drawing magic squares.pdf
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Drawing Magic squares: The magic of balanceand symmetry
Eduardo Saenz de CabezonUniversidad de La Rioja
May-2003
Abstract
Many different types of magic squares have been considered trough-out the long history of these mathematical objects. With the helpof techniques that associate polynomical series to discrete functions,we can “draw” magic squares by means of the graphic expression ofthese polynomical functions. The graphics obtained express in a veryclear manner both the mathematical properties of the different typesof magic squares and the beauty underlying its main caractheristic:symmetry.
1 Different types of magic squares.
A magic square of order n is an n × n matrix which has the same sumfor every row, column and full diagonal. This sum is called the magic sumor magic constant of the square. In the following pages we will considermagic squares containing the consecutive numbers 1 to n2 (although someexceptions will be considered); these are called normal magic squares, and
their magic constant is Sn = n(n2+1)2
.During the long and sometimes complicated history of magic sqares, d-
ifferent types of them have been considered, adding new properties to theusual magic condition. For a wide review on the topic, see [?]. Some of themain types of magic squares considered are the following:
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- Pandiagonal Magic Squares:A magic square is pandiagonal (or diabolic) if the sum for every full or
broken diagonal equals the magic constant.- Maxi-perfect Magic Squares:A pandiagonal magic square is maxi-perfect if any 2× 2 subsquare sums
2n2 − 2.- Bordered Magic Squares:If we remove the outer lines and columns of a bordered magic square of
order n, the remaining square (of order n− 2) is still magic.- N-magic Squares:A magic square is n-magic if the square formed substituing a number x
in the original square by xn is still magic. 2-magic squares are called bimagicand 3-magic squares are called trimagic.
- Associative Magic Squares:A magic square of odd order is associative if the numbers in associate
cells are complementaries. The associate of cell at position i, j is the cell atposition n+ 1− i, n+ 1− j. Two numbers are complementaries if they sumn2 + 1.
These are the main types considered among a great variety of them, evenwithin some of these “classes” of magic squares, various “subclasses” havebeen considered.
2 Polynomical series associated to discrete
functions.
For this section, we will follow the ideas in [?] concerning partial derivationof discrete functions and how to interpolate them via polynomic series.
Let si,mi: Zn → Zn given by
si(z1, · · · , zi, · · · , zn) = (z1, · · · , zi + 1, · · · , zn)mi(z1, · · · , zi, · · · , zn) = (z1, · · · , zi − 1, · · · , zn)
using these functions, we can define for all 1 ≤ i ≤ n the following partialderivations (left and right derivation respectively):
d−i f = f − fmi dif = fsi − f ∀f : Zn → Zn
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These operators can be iterated to obtain different derivation schemesfor discrete functions. Some poynomials are associated to these derivationschemes, related to the points used in the computations of the derivatives.
Now given any function f : Zn → R, and a subset S of Zn verifyingsome conditions, we can associate a polynomical function q to f so thatq(x) = f(x)∀x ∈ S, using the derivation schemes and associated polynomialsmentioned above.
For example, if the derivation scheme is an iteration of the right deriva-tion on every component, we obtain the following polynomical series; if weconsider the terms for k less or equal a given number, it is valid for a certainromboid centered on x0
f(x0 + x) =∞∑k=0
∑j1+···+jn=k
dj11 · · · djnn f(x0)
j1! · · · jn!pj1(x1) · · · pjn(xn)
where d0f = f , dnf =n· · · df , and p0(t) = 1 , p1(t) = t , p2(t) = t(t − 1) ,
p3(t) = t(t− 1)(t− 2) , p4(t) = t(t− 1)(t− 2)(t− 3) ,· · · .We will be interested in polynomials interpolating discrete functions in
Z2 valid for square grids Sn containing n2 points and being its corners(1, 1), (n, 1), (1, n) and (n, n). In this case, we have that the following seriesis valid for our purposes
f(x− 1, y − 1) =2(n−1)∑k=0
∑j1 + j2 = k(j1, j2) ∈ S
dj11 dj22 f(1, 1)
j1!j2!pj1(x− 1)pj2(y − 1)
3 Drawing Magic Squares
For “drawing” a magic square M of order n we shall consider the followingdiscrete function fM :
The set S in which fM is defined is the grid of entire numbers in thesquare [1, n] × [1, n] and fM(x, y) = M(x, (n + 1) − y) if 1 ≤ x, y ≤ n, thatis, fM transposes M onto the grid S.
Now we have a discrete function that “translates” the magic square,and we can apply what we have seen in the precendent section. So, a
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polynomical function can be defined which interpolates the magic square-we’ll give it the same name fM(x, y)-, and we can give its graphical rep-resentation in the three-dimensional space, by means of the graph of fM ,GfM = (x, y, fM(x, y))∀(x, y) ∈ S. A colored two-dimensional representationof GfM can be given associating colors to the values of fM(x, y).
In the next pages, we will se through some examples, how the graphicexpressions of GfM show the properties of a magic square. The 3D graphicswill show balanced surfaces defined on a square that express the balance ofnumbers around the center of a magic square, being this center a single pointif the square has even order, or a cell if the square has odd order; symmetryis also present in these graphics as it is the main characteristic of the discretefunctions -magic squares- that underlie the polynomials we are drawing. The2D graphics will clearly show the beauty of symmetry, in which radicates thecharm of magic squares. The drawings will show many different possibilitiesfor symmetry in the limits of a square. Additional properties of the differenttypes of magic squares will also be revealed in the graphs, giving variety andmeaning to the symmetry which is the reason for calling magic to certainarrangements of numbers on a square. Sometimes numbers cannot expressvisually this symmetry and its beauty; in ths case, drawings can, expressingat the same time in a visual manner the arithmetic relations underliying.
Example 1 Lo-ShouThe most famous magic square, and probably the first magic square in
history is that of order 3. Known from some centuries before christian era,it has been used as a magic and even religious talisman. Also known as Lo-Shou, due to a legend that places its origin in the back of a turtle that cameup out of the chinese river Lo, it is the only magic square of order three -ifwe consider symmetries and rotations as the same square-. Lo-Shou is thesimplest -non trivial- magic square, but has many mathematical properties,which can be “seen” through the graphic expressions of the polynomial thatinterpolates it.
The square is the following:
6 1 87 5 32 9 4
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