Associated Hypercubes

Each dimension of magic hypercube has varying number of classes based on types and number of summations. But all dimensions of magic hypercubes are also divided into two main groups which are determined by how the numbers are arranged. These two groups are Associated and Unassociated. Normally, if the hypercube is not associated, the fact simply is not mentioned.

Definition

A magic hypercube where all pairs of cells diametrically equidistant from the center of the hypercube equal the sum of the first and last terms of the series, or mn + 1 for a normal magic square, cube, etc. is called associated (m = order, n = dimension). These number pairs are said to be complementary. The series used is consecutive and starts at 1 in a normal magic hypercube. However, even if the series consists of non-consecutive numbers, the complement of a particular number is found by subtracting it from the sum of the first and last numbers in the series. This type of magic hypercube is often referred to as center-symmetric.
There are other types of symmetrical hypercubes, but only center-symmetric is synonymous with associated.

This is one of the eight aspects of the one order-3 magic square.
I have indicated the 4 pairs of complementary numbers using different colors. Notice that the members of the pair are on each side of the central cell as per the definition for associated.

The second square is obtained by reversing the position of the numbers in each pair. It is another aspect of the same magic square i.e. self-similar.

To the right is one of the 48 aspects of Index1 of the 4 basic magic cubes.

I have indicated several pairs of complementary numbers by color. Reverse the position of these numbers and reverse the positions of the numbers in the other pairs to obtain another aspect of this same magic cube.

All order-3 magic hypercubes are associated. All associated magic hypercubes are self-similar. This is because when all numbers in the hypercube are complemented, the result is a different aspect of the same hypercube. This is illustrated in the tesseract example below.

This illustration is of a larger order, to demonstrate that complementary number pairs are indeed located "diametrically equidistant from the center" as per the definition.

I have complemented the numbers in this first magic square to illustrate another aspect of the same square.

I have used this particular square for my example because it has an extra feature. Note the location of the odd numbers. This is called a lozenge magic square.

Odd order magic squares can only be associated if the center number of the series occupies the center position of the square (or cubes, etc.).
For even order magic hypercubes, the center of the figure is a point, rather then a number.

This magic cube was published by Schubert in 1898. It is a simple magic, and contains no magic squares.

It is associated, as is evident when you notice that the 2 numbers of each complement pair appear at equal distances on opposite sides of the center point of the cube. I have used separate colors on 4 pairs to make them easier to see.

Numbers in all complement pairs have been exchanged to produce the self-similar, but different aspect of the original cube.

I have gone into quite a bit of detail on the associated features of the dimension 2 and 3 magic hypercubes, because as the dimension increases, so does the complexity, and the ability to see the basic features clearly. (Can you easily see that both cubes above are actually different aspects of the same cube?)

Now it is time to consider look at the dimension 4 magic hypercube, the tesseract. This time we will consider only the order-3. Features we have discussed above for higher order squares and cubes, apply also to higher orders of tesseracts.

Above is an order-3 magic tesseract (Index # 54 of 58). Because all order-3 magic hypercubes are associated (center symmetric), it is also self-similar. By complementing each number in this figure you obtain another aspect of the same magic tesseract. To complement each number, subtract it from the sum of the first and last number in the series (in this case 1 + 81 = 82). In the above figure, B. shows the complement of A.

For much more information on Self-similar hypercubes, see my self-similar squares and cubes.
These pages will also have more information on even order associated squares and cubes, and also other types of symmetry in hypercubes.

Because all order-3 magic hypercubes are associated, they contain n order-3 magic hypercubes of the next lower dimension.
This illustration shows the four magic cubes contained in the # 54 tesseract above. Each of these cubes, in turn, contain 3 magic squares, I show one of these for each cube. The square shown in the fourth cube is the same orientation as one of the other cubes. Because a cube has only three dimensions, there can be only three orientations of squares.

These cubes do not contain consecutive numbers (so are not normal), because they form part of a normal tesseract which contains the consecutive numbers from 1 to 81. For the same reason, the magic squares that they contain are not normal.

Following are the text listings for these four cubes. In each case, rows and columns of all 3x3 arrays sum correctly to 123, but the diagonals are correct in only the middle array of each cube.

Horizontal          vert_B2F            vert_L2R           Central
36   67   20        30   68   25        34   68   21       37   80    6  
65   27   31        67   27   29        65   27   31       78    1   44  
22   29   72        26   28   69        24   28   71        8   42   73  
                                    
77    3   43        80    1   42        78    1   44       77    3   43  Only these middle arrays
 7   41   75         3   41   79         7   41   75        7   41   75    have correct diagonals,  
39   79    5        40   81    2        38   81    4       39   79    5      so are magic squares.  
                                    
10   53   60        13   54   56        11   54   58        9   40   74  
51   55   17        53   55   15        51   55   17       38   81    4  
62   15   46        57   14   52        61   14   48       76    2   45

Be aware that there are many more cubes in a tesseract. Most if not all of these will not be magic though, because their triagonals do not sum correctly.
For example, all tesseracts are bounded by 8 cubes, just as all cubes are bounded by 6 squares, and all squares are bounded by 4 lines!
A perfect magic tesseract has 64 magic cubes!

By extension an order-3 dimension 5 magic hypercube would contain 5 magic tesseracts in it’s center.
Each of these 5 tesseracts would contain 4 magic cubes, etc.