Geometric Magic Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers
160
Geometric Magic Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers
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Product Details
| ISBN-13: | 9780486290027 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 10/03/2013 |
| Series: | Dover Math Games & Puzzles |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 160 |
| File size: | 16 MB |
| Note: | This product may take a few minutes to download. |
| Age Range: | 12 - 18 Years |
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Geometric Magic Squares
A Challenging New Twist Using Colored Shapes Instead of Numbers
By Lee C. F. Sallows
Dover Publications, Inc.
Copyright © 2013 Lee C. F. SallowsAll rights reserved.
ISBN: 978-0-486-29002-7
CHAPTER 1
Part I
Geomagic Squares of 3×3
There is mystery in symmetry. With an m to spare.
1 Introduction
I expect most readers will be familiar with the traditional magic square: a chessboard-like array of cells in which numbers, usually but not always consecutive, are written so that their totals taken in any row, any column, or along either diagonal, are alike. Figure 1.1 shows the best known example of size 3×3, the smallest possible, a square of Chinese origin known as the Lo shu. The constant sum of any three entries in a straight line is 15. The diagram at left shows the Lo shu in traditional form, an engaging device nowadays identified by sinologists as a pseudo-archaic invention of the 10th century A.D.; see Cammann. The dot-and-line notation was intended to suggest an origin of extreme antiquity.
The history of magic squares is a venerable one, earliest writings on the topic dating from the 4th century BC. Abstruse as they may appear, these curiosities have long exercised a peculiar fascination over certain minds, attracting over the centuries a steady following of devotees, by no means all of them mathematicians. As Martin Gardner has written, "The literature on magic squares in general is vast, and most of it was written by laymen who became hooked on the elegant symmetries of these interlocking number patterns."
It is true. I myself am such a layman; a mathematical amateur with an irrational fondness for the crystalline quality of these numerical prisms (see, for example, and.) But with that humble position owned up to, my purpose in the present essay is in fact decidedly less timid.
My thesis is that the magic square is, and has ever been, a misconstrued entity; that for all its long history, and for all its vast literature, it has remained steadfastly unrecognised for the essentially non-numerical object it really is. Just as a cylinder may be mistaken for a circle when observed from a single viewpoint, so may a familiar object turn into something quite unexpected when seen from a new perspective. In a similar vein, I suggest the numbers that appear in magic squares are better understood as symbols standing for (degenerate instances of) geometrical figures. Hence the prefix geometric to distinguish the wider genus of magic square that will turn out to include the old species within it. For, as we shall see, the traditional magic square is really no more than that special instance of a geometric magic square in which the entries happen to be one-dimensional. But once we are introduced to squares using two-dimensional entries the scales fall from our eyes and we step into a wider, more exhilarating world in which the ordinary magic square occupies but a humble position.
2 Geometric Magic Squares
Consider a graphical representation of the Lo shu as seen in Figure 2.1 at right, in which straight line segments of length 1, 2 ,3, ... replace like-valued numbers in each cell. The orientation of these segments in their cells is unimportant; they may be horizontal, vertical, or slanted at any angle. The constant sum, 15, as represented by 8+1+6 in the bottom row, say, then becomes three segments of length 8, 1, and 6 that are joined head to tail so as to form a single straight line of length 15.
We note that the order in which 8, 1, and 6 are abutted is non-critical, the important thing being only that they fit together end-to-end so as to fill or 'pave' a straight line of length 15. And similarly for the seven other sets of three line segments occupying the remaining rows, columns, and main diagonals, collectively known as 'magic lines.' Hence more generally:
(1) The numbers that occur in magic squares can be seen as abbreviations for their geometrical counterparts, which are straight line-segments of appropriate length.
(2) The process of adding numbers so as to yield the recurring constant sum is then easier to interpret as the arithmetical counterpart of partitioning or tiling a space with these line segments.
The advantage of this view now emerges in an entirely novel contingency it immediately suggests. For just as line segments can pave longer segments, so areas can pave larger areas, volumes can pack roomier volumes, and so on up through higher dimensions. In traditional magic squares, we add numbers so as to form a constant sum, which is to say, we 'pave' a one-dimensional space with one-dimensional 'tiles.' What happens beyond the one dimensional case?
Geometric or, less formally, geomagic is the term I use for a magic square in which higher dimensional geometrical shapes (or tiles or pieces) may appear in the cells instead of numbers. For the moment we shall dwell on flat, or two-dimensional shapes, although non-planar figures of 3 or higher dimensions may equally be used. The orientation of each shape within its cell is unimportant. Such an array of N × N geometrical pieces is called magic when the N entries occurring in each row, each column, as well as in both main diagonals, can be fitted together jigsaw-wise to produce an identical shape in each case. In tessellating this constant region or target, pieces are allowed to be flipped. As with numerical, or what I now call numagic squares, geomagic squares showing repeated entries are denoted (and deemed) trivial or degenerate, which are terms we shall have need of more often. Rotated or reflected versions of the same geomagic square are counted identical, as are rotations and reflections of the target. A square of size N × N is said to be of order N. We say that a geomagic square is of dimension D when its constituent pieces are all D dimensional. This is an informal introduction to geomagic squares; for a formal definition see Appendix I. In the following, our concern will be almost exclusively with 2-D, or two-dimensional squares.
Figure 2.2 shows a 3×3 two-dimensional geomagic square in which the target is itself a square. Any 3 entries in a straight line can be assembled to pave this same square-shape without gaps or overlaps, as illustrated to right and below. Note how some pieces appear one way in one target, while flipped and/or rotated in another. Thin grid lines on pieces within the square help identify their precise shape and relative size.
At the top is a smaller 3 × 3 square with numbers indicating the areas of corresponding pieces in the geomagic square, expressed in units of half grid-squares. Since the three pieces in each row, column, and diagonal tile the same shape, the sum of their areas must be the same. This is, therefore, an ordinary numagic square (or one-dimensional geomagic square) with a constant sum equal to the target area. Analogous area squares for many geomagic squares are often degenerate because differently shaped pieces may share equal areas.
The concept of geometric magic squares grew out of an original impulse to create a pictorial representation of the algebraic square shown in Figure 2.3, a formula due to the 19th century French mathematician Édouard Lucas that describes the structure of every 3×3 numagic square. The Lo shu, for example, is that instance of the formula in which a = 3, b = 1, and c = 5. From here on the terms formula and generalization will be used interchangeably.
The idea underlying this pictorial representation was as follows. Suppose the three variables in the formula are each represented by a distinct planar shape. Then the entry c + a could be shown as shape c appended to shape a, while the entry c - a would become shape c from which shape a has been excised. And so on for the remaining entries. A back-of-the-envelope trial then lead to Figure 2.4, in which a is a rectangle, b a semi-circle, and c a (relatively larger) square, three essentially arbitrary choices.
This result was more effective than anticipated, the match between protrusions and indentations ("keys" and "keyholes") making it easy to imagine the pieces interlocking, and thus visually obvious that the total area of any three in a straight line is the same as a rectangle of size 1×3, or three times the area of the central piece, in agreement with the formula. However, the fact that the 3 central row and 3 central column pieces will not actually fit together to complete a rectangle, as the pieces in all other cases will, now seemed a glaring flaw. The desire to find a similar square lacking this defect was then inevitable, and the idea of a geometric magic square was born.
It was not until later, however, that the relationship between geometric and traditional magic squares became clear. For, as we have seen, although the term geometric magic square may seem to suggest a certain kind of magic square, in fact things are the other way around. On the contrary, it is ordinary magic squares that turn out to be a special kind of geometric magic square, the kind that use one-dimensional pieces.
The problem of how to actually produce such a square now took centre stage. Following a lot of thought on this question, thus far two approaches have suggested themselves: (1) pencil and paper methods based on algebraic formulae, along the lines just mentioned. (2) in the case of squares restricted to polyforms or shapes built up from repeated atoms, brute force searches by computer. For short, I call the latter polymagic squares, which is probably a misnomer, but no matter. Foremost among the polyforms are polyominoes (built up from unit squares), polyiamonds (equilateral triangles) and polyhexes (regular hexagons). Figure 2.5 shows 'Magic Potion', an example using nine hexominoes. I'm afraid I have been unable to resist the temptation of assigning titles to some of the better specimens. In searching for such a square different target shapes must be tried. In this case, the result was felicitous. In general, both construction methods have proved fruitful. Some simple inferences that follow from Lucas's formula are an essential basis for both.
3 The Five Types of 3×3 Area Squares
Let G be a geomagic square, where G' is the square that results from replacing each piece in G by a number representing its area. Then, by the definition of a geomagic square, G' is a numagic square, although perhaps degenerate, since piece areas may repeat. Thus, if G' is of order-3, its entries must satisfy the relations expressed in Lucas's formula, and, if G is a polymagic square, these entries will be whole numbers. It is easily verified that these are distinct if, and only if, a [not equal to] ± b or ± 2b, or 0.
Consider now the possible forms that a degenerate magic square may take. We shall use Figure 3.1 as our standard for identifying cells. Suppose now that Figure 3.1 is a trivial square in which A = B.
Then by Lucas's formula, c + a = c - a - b, or b = - 2a, which on substitution in the formula yields the type 1 square of Figure 3.2. Here we see the full set of relations implied by A = B, or equivalently, of B = C, C = F, F = I, I = H, H = G, G = D, and D = A, when rotations and reflections of the same square are in turn considered. Repeating this process for the cases A = C, A = E, ... , B = D, ... etc, we discover just three further possible forms of a degenerate square, as seen in the remaining instances of Figure 3.2.
Thus, for every geomagic square G, either G' is a magic square in which every number is distinct or non-degenerate (call this type 0: Lucas's formula with a ≠ ± b or ± 2b or 0), or G' is a degenerate magic square showing one of the four structures of Figure 3.2. We are now ready for a look at the first method for producing geomagics.
4 Construction by Formula
As discussed previously, every numerical magic square has a primitive geometrical analog using straight line-segments. We have only to broaden these lines into strips or rectangles of same height to result in a two-dimensional geomagic square, the target then being a longer strip that is formed simply by concatenating the shorter ones occupying each magic line (i.e., each row, column, and diagonal). By suitable choice of rectangle height, the target can even be made a square, as in the example based on the Lo shu shown in Figure 4.1.
Similarly, just as any set of contiguous points along the real number line can be mapped one-to-one onto another set of contiguous points around the circumference of a circle or part-circle, so numerical magic squares have another primitive geometrical analog using circular arcs or sectors of appropriate angle, the target then being the circle or part circle formed by subjoining these arcs or sectors. Figure 4.2 shows such a representation of the Lo shu using sectors. Since the
constant sum in the Lo shu is 15, the smallest sector subtends an angle of 360 ÷ 15 = 24°, the angles of the other sectors being multiples of 24°, up to 9 × 24 = 216°.
It is easy to see that this circular target could be replaced by a regular 15-gon, the sectors then changing to 15-gon segments of corresponding size. Likewise, the sectors in Figure 4.2 could be changed into annular segments, the target then becoming a ring with a central hole, or central 15-gon hole. Further variations may occur to the reader. By combining the straight line segment and circular arc interpretations, numerical squares could equally be mapped onto 3-D helical segments.
The rectangles and sectors in Figures 4.1 and 4.2 can be further elaborated. Earlier I spoke simplistically of 'broadening the line segments into strips of same height.' A better way of conceiving this is to think of the broadened strip as just two 1-D segments of same length, one above the other, their ends joined by two straight vertical lines so as to form a rectangle. However, it is not necessary that these lines be straight, only that they be congruent. Imagine a piece formed by a pile of contiguous line segments, all parallel to each other, and yet shifted to left or right so that their ends describe some non-linear profile, as in Figure 4.3.
Provided all are contructed similarly, differing only in their lengths, pieces constructed in this way can again be concatenated to form a long, thin target whose ends are sculpted with the same curve. Similar remarks apply to circular segments, a striking example of the kind of profile just mentioned being realized in Figure 20.13 in the section on picture-preserving geomagic squares in Part 3.
This view of 2-D shapes as a stack of parallel straight line-segments appropriately aligned might seem to preclude shapes with re-entrant angles such as Figure 4.4 because the 1-D segments become broken. Happily, however, it turns out that this doesn't matter. In fact, it wouldn't matter if the projecting lug were entirely detached from the main body of the piece to become an island, so that that its corresponding indentation became an isolated hole. This brings us to disconnected pieces.
Previously we saw that every numerical magic square corresponds to a 1-D geometrical magic square written in shorthand notation. But this is not to say that numerical squares account for all possible 1-D geomagic squares. In fact, they account only for that subset of 1-D squares using connected line segments. Figure 4.5 shows a 1-D geomagic square of order-3 that includes disjoint pieces, or pieces composed of two or more separated islands bearing a fixed spatial relation to each other. The overall shape of the compound piece is thus preserved even when moved. Here, the 1-D lines have been broadened and colored to enhance clarity, a trick that could obviously be extended so as to yield a true 2D geomagic square sporting rectangular targets. However, the point to be made here is that Figure 4.5 is a 1-D geomagic square for which there exists no corresponding numerical magic square. Magic squares using numbers thus account for no more than a small fraction of all 1-D geomagic squares.
Just as with linear pieces, so circular arc pieces do not have to be connected. Figure 4.6 shows a 3×3 square using disjoint arcs, their unit segments here simplified into single colored dots. Once again, such disconnected pieces cannot be represented by single numbers.
Of course, the trouble with geomagic squares of the type seen in Figures 4.1 and 4.2 is that they are really nothing more than the same old numerical magic square in alternate guise. The question is: how do we go about producing more interesting 2-D geomagics such as the first one looked at in Figure 2.2, which are something other than just a geometrical rehash of an arithmetical square? One approach is to start with a trivial geomagic square based on a trivial algebraic formula, and then to de-trivialise this by adding appropriate keys and keyholes. I call algebraic squares, trivial or otherwise, that are used in this way, templates. An example will clarify.
(Continues...)
Excerpted from Geometric Magic Squares by Lee C. F. Sallows. Copyright © 2013 Lee C. F. Sallows. Excerpted by permission of Dover Publications, Inc..
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