Cube faces with orientation

Exploring the influence of a cube's faces orientation on the corners and edges
Rubik's Cube Group theory Geometry Platonic solids

As I was working further on the Sudokube problem (I realized my previous solution was ot optimal, as there were some duplicate edge pieces) and I really wanted to look deeper into the consequences of merging two quite tough problems, I set myself to make a catalog of the different edge and corner configurations of a cube with oriented faces. This is not exhaustive, because there are a few (at least one) other configurations that will look a little less ordered than those below.

I found it interesting to notice how some configurations have few types of either corners or edges (it’s usually one or the other). While naming them, I took care to have a consistent naming scheme and to avoid duplicates between each orientation set, so I could reuse them at a later time (at the time of writing, I am still not sure how to fit this in the new script I’m working on).

Also, it must be noted that I took as equivalent the mirror versions of these ; it is interesting to see that there are different (and not just one) ways to apply a standard “mirror” transform : I considered equivalent the case where all arrows are inverted (up becomes down and vice-versa, same for left and right : this means each face is by rotated 180° or 2𝞹) ; it could be worth investigating what happens with quarter-turns - and also with all platonic solids).

Base configurations

These are the “nice” ones, pretty musch as symmetrical as they can get.

Note: I just realized arrow heads are not displayed as expected.. fix will come soon.

Double-helix configuration

This is what seemed, by instinct, the cleanest way to go until I realized it was a quite arbitrary choice. In this configuration, there is one clockwise corner and one counter-clockwise corner, opposed to each other. There are two other types of corners, which I nicknamed goofie and regular (yes, I do snowboard and/or skateboard at times) which are not the mirror of one another : that would mean there should exist at least one other variant of this one, which might be called inverted double-helix configuration.

This configuration is pretty straight-forward and intuitive, with 4 different edge types and three of each, along with two additional types of corners (besides clockwise and counter-clockwise) and three of each too.


Single-helix configuration

I obviously wanted a cube where the two opposing corners are both either clockwise or counter-clockwise: this is the single-helix configuration. Interstingly enough, there is only one type of corners here (besides the two opposing CW or CCW corners)! Mirroring the cube will change the orientation of the two opposing corners, but not whether they are diverging or converging. Edge configuration surprised me here, as there are 5 groups of 1, 1, 2, 3 and five members.


Bilinear configuration

In this case, we consider two continuous stripes of continous orientation covering three faces each, sort-of like the way a tennis ball is made. This one is interesting, as it has four types of corners (two each) ; there are six types of edges (1, 1, 2, 2, 2 and 4 members).


Trilinear configuration

I couldn’t stop there, and added a configuration where there are three stripes of continuous orientation. Six types of corners here, of which one (goofie) we have already met with the double-helix configuration ; another is completely new (dubbed dumbo, occurs once), then two pairs of additional types (one of which we already encounted in the bilinear configuration. As for the edges, there are no less than 8(!) different types of edges (1, 1, 1, 1, 1, 2, 2 and 3 members).


Other configurations

As stated before, the configurations shown above have variants (with face rotations) that need to be investigated. I suspect there are additional variants, but this would require a systematic search that takes into account the equivalence of each face : using the normalized edge and corner configurations to explore them seems to be a good way to go.

Prismatic configuration

In this configuration (or rather, family of) there is a stripe of continuous orientation around the cube ; the bottom side can be oriented either way, but the top side can have 3 different orientations relative to the bottom (remember we are not counting the mirror transform). This is still to be done.

The question that remains

We now have 7 distinct configurations, but I am pretty sure there is at least one other. Care to help?