ConwayLife.com

I assume that in general, point reflection is isomorphic to the process of reflecting an n-dimensional object across n perpendicular planes, with the point everything is reflected through represented by the point at which every plane intersects? Since an even number of reflections cancel each ...

I'm not sure if this would be overly complex to make, but a graph similar to File:Symmetries.png may be useful to include on the page to demonstrate which symmetries are class subclasses of other symmetries, since with the current layout it is entirely plausible that one symmetry could be ...

Is there a way to automate soup search in 3D rules?
I want so see if there is anything beyond the period 2 and period 3 oscillators that show up in 3D3..6/6,13,15,17,19.


Well, the SAT-solver based code I used to fill up the OmniOsci Oscillator table a while ago theoretically works in any ...

Slightly off topic, but this is the closest thread topic to anything of this nature.
Has anyone investigated symmetries in the other honeycombs/3D tesselations with congruent cells and vertices, namely Hexagonal Prismatic, Triangular Prismatic, Gyrated Triangular Prismatic, and Bitruncated Cubic ...

Is point reflection counted separately from 180-degree rotation due to it also turning the object into its mirror image,


Point reflection is ±C1 , 180-degree rotation is C2 variously decorated to indicate where the point or line is.


and does this distinction disappear in 4D since double ...

Ok, this list starts with the nontrivial elements of the symmetry group modulo translations.
Then we have XOrY type transformations for composite symmetries where the static symmetry is nontrivial.

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?

speedydelete wrote: December 24th, 2025, 12:23 pm I've created a 5S website! Please do not put in large spaceships, or large amounts of spaceships. It is at https://speedydelete.com/5s/. Additionally, everything posted in this thread should be added, as well as the >1000 NRSS rule's ships.

How does this page relate to the old 5S page?

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 ...

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 ...

Is there a general way to calculate the symmetries (static and kinetic) of any 2D hyperbolic riling {p,q}?

For prime p and q (e.g. {7,3}, {5,7}, {11,3}, ...) I assume we just need to account for mirror symmetries (all valid lines pass through a vertex and opposing edges of cells), as well as ...

Is there a general way to calculate the symmetries (static and kinetic) of any 2D hyperbolic riling {p,q}?

For prime p and q (e.g. {7,3}, {5,7}, {11,3}, ...) I assume we just need to account for mirror symmetries (all valid lines pass through a vertex and opposing edges of cells), as well as ...

In 4D, the eight cell-adjacent cells once again just return the same tesseract honeycomb. I assume the 24 face-adjacent cells result in the 24-cell tiling?


Correct.


What do the 32 edge-adjacent cells and 16 vertex-adjacent cells result in?


The 32 edge-adjacent cells give the original ...

This neighbourhood is equivalent to the truncated octahedral tiling of space.
This is also what the 8-cell "vertices only" neighbourhood corresponds to, is it not (assuming we only consider cells adjacent to hexagonal faces, and there are eight non-interacting universes)?

Correct. However, the ...

New neighbourhood suggestion: currently hexagonal rules are simulated using a square grid neighbourhood which resembles the intersection of two 2x2 squares. How about adding a new neighbourhood which comes about as the intersection of two 2x2x2 blocks (with the one intersecting cell being the ...

muzik wrote: September 29th, 2025, 2:28 pm Are the "edges-only" and "hexahedral" neighbourhood identical, in the sense that they both emulate the rhombic dodecahedral honeycomb, and the former just does it in a way that results in the grid being partitioned into two disconnected checkerboards?

Yes.

Is it at all possible to reduce the bounding boxes for the nrc and nrk examples?

Ok I got the nrc down to a 3x3 bounding box for period up to 56 (see the Google doc).
For period 60 the SAT solver assures me there are no min-pop 3 oscillators that fit within a 3x3 box.

Edit: Same for periods 64 ...

I hooked up a SAT solver onto this problem and it turned out to be rather effective.

#C C1 --> C2_1
x = 5, y = 5 rule = B6/S1234L
$b2obo$3bo$b2o$ob2o!

#C C1 --> C2_4
x = 6, y = 5, rule = B6/S123L
$2bo$b3o$4b2o$4bo!

#C D2_xo --> D4_x1
x = 8, y = 5, rule = B7/S1234L
$4bo$bob2o2bo$bob2o2bo$4bo!

#C D2_xo --> D4_x4
x = 7, y = 5, rule = B4/S013L
$o5bo$3b2o$3b2o$o5bo!

#C D2_x ...

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

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 ...

muzik wrote: October 9th, 2025, 8:10 pm 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.

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 ...