# Message #56

From: rbreiten <rbreiten@yahoo.com>

Subject: Re: [MC4D] Orientations of the centre cubes

Date: Mon, 24 Nov 2003 22:42:22 -0000

— In 4D_Cubing@yahoogroups.com, David Vanderschel <DvdS@A…> wrote:

> So did I. Oddly, Jay had posted the same comment back

> in September:

It seemed so familiar that I checked to make sure I wasn’t replying

to an accidental double post.

Anyway, let u f r d b l refer to clockwise quarter turns of up,

front, right, down, back, and left faces of 4^3 cube as usual, and

let U F R D B L refer to clockwise quarter turns of the layer

adjacent to the face with lowercase name (so uU == dD plus a spatial

reorientation). Let u’ and u2 refer to anticlockwise quarter turn

and half turn of the u face, etc.

To rotate just the up face of the internal cube a half turn, perform

U2.f2r2.U2.rl’du’fb’r2f’bd’ur’lf2.U2.r2.U.r’f2r2fr’f’r’f2ldrd’l’.U2.d’

f2d2fd’f’d’f2uldl’u’.U.f2

The first U2 is our half turn of internal u face along with some

naughty stuff in the shell, and the rest unscrambles all the naughty

stuff without permanently affecting the internal cube.

The first longish subsequence rotates the center four 1-color squares

of the f and r faces a half turn while leaving the rest of the cube

fixed. When conjugated by U2, this restores the 1-color cubies to

their original positions. The only tricky point is that exponent

counting seems to show that f and r faces end up rotated a half turn

total, but the conjugation by U2 kind of distributes these half turns

to the l and b faces in a way that makes everything work out (try it

on paper, or if you can bear to look at it, mark up your 4^3). The

last U2 is really U’U’, and the last two longish subsequences are

edge pair flips (the same one in the bottom face and two in the

middle layer), that when conjugated by U and U’ respectively, restore

the middle edges that were disrupted by the first U2.

There is still a bug in my sequence for rotating two adjacent faces

of the internal cube a quarter turn each (wrote it down late late

last night). I’ll work it out tonight and post it tomorrow.

I’m testing these at work via

http://www.nrr.co.uk/rubik/cuboid/?planes=444

which has the nice property of taking keyboard input (left my cube at

home). It uses shift for anticlockwise and ctrl for inner layer.

But I do need to get *some* work done today.

rb