The Magic Encyclopedia ™

Aspects
(by Aale de Winkel)

With this article I try to dispense with the historic mishap of 'rotation' as being a factor in the number of aspects.
The n!2n aspects of a hypercube are usually not counted as it is regularly seen as being a trivial variation of the hypercube. As the factor 2n is rightfully attributed to reflection (negation of a coordinate) the factor n! is often attributed to some compound operation involving rotation. The factor n! has however a natural association with axial-permutation aka the permutation of the coordinates, just as in a square the transposed version simply exchange the coordinates, the same thing can be done in the higher dimensions.

Aspects
the axes are numbered 0..n-1 (0 usually refered to as 'x', 1 as 'y' etc)
Reflection
~R
factor: 2n
reflection of an axis (x => -x) (y => -y) etc.
reflection is effected by negation of the coordinate
R = j=0n-1(reflected(j) ? 2j : 0)
reflected(j) is true if axis j is reflected; false otherwise
usually the hypercube is shifted back on its spot.
aternate notation: ~R = _R[n-1,..,0]
axial permutation
^[perm(0..n-1)]
factor: n!
permutation of the axes
the axes are permuted amongst on another effecting permutation of the coordinates
for a square (S) this is regularly called transposing and denoted by St
(see pe Excell's function transpose as well as it's special copying option)
nN2^[perm(0..n-1)] : [ j1 ; #j=1 ] = 2perm(j)


samples
this table works out in full the aspects of the order 2 square and cube
the general formula in the table above is shown in the red fields below
square N
0 1
2 3
N~1
1 0
3 2
N~2
2 3
0 1
N~3
3 2
1 0
N[1,0]
0 2
1 3
N[1,0]~1
2 0
3 1
N[1,0]~2
1 3
0 2
N[1,0]~3
3 1
2 0
cube N
0 1 | 4 5
2 3 | 6 7
N~1
1 0 | 5 4
3 2 | 7 6
N~2
2 3 | 6 7
0 1 | 4 5
N~3
3 2 | 7 6
1 0 | 5 4
N~4
4 5 | 0 1
6 7 | 2 3
N~5
5 4 | 1 0
7 6 | 3 2
N~6
6 7 | 2 3
4 5 | 0 1
N~7
7 6 | 3 2
5 4 | 1 0
N[1,0,2]
0 2 | 4 6
1 3 | 5 7
N[1,0,2]~1
2 0 | 6 4
3 1 | 7 5
N[1,0,2]~2
1 3 | 5 7
0 2 | 4 6
N[1,0,2]~3
3 1 | 7 5
2 0 | 6 4
N[1,0,2]~4
4 6 | 0 2
5 7 | 1 3
N[1,0,2]~5
6 4 | 2 0
7 5 | 3 1
N[1,0,2]~6
5 7 | 1 3
4 6 | 0 2
N[1,0,2]~7
7 5 | 3 1
6 4 | 2 0
N[0,2,1]
0 1 | 2 3
4 5 | 6 7
N[0,2,1]~1
1 0 | 3 2
5 4 | 7 6
N[0,2,1]~2
4 5 | 6 7
0 1 | 2 3
N[0,2,1]~3
5 4 | 7 6
1 0 | 3 2
N[0,2,1]~4
2 3 | 0 1
6 7 | 4 5
N[0,2,1]~5
3 2 | 1 0
7 6 | 5 4
N[0,2,1]~6
6 7 | 4 5
2 3 | 0 1
N[0,2,1]~7
7 6 | 5 4
3 2 | 1 0
N[2,0,1]
0 4 | 2 6
1 5 | 3 7
N[2,0,1]~1
4 0 | 6 2
5 1 | 7 3
N[2,0,1]~2
1 5 | 3 7
0 4 | 2 6
N[2,0,1]~3
5 1 | 7 3
4 0 | 6 2
N[2,0,1]~4
2 6 | 0 4
3 7 | 1 5
N[2,0,1]~5
6 2 | 4 0
7 3 | 5 1
N[2,0,1]~6
3 7 | 1 5
2 6 | 0 4
N[2,0,1]~7
7 3 | 5 1
6 2 | 4 0
N[1,2,0]
0 2 | 1 3
4 6 | 5 7
N[1,2,0]~1
2 0 | 3 1
6 4 | 7 5
N[1,2,0]~2
4 6 | 5 7
0 2 | 1 3
N[1,2,0]~3
6 4 | 7 5
2 0 | 3 1
N[1,2,0]~4
1 3 | 0 2
5 7 | 4 6
N[1,2,0]~5
3 1 | 2 0
7 5 | 6 4
N[1,2,0]~6
5 7 | 4 6
1 3 | 0 2
N[1,2,0]~7
7 5 | 6 4
3 1 | 2 0
N[2,1,0]
0 4 | 1 5
2 6 | 3 7
N[2,1,0]~1
4 0 | 5 1
6 2 | 7 3
N[2,1,0]~2
2 6 | 3 7
0 4 | 1 5
N[2,1,0]~3
6 2 | 7 3
4 0 | 5 1
N[2,1,0]~4
1 5 | 0 4
3 7 | 2 6
N[2,1,0]~5
5 1 | 4 0
7 3 | 6 2
N[2,1,0]~6
3 7 | 2 6
1 5 | 0 4
N[2,1,0]~7
7 3 | 6 2
5 1 | 4 0