Magic Hypercubes  Overview

Editors note:
Welcome to Magic Hypercubes Terminology Introduction Dimensions 0  ∞ In a 1dimensional space In a 2dimensional space In a 3dimensional space In a 4 dimensional space In a 5 or 6dimensional space Conclusion
WELCOME TO THE WORLD OF MAGIC HYPERCUBES  (AN OVERVIEW) Hypercubes are an extension of magic squares and cubes into higher dimensions. It is also the general name for the family of objects such as point, a line segment, a square, a cube, a tesseract, etc. A cube (3dimensions) may be projected onto a piece of paper by introducing some distortion. Likewise, one can show the projection of a 4dimensional object onto a piece of paper by introducing even more distortion. A four dimensional open lattice, shown in Figure 1, of a tesseract ready to place the numbers 1, 2, ..., 81 at each intersection of lines, so that every line will sum the same sum S, which is called the magic sum. S = 123. Figure 1 shows the open lattice of a 3dimensional cube and a 4dimensional tesseract. In both cases, the outline is shown in black. Numbers are placed at the intersections of the lines. Magic tesseracts are often shown without the center lines, resulting in a clearer image. See my Hypercube Presentations 1 and 2 pages for more on illustrating these objects. hh
Figure 1. A cube and tesseract illustrating the Lattice.
Some new words have had to be introduced into the mathematical language and some words may not be wellknown. One keeps track of where any given number is in the hypercube by means of either a set of {w, x, y, z} style coordinates for smaller dimensional spaces, or a set like (X1, X2, X3, ..., x_{p}, X_{q}, ..., x_{n}) for the higher dimensions. One appreciates that rows and columns, etc. of a square, or cube can be made to run parallel to a set of coordinate axes, But as the diagonals of a cube, for example, can never be perpendicular to each other (true in any odddimensional space), they will run obliquely across the diagram. One tends to align the edges of the hypercube along the coordinate axes. The term "nagonal" is a shortened version of "ndimensional diagonal" So that you would find in a 3dimensional structure a 1agonal, or monagonal; a 2agonal, or diagonal; and a 3agonal, or triagonal. If 5 coordinates change in an 8dimensional hypercube, while the others remain constant, as you travel along it, then this would constitute a pentagonal, or 5agonal. A monagonal is customarily known as a row, a column, a file, or a pillar. One soon runs out of names. A monagonal is sometimes called an Irow. Figure 2. A Magic Square of Order four. (torus and continuous field) An assumption is made that readers understand that the mathematics involved is not in the conventional infinite space taught in high schools, but is instead a modular space which is bounded by the order (m). A magic square, for example is best represented on the surface of a donut because all the broken diagonals become continuous. Higher spaces are on hyperdonuts. A 1dimensonal line and a 2dimensional square can be shown on a sheet of paper (or computer screen) with no distortion. A 3dimensional cube can also be shown on a sheet of paper, but as it is a projection from three to two dimensions, distortion is required. If we show the cube as a wireframe lattice, our minds can visualize the physical location of the numbers at the intersections of the lattice. In 1950, John R, Hendricks designed the open lattice system of visualizing a 4dimensional tesseract on a sheet of paper [1]. This was finally published in a mathematical journal in 1962, and is now the accepted way of illustrating magic tesseracts. Presumably higher dimension hypercubes could also be projected onto a flat surface in a similar manner. However the complexity of the resulting diagram would make it too difficult to comprehend. The only practical way of illustrating these higher dimension hypercube is by listing the numbers in square arrays (which is still often done for cubes and tesseracts [1]John R. Hendricks, The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, vol. 5, no. 2, 1962, pp 171189
DIMENSIONS FROM ZERO TO INFINITY INTRODUCTION When one extends the notion of a magic square and cube to higher dimensional spaces, one is literally at a loss for words. One used to talk about the long and short diagonals of a square. That is no longer adequate. For this attempt to illustrate the comparison between dimensions, we will use two examples, the simple magic hypercube and the perfect magic hypercube. The simple magic hypercube is just that. It contains only the minimum requirements to qualify it as being magic. The order 3 normal magic cube associated. Because of this, and the fact that it is an odd order, it contains 3 central magic squares. Because not all squares are magic, it is still classed as simple. Likewise for the higher dimension hypercubes. The
perfect magic hypercube is required to sum to the Magic Constant (S)
in all possible ways. Perfect also means that you may sum the Magic Sum in 3^{n
} 1 possible directions through every point in modular space. Perfect is
the most magical because all the hyperplanes are also perfect. The differences
are shown in a sidebyside comparison. Of course there are other classes in between those two (except for the magic square). Also, within each class of hypercube are special features such as multimagic, inlaid, bordered. These features (and other classes) are not considered in this comparison. Let us examine what kinds of changes occur as the dimension increases by looking at the specifications.
IN A ZERO DIMENSIONAL SPACE The simplest regular and perfect magic hypercubes reduce to a point with the number "1" assigned and it is considered to be a trivial case.
IN A ONEDIMENSIONAL SPACE The SIMPLE MAGIC HYPERCUBE is a Magic Line of Order 2, containing the numbers 1 and 2 and the Magic Sum is 3.
Figure 3a illustrates what may be considered the REGULAR MAGIC (Dimension 1) HYPERCUBE of order 2. It contains the numbers 1 and 2 placed at positions along a line so that they are balanced. One is at 1 and Two is at + ½ and 1(1) + 2(.5)= 0 . The PERFECT MAGIC (Dimension 1) HYPERCUBE (Figure 3b) is an evenly balanced Magic Line of Order 4. It contains the numbers 1,2,3, 4 and are so arranged so that if the numbers were considered as weights, then the balance is achieved. The Magic sum is 10. This has an extra feature in that it is bimagic, meaning that if one squares all the numbers, where the sum is 30, it still balances. The interval shown on the scales is ½ so we have: 2(1.5) + 4(.5) + 1(.5)+3(1.5) = 0 and 4(1.5) + l6(.5) + 1(.5) + 9(1.5) = 0 Not withstanding the above examples, the smallest magic hypercube is normally considered to be the magic square. IN A TWO DIMENSIONAL SPACE
The SMALLEST PERFECT (NASIK) MAGIC SQUARES are Order 4. They are now commonly called pandiagonal. They were wellknown to the Jaina priests at Nasik, India around 1100 A.D. All the numbers from 1 to 16 are arranged in such a way that the Magic Sum is 34 in,
4 rows (parallel
to xaxis)
IN A THREE DIMENSIONAL SPACE
The SMALLEST SIMPLE
MAGIC CUBE is Order3. Figure 6. There are 4 basic magic cubes of order 3 All the numbers from 1 to 27 are arranged in such a way that the Magic Sum is 42 in 31 ways,
9
rows (parallel to xaxis).
The blue numbers in Figure 7
indicate one of the three magic squares contained in each order 3 magic cube.
It is believed that Fermat
discovered the first magic cube, but it was of order 4 (and not fully magic by
present day definitions).
The SMALLEST PERFECT
(NASIK) MAGIC CUBE is Order 8. The perfect magic cube of order8 uses all the numbers from 1 to 512, arranged in such a way that the Magic Sum is 2,052 in 832 ways.
64 rows (parallel to xaxis) There are 6 main classes of magic cubes. See the table at the end of this page for more information. [1] A.H. Frost,
On the General Properties of Magic Squares (and Cubes), Quarterly Journal of
Mathematics, vol.15, 1878, pp 3449 and 93123
IN 4DIMENSIONAL SPACE
The SMALLEST SIMPLE MAGIC TESSERACT is
Order 3. John R. Hendricks redrew the tesseract in 1950 and finally had it published in 1962.
The
SMALLEST PERFECT (NASIK) MAGIC TESSERACT is Order 16.
4,096 rows (parallel to xaxis). Mitsutoshi Nakamura has determined there are 18 main classes of magic tesseracts and has constructed an example of most of them. [1] John R. Hendricks,
Magic Squares to Tesseracts by Computer, Selfpublished 1998, 0968470009
page 114 IN 5DIMENSIONAL SPACE The SMALLEST SIMPLE MAGIC HYPERCUBE of dimension 5 is of Order 3. Published by Hendricks in May 1962 [1]. All the numbers from 1 TO 243 are arranged in such a way that the Magic Sum is: 366 in 421 ways.
81 rows
(parallel to xaxis) The number of dimension 5 hypercubes is not known (for any order), but there are 3840 variations of each due to rotations and reflections. The SMALLEST PERFECT (NASIK) MAGIC HYPERCUBEs of dimension 5 are of Order 32. The first one was published by John R. Hendricks in May 1999 (in program form) [2]. All the numbers from 1 to 33,554,432 are arranged in such a way that the Magic Sum is 536,870,928 in 126,877.696 ways.
1,044,576 rows (parallel
to xaxis) The booklet, Perfect n Dimensional Magic Hypercubes of Order 2" by Hendricks shows how to make the smallest Perfect hypercube of any dimension. [1] John R. Hendricks, The Five and Six
Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin,
vol. 5, no. 2, 1962, pp 171189
IN 6DIMENSIONAL SPACE The
SMALLEST SIMPLE MAGIC HYPERCUBE is of Order 3. The first one was
published by Hendricks in May 1962 [1].
243 rows (parallel to xaxis) David M. Collison constructed a 7 and an 8Dimensional magic Hypercube of Order 3 before 1995. Meredith Houlton calculated to the ninth dimension! ad infinitum
EVEN THOUGH THE MATHEMATICS CAN PRODUCE THESE
LARGE CONSTRUCTIONS, HOW ARE THEY BEST DISPLAYED? However, for the Dimension 5 and 6 magic hypercubes of order3 described in the footnote, he did use his tesseract diagram. Three of these were required to show the dimension 5 magic hypercube, and 9 were required to show the dimension 6. [1]John R. Hendricks, The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, vol. 5, no. 2, 1962, pp 171189
CONCLUSION
Besides
the regular and perfect magic hypercubes, there are many other classes
For the next dimension, (4 the tesseract) there will be many more classes. At this time (August, 2007) Mitsutoshi Nakamura [1] has found that there are 18 classes of tesseracts. [2] Within these major classes, there are many varieties. Inlaid Magic hypercubes have been made for squares for a long time. That is where one finds smaller magic squares embedded within larger ones. John R. Hendricks made the world's first Inlaid magic cubes and inlaid magic tesseracts. Another variety is the bimagic hypercube [3]. The first bimagic cube of order 25 was made by Hendricks [4]. It contains the numbers 1, 2, 3,..., 15625 and sums 195325 in 625 rows, 625 columns, 625 pillars and in the four continuous triagonals. The sums of the squares in each of these turns out to be 2,034,700,525. Bimagic Tesseracts are now the new challenge. [1] Mitsutoshi Nakamura's web
site with examples is at
http://homepage2.nifty.com/googol/magcube/en/
Counting Basic Magic Hypercubes [1]
[1] Walter Trump has
done much work on this subject. His results are at
http://www.trump.de/magicsquares/howmany.html
