Date: 30 DEC 1980 0152-EST
From: DCP at MIT-MC (David C. Plummer)
Subject: Notes on transforms on the 4x4x4 and 5x5x5 cubes. (65 lines)
To: CUBE-LOVERS at MIT-MC
First we have to define what a twist is. I propose that a twist
is a twisting of face or faces about an axis such that there is
only one plane on which sliding friction happens. This means that
slice turns are not single turns, but rather multiple turns. This
is consistent with the 3x3x3 definition, and it is generalizable
to nxnxn. Also, I think it is wisest to consider quarter twists
as the only single twist "move" (don't count half twists as one!!)
One thing to note: Even order cubes should have a STANDARD
coloring. This is nice to have since there is now visible axis to
align the corners on. I know it isn't necessary, but if I own a
cube and you own a cube, and they are colored differently, and we
swap for a day, both of us will probably have a hard time trying
to get used to a different color arrangement than what each is
used to. May I suggest
Front Right Up Back Left Down
RED WHITE BLUE ORANGE YELLOW GREEN
The general PONS can be described as follows (for cubes, not
necessarily dodecahedra):
Pick an axis. Make 180 twists along all twistable planes.
Pick another axis. Do the same.
Do the same on the last axis.
This will produce a checkerboard pattern on all faces using
complementary (from the opposite side of the cube) colors. On the
4x4x4 (and in general on even order cubes), to corner cubies do
not move (but then again it is hard to tell without a fixed
reference).
I have done more thinking about the 5x5x5 than the 4x4x4 since
the 5x5x5 offers several immediate advantages. First of all, from
solved we can do things like: do a transform as we would on a
3x3x3 and consider the 3x3 set of cubies in the center of the
faces as the center of our favorite 3x3x3 Hungarian. Another way
to view this is to do normal Hungarian rotations using only one
level deep of cubies. Then run the algorithm backwards doing
twists using two levels deep (considering theface center as a
center and 2x2 at the corners as corners, and 1x2 around the
edges as edges. (Bad explanation, but it is 1:30 in the morning).
I believe doing the traditional SIX-DOTS pattern forward then
backward using this method will produce something that looks
like:
+++++ +++++
+@@@+ +@@@+
+@+@+ +@x@+
+@@@+ +@@@+
+++++ +++++
The one on the right is what happens if you do it again in the
same direction. I think the Plummer's Cross will look like:
+@+@+
@@+@@
+++++
@@+@@
+@+@+
And if I am not mistaken, the approriate EXTENDED-SIX-DOTS
mentioned above performed on this will produce a 3-cycle
checkerboard. And the PONS on top of that will produce a 6-cycle
checkerboard. Of course, I could be wrong, but we wont know until
(a) somebody carfully works it out on paper (b) somebody writes a
computer program to simulate the crazy thing, or (c) somebody
actually builds one.
That's it for tonight folks, aren't you glad!!