/ or Flip\.

For higher dimensions, I assume there would be some other distinctions to be made. For example, on 4D grids with 8-fold and 12-fold rotational symmetry, an object could rotate 1/8 or 1/12 of a full turn in that fraction of its period, but it could also rotate 3/8 or 5/12 of a full turn in p/8 or p/12 which would result in the same kinetic symmetry but a different kind of transformation that yields that symmetry.

Do these things have a name? I don't remember Chris giving them one.

[gzaytman]

Counting symmetries is easy enough, but is it possible to compute the number and types of unique "transformations", i.e. the things LifeViewer provides when identifying something?

I'm not quite sure what you mean. Do you just mean the order symmetry group modulo translations or something else?