From:

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Allan Wechsler's message of 17 May 1982 contains some interesting

comments on the four-dimensional hyper-cube, or tesseract. I will

expand on them, and offer a correction.

The tesseract has eight cubical sides, labeled Back, Front, Up,

Down, Left, Right, Out, and In. Each side may be twisted in any of

the twenty-four ways that a cube may be rotated in three-space.

Since these twenty-four twists are generated by repeated

application of the six quarter-twists of the cube, I consider a

move to be a single quarter-twist of one of the cubical sides.

I have picked three of the quarter-twists of the Out side to be the

``clockwise'' twists, given as Of, Ou, and Or below. Given the

constraint that clockwise twists must be conjugates of each other

with respect to the movement group of the tesseract in four-space,

the remaining clockwise quarter-twists are determined. In the

following list, the upper-case letter denotes the side to be

twisted, and the quarter-twist is displayed as a permutation on the

(square) faces of that (cubical) side.

Of=(URDL) Ou=(RFLB) Or=(FUBD) If=(RULD) Iu=(FRBL) Ir=(UFDB) Ro=(UFDB) Rf=(OUID) Ru=(FOBI) Lo=(FUBD) Lf=(UODI) Lu=(OFIB) Ur=(OFIB) Uo=(FRBL) Uf=(ROLI) Dr=(FOBI) Do=(RFLB) Df=(ORIL) Fu=(ORIL) Fr=(UODI) Fo=(RULD) Bu=(ROLI) Br=(OUID) Bo=(URDL)

These twists have the satisfying property that when two different

twists move an edge from position E1 to position E2, then one of

the twists is clockwise and the other counterclockwise. For

instance, both the Dr and the Fr' twists move an edge from FID to

FOD. Another property that mimics the three dimensional cube is

that clockwise twists on opposite sides are reversed: The action

of Of on the O side is the inverse of the action of If on the I

side.

To see how the table above is constructed we must describe the

movement group of the tesseract (the group of whole-tesseract moves

in four-space). I look at it as operating on quadruples VWXY of

mutually adjacent sides. To see if VWXY->V'W'X'Y' is in the group,

replace all occurrences of B, D, L, and I with F, U, R, and O,

respectively. The resulting permutation must have the same parity

as the number of replacements performed. Thus FLOD->UROB is in the

group, because we perform three replacements to form FROU->UROF, an

odd permutation.

To tell whether a quarter-twist is clockwise or not, take the side

V to be twisted, two consecutive letters WX from the permutation,

and a fourth, orthogonal letter Y from {F, U, R, O}. If VWXY->OURF

is in the movement group of the tesseract, then we have the

clockwise quarter-twist Vy, otherwise the counterclockwise

quarter-twist Vy'. For instance, if we twist the U side as (LFRB),

then VWXY=ULFO->OURF is in the movement group (one replacement

creates the four-cycle URFO->OURF), so we have the clockwise twist

Uo.

Let us now examine the reachable configurations of the corners of

the tesseract. Every quarter-twist moves eight corners in two

four-cycles, so only even permutations of the corners are

achievable. The orientations of the corners are more complex. If

we move corner VWXY to V'W'X'Y', then VWXY->V'W'X'Y' must be in the

movement group of the tesseract. Thus only half of the twenty-four

permutations of {V', W', X', Y'} are achievable, because of the

permutation parity constraint.

To define the orientation of the corners, we label the sides of

each corner and each corner position with the letters VWXY, and

read the orientation of a cubie as the letters it has in the V, W,

X, and Y sides of its position. It is important here to obey the

the permutation parity constraint when doing the labelling, so that

each cubie may be placed in the home (VWXY) orientation in any

position. For instance, one possible labelling is as follows,

where each column refers to a corner:

V F F F F F F F F B B B B B B B B W U U U U D D D D U U U U D D D D X R O L I O L I R O L I R R O L I Y O L I R R O L I R O L I O L I R

Thus if the FURO corner (column 1) is in the FLUO position (column

2), then its orientation is VXYW. Any orientation that is an even

permutation of VWXY is possible. The group of orientations, A4, is

the same as the movement group of the tetrahedron (with vertices

labeled VWXY) in three-space.

As I suggested in my message of 23 September 1982, this orientation

group is not Abelian, so the orientation of the last corner is not

completely determined by the orientations of the other fifteen. To

see what is determined, let us look at the tetrahedron with

vertices at half of the corners of a three-dimensional cube, say

the FUR, FDL, BUL, and BDR corners. As I reported on 15 June 1982,

the twelve movements of the tetrahedron consist of the identity,

three 180-degree rotations, and eight 120-degree rotations. The

June message also mentions that if those corner cubies are moved as

a unit, preserving their positions and orientations relative to

each other, then the 180-degree rotations are achievable on the 3^3

(they are the corners of the Zig-Zag pattern) but the 120-degree

rotations violate the corner twist invariant. Of course, four of

them perform a net clockwise twist, and four of them perform a net

counterclockwise twist. Define the twist of a tetrahedron movement

to be the net clockwise twist it applies to the corners of the

cube. Thus the twist of the movements of a tetrahedron VWXY is

given by the following table.

Twist 0: VWXY, WVYX, XYVW, YXWV

Twist 1: VXYW, WYXV, XVWY, YWVX

Twist 2: VYWX, WXVY, XWYV, YVXW

By reasoning about the actions on corners of the cube, it is clear

that the twist of the product of two movements is the sum of their

twist, modulo three. Thus the twist group is an Abelian quotient

of A4, isomorphic to the cyclic group on three elements.

Since the orientation group of the tesseract corners is also A4, we

may use the twist group to construct an orientation invariant of

the corners of the tesseract. As described in the September

message, each qtw moves the corners in untwisted cycles, so the sum

of the twists of the orientations of the corners must be zero,

modulo 3.

I ran the Furst, Hopcroft, and Luks algorithm on the 2^4 tesseract

and found that this is the only invariant of corner orientation.

Therefore, the number of reachable positions of the tesseract is

(15! / 2) (12^15 / 3) ~ 3.358 x 10^18. This is larger than Allan

Wechsler's upper bound because he thought there were only six

orientations of each corner.