Date: Sat, 19 Jul 80 01:42:56 +0400
From: Allan C. Wechsler <ACW@MIT-AI >
~~~ Subject: Re: Where to get them in the Boston Area, Cube Language.

"Games People Play" on Mass Ave, near the fork where Mt. Auburn
St. branches off, carred them the last I knew. According to
rumor, the domestic product ("Rubik's Cube"), selling for
about \$7, is mechanically inferior to the Hungarian import,
costing \$15. I don't know how everyone feels about money,
but to me, not having to fight with the thing would be worth
the extra \$8.

Bernie's explanation of how to achieve the Plummer and Christman
Crosses is a prime example of why we need a cube language.

Since no one has proposed anything, I will jump into the fray.

Center the cube at the origin of a 3-d coordinate system. The
axes of the coordinate system protrude from the centers of the faces.

Make a hitch-hiker's gesture with your right hand and point the thumb
along the X axis. Imagine rotating the WHOLE CUBE one quarter turn
in the direction your curled fingers are pointing. I call this operation
"I". (The X axis is the horizontal axis that does not skewer you.)

If the cube was lying on a table, "I" would roll it toward you.

Now point up, along the Y axis, with your thumb. A quarter-rotation
in the direction your curled fingers point is the operation I call "J".

The Z axis goes right through your belly. A quarter turn around it
I call "K". Actually, K=IIIJI.

To simplify things a little, we define I'=III, J'=JJJ, and K'=KKK.
In general, M' is M done backwards. If we call the do-nothing
manipulation "1", then MM'=M'M=1.

For my own nefarious reasons I define "H" as the operation (unachievable
in real life) of REFLECTING the cube right-to-left through the YZ plane.
We note H'=H.

Twisting the front (Z=1) face 90 degrees counter-clockwise I call
"T".

One more piece of notation: For any manipulations M and N, I write
M'NM as N[M], reading this as if N were a function: "N of M".

```Note M[1]=M.
```

Examples:

T[I] means "Twist the top face"
T[II]=T[JJ] means "Twist the back face"
T[I'] means "Twist the bottom face"
T'[J] means "Twist the left face CLOCKWISE"
T[I] T'[I'] J' means "Rotate the floor-parallel center-slice a
quarter turn counter-clockwise as seen from above."

```Note that (MN)'=N'M', and (N[M])'=N'[M].

Also notice that (MN)[P] = M[P] N[P].
```

To do the Pons Asinorum checkerboard:

Set Q= (TT)[J] (TT)[ZJ] "Half turn body-slicing center-slice."

```Then the Pons is Q Q[J] Q[K].
```

Does anybody see what I'm getting at or am I a lone, mad
genius?

```Set Q= T[J] T[J'].

Then (Q Q[J])^3 is quite pretty.
```

---Wechsler