Date: Mon, 16 Feb 81 23:27:00 -0500 (EST)
From: Dan Hoey <Hoey@CMU-10A >
~~~ ~~~ Subject: Four colors suffice

Douglas Hofstadter, in the Metamagical Themas column in
Scientific American this month, shows two alternate ways of
coloring a cube. Both suffer from two drawbacks: They fail to
distinguish all cube positions, and they use more than six colors.
This seems inefficient to us, since there is a coloring of the cube
which distinguishes all elements of the Supergroup and uses
only four colors (and which, like Hofstadter's colorings and
the standard coloring, satisfies the restriction that every
whole-cube move is a color permutation, as discussed in point 2
below).

Our coloring, called the Tartan, is formed by assigning the
colors blue, green, red, and yellow to the four pairs of antipodal
corners of the cube. Thus for each face of the cube, the four
corners of the face are assigned four different colors. We use the
term ``plaid'' to denote such an assignment of colors to the
corners of a square. To color the cube, divide each facelet of each
cubie into four squares, and color the squares so all facelets on a
side of the cube display the plaid associated with that face. The
result is shown below, with the initial assignment of colors to
corners in lower case.

```		  (r)---------------(y)
| R Y   R Y   R Y |
| B G   B G   B G |
|		     |
| R Y   R Y   R Y |
| B G   B G   B G |
|		     |
| R Y   R Y   R Y |
| B G   B G   B G |
(r)---------------(b)---------------(g)---------------(y)
| R B   R B   R B | B G   B G   B G | G Y   G Y   G Y |
| G Y   G Y   G Y | Y R   Y R   Y R | R B   R B   R B |
|		   |		     |		       |
| R B   R B   R B | B G   B G   B G | G Y   G Y   G Y |
| G Y   G Y   G Y | Y R   Y R   Y R | R B   R B   R B |
|		   |		     |		       |
| R B   R B   R B | B G   B G   B G | G Y   G Y   G Y |
| G Y   G Y   G Y | Y R   Y R   Y R | R B   R B   R B |
(g)---------------(y)---------------(r)---------------(b)
| Y R   Y R   Y R |
| G B   G B   G B |
|		     |
| Y R   Y R   Y R |
| G B   G B   G B |
|		     |
| Y R   Y R   Y R |
| G B   G B   G B |
(g)---------------(b)
| G B   G B   G B |
| R Y   R Y   R Y |
|		     |
| G B   G B   G B |
| R Y   R Y   R Y |
|		     |
| G B   G B   G B |
| R Y   R Y   R Y |
(r)---------------(y)
```

To understand the importance of the Tartan, there are
several points to consider:

1. By reading off the four colors of a plaid in clockwise
order, starting at an arbitrary point, we obtain four permutations
disjoint, so all 24 permutations of the four colors appear on the
Tartan, once each.

2. Every motion in the group C of whole-cube rotations is a
permutation of the pairs of antipodal corners, and so corresponds to
a recoloring of the Tartan. Some restriction of this sort is
necessary to prevent us from simply drawing a different
black-and-white picture on each facelet and calling that a
two-coloring.

3. Point 2 implies that C is isomorphic to a subgroup of
S4, the group of permutations on the four colors. But both C and S4
have 24 elements, so C is isomorphic to S4 itself (a fact
well-known to crystallographers).

4. Since every color permutation is realizable by a
whole-cube move, there is only one Tartan (up to whole-cube moves).
This is why we use colors as labels, rather than some FLUBRDoid
positional scheme. [The actual choice of colors and the name
``Tartan'' arise from the DoD Ironman project.]

5. Every reflection of the Tartan is color-equivalent to a
rotation. In particular, the identity is color-equivalent to a
reflection through the center of the cube. If you were to lend your
Tartan to someone who ran it through a looking-glass, you could not
discover the fact except by removing the face-center caps and

We have constructed a Tartan from a Rubik's cube and
colored tape. Due to the similar appearance of the plaids, it takes
us several times as long to solve the Tartan as it takes to solve
Rubik's cube.

Our search for pretty patterns has not been particularly
rewarding. Part of the reason seems to be that the cube's
appearance is strongly constrained by the Tartan's coloring. On
Rubik's cube one may make a particular face pattern (e.g. orange T
on white background) using any of several identically colored
facelets. On the Tartan, however, the plaid on any facelet of a
cubie, together with the orientation of the plaid relative to the
cubie, determines the plaid and orientation of the other facelet(s)
of the cubie.

The one nice pattern we have is in fact the conceptual
precursor to the Tartan. It is Pons Asinorum (FFBBUUDDLLRR) applied
to the position shown in the diagram above. In this position, the
plaids of adjacent facelets line up with each other to display the
same arrangement of plaids, magnified by a factor of two. Each face
looks like the following, for some assignment of colors to the
numbers 1 through 4:

```(1)---------------(2)
| 1 2   2 1   1 2 |
| 4 3   3 4   4 3 |
|		   |
| 4 3   3 4   4 3 |
| 1 2   2 1   1 2 |
|		   |
| 1 2   2 1   1 2 |
| 4 3   3 4   4 3 |
(4)---------------(3)
```