I've been recently investigating a set of patterns that I call
Swirl patterns for lack of a better name. In a Swirl, each face
looks like one of these:
X X X X X X X Y Z (Left-hand swirl) Z Y X (Right-hand swirl) Z Z Z Z Z Z
where X and Z are complementary colors, and Y is something else.
I've classified them roughly into 6 classes, based on handedness
of swirl and relative alignment of faces.
If you look at the 3 faces adjacent to a corner, they may have the
same handedness, or they may have mixed handedness. In addition,
two adjacent faces may have parallel or perpendicular swirls.
(Parallel swirls have their XYZ columns parallel; perpendicular
swirls have thir XYZ columns perpendicular.)
Here are my 6 classes; there are another 6 which are mirror images
of these, but I don't count them. Nor do I care (at the moment)
about color pairings - though I know they are important - or about
the colors of the face cubies, which probably aren't important.
1. Same handedness. Two of the three faces have parallel swirls.
2. Same handedness. All three faces have mutually perpendicular
3. Mixed handedness. The two same-handed faces are parallel, with
thir XYZ columns in contact (i.e. forming a belt around the cube).
4. Mixed handedness. The two same-handed faces are perpendicular;
the opposite-handed face is perpendicular to both.
5. Mixed handedness. The two same-handed faces are perpendicular;
the opposite-handed face is parallel to one.
6. Mixed handedness. The two same-handed faces are parallel, with
their XYZ columns pointing towards the third face.
Four of these classes are in the antislice group and are a short
distance (8 qtw) away from SOLVED. They are classes 1, 3, 5 and 6.
They are also the antislice group's analogues of the slice group's
"6-spot" or "twelve-square" patterns.
What got me started on this is a problem that I still have. One
day while playing aimlessly in the antislice group (I thought I
had remained in it) I ran across a class 2 Swirl, which was (a)
quite pretty (when looked at from the correct corner it looks
like a pinwheel) and (b) a bear to solve. (Clearly I thought it
was one of the "easy" Swirls.)
Having solved it, I wanted to get back to it and found I didn't
know how. I tried solving to it and came up with an impossibility
- that's how I know the color arrangements must be important - and
I haven't found it in my searches yet - nor have I found class 4.
Questions: what's the fastest way to get to a class 2 Swirl? What
color arrangements are permissible? Is it really in the antislice
group? (I now believe not.) Is any class 4 Swirl achievable? How
quickly? Is there anything else interesting about Swirls?
I'm still playing with these - will give more data as I get it.