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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. |
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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.
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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.
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| 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. |
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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. |
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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!
An order-16 Nasik (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. And each of these
4 cubes would have 3 magic squares!
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