3D cubic grid symmetries and naming

[muzik]

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

If those are the same thing as what LifeViewer gives for identified oscillators and spaceships, yes. Here's a list for oscillators on a square tiling:

Code: Select all

FlipX: n-c, n-e
FlipY: n-c, n-e
Flip⟍: n/
Flip⟋: n/
Rot180: n.c, n.e, n.k
Rot90CW: nrc, nrk
Rot90CCW: nrc, nrk
FlipXOrRot180: -c+c, -c+e, -e+e, -e+k
FlipYOrRot180: -c+c, -c+e, -e+e, -e+k
Flip⟍OrRot180: /xc, /xk
Flip⟋OrRot180: /xc, /xk
FlipOrth: .c+c, .e+e, .k+k
FlipDiag: .cxc, .kxk
Rot90: .crc, .krk
FlipOrthOrRot90: xc*c, xk*k
FlipDiagOrRot90: +c*c, +k*k
FlipOrthOrDiag: rc*c, rk*k

There isn't such a system for the hexagonal or triangular grid (because LifeViewer identification doesn't currently support mod calculation). Perhaps I should invent one ahead of time...

For the cubic grid, we'd have cases such as "FlipX", "FlipY" and "FlipZ", which would correspond to mirror transformations on any of three orthogonal planes, regardless of whether that plane is one that passes through cell centers or one that passes through cell vertices.

[gzaytman]

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.

[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.

This should have been completed months ago but I only ended up finalizing this yesterday. I may also add images at some point for ease of viewing (since we don't have a 3D viewer).

The main thing I want to know at this point are 4D spaceship symmetries - I assume there are more of these than there are 3D oscillator symmetries, is is the case for 3D spaceship symmetries versus 2D oscillator symmetries, or 2D spaceship symmetries versus 1D oscillator symmetries?

[muzik]

An image has now been uploaded for each of the 92 distinct symmetry types. I don't know if they're all visually clear enough, especially since it's hard to position the more complex objects such that their symmetries are obvious, but this should be sufficient for now.

Is point reflection counted separately from 180-degree rotation due to it also turning the object into its mirror image, and does this distinction disappear in 4D since double rotations permit further manipulation?

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 included on multiple tables (e.g. C1 doesn't fix a cell, edge, face or vertex, so therefore could have been placed in any, and only appears in the cell-fixing table because it's sequentially the first), and a graph-based visual classification approach would demonstrate these relationships more understandably.

[Rhombicubocta]

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 honeycombs.

[gzaytman]

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 rotations permit further manipulation?

In 4D double 180-degree rotation is the same as point reflection.it's denoted ±[C1×C1], there's also single 180-degree rotation +1/2[C2×C2].

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 included on multiple tables (e.g. C1 doesn't fix a cell, edge, face or vertex, so therefore could have been placed in any, and only appears in the cell-fixing table because it's sequentially the first), and a graph-based visual classification approach would demonstrate these relationships more understandably.

I considered making one but decided a graph with 92 nodes would be too visually cluttered. I could still make one if there's interest.

[gzaytman]

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 honeycombs.

I investigated the Bitruncated Cubic Honeycomb, a.k.a. the Truncated Octahedron Honeycomb, a.k.a. the Body-Centered Cubic, a.k.a. D3*, a.k.a. A3*, a.k.a. 3D.lua's "vertices only" neighborhood, it has
61 static symmetries,
132 with period/mod=2,
5 with period/mod=3,
8 with period/mod=4, and
3 with period/mod=6.

I also investigated its dual lattice the Rhombic Dodecahedral Honeycomb, a.k.a. the Face-Centered Cubic, a.k.a. D3, a.k.a. A3, a.k.a. 3D.lua's "edges-only" neighborhood, a.k.a. 3D.lua's "hexahedral" neighborhood, it has
69 static symmetries,
147 with period/mod=2,
8 with period/mod=3,
8 with period/mod=4, and
4 with period/mod=6.

[gzaytman]

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 included on multiple tables (e.g. C1 doesn't fix a cell, edge, face or vertex, so therefore could have been placed in any, and only appears in the cell-fixing table because it's sequentially the first), and a graph-based visual classification approach would demonstrate these relationships more understandably.

I considered making one but decided a graph with 92 nodes would be too visually cluttered. I could still make one if there's interest.

Yes, I just made the graph File:3d Symmetries.png. It's not very helpful.

[muzik]

In 4D double 180-degree rotation is the same as point reflection.it's denoted ±[C1×C1], there's also single 180-degree rotation +1/2[C2×C2].

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 other out, in 2D, 4D, 6D and so on a point reflection is identical to performing n/2 180-degree rotations in n/2 different 2D planes, whereas in 1D, 3D, 5D, ... the fact that there's an odd number of reflection planes means that the object is mapped to its enantiomorph, making point reflection its own distinct thing?

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 honeycombs.

I investigated the Bitruncated Cubic Honeycomb, a.k.a. the Truncated Octahedron Honeycomb, a.k.a. the Body-Centered Cubic, a.k.a. D3*, a.k.a. A3*, a.k.a. 3D.lua's "vertices only" neighborhood

I also investigated its dual lattice the Rhombic Dodecahedral Honeycomb, a.k.a. the Face-Centered Cubic, a.k.a. D3, a.k.a. A3, a.k.a. 3D.lua's "edges-only" neighborhood, a.k.a. 3D.lua's "hexahedral" neighborhood

For the rhombic dodecahedral honeycomb is this equivalent to 2-coloring the cubic honeycomb and discarding all of the cubic grid symmetries that would swap one colour for another? Likewise, would the truncated octahedron honeycomb be equivalent to doing the same thing for a vertex-based 4-coloring?

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 included on multiple tables (e.g. C1 doesn't fix a cell, edge, face or vertex, so therefore could have been placed in any, and only appears in the cell-fixing table because it's sequentially the first), and a graph-based visual classification approach would demonstrate these relationships more understandably.

I considered making one but decided a graph with 92 nodes would be too visually cluttered. I could still make one if there's interest.

Yes, I just made the graph File:3d Symmetries.png. It's not very helpful.

As convoluted as this looks it definitely makes things clearer than before. I wonder if an approach more akin to File:Kinetic symmetry diagram.png would be better, such that symmetries of the same degree are colour-coded rather than positioned at the same y level.

There is one question that was raised on the polytope discord that may be relevant to ask here: while all of the minimal example patterns are connected at least by their vertices or edges, what would the minimal possible polycube be for each of the 92 symmetry types?

[gzaytman]

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 other out, in 2D, 4D, 6D and so on a point reflection is identical to performing n/2 180-degree rotations in n/2 different 2D planes, whereas in 1D, 3D, 5D, ... the fact that there's an odd number of reflection planes means that the object is mapped to its enantiomorph, making point reflection its own distinct thing?

Correct.

For the rhombic dodecahedral honeycomb is this equivalent to 2-coloring the cubic honeycomb and discarding all of the cubic grid symmetries that would swap one colour for another? Likewise, would the truncated octahedron honeycomb be equivalent to doing the same thing for a vertex-based 4-coloring?

Both of those are correct

As convoluted as this looks it definitely makes things clearer than before. I wonder if an approach more akin to File:Kinetic symmetry diagram.png would be better, such that symmetries of the same degree are colour-coded rather than positioned at the same y level.

Well, I split the 92-node graph into four subgraphs corresponding to the groups that fix a cell, face, edge, or vertex.