The Magic of Magic Squares

Eli Ess

Senior Thesis in Mathematics Middlebury College

December 2005

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c Copyright by Eli Ess, 2005.

All Rights Reserved

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Abstract Even after 4000 years from when they were first discovered, new characteristics about magic squares are still being revealed. Recently, it was shown how a simple third order magic square is similar to the nine points of inflection on an elliptic curve. In this thesis I hope to demonstrate this similarity and shed light on the beauty of magic squares.

vi

Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Magic Squares

v 1

1.1

Background and Origins . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

What is a Magic Square? . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

4 x 4 Nasik Square . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

Benjamin Franklin’s Square . . . . . . . . . . . . . . . . . . . . . . .

4

1.5

The Simplest Magic Square . . . . . . . . . . . . . . . . . . . . . . .

5

1.6

Magic Squares Geometrically

6

. . . . . . . . . . . . . . . . . . . . . .

2 Elliptic Curves

9

2.1

What is an Elliptic Curve? . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

Chord-and-Tangent Method . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Counting Points Correctly . . . . . . . . . . . . . . . . . . . . . . . .

13

2.4

The Chord-and-Tangent Method Algebraically . . . . . . . . . . . . .

14

3 Tangent Lines and Points of Inflection

17

3.1

Points of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3.2

Points of Inflection on an Elliptic Curve . . . . . . . . . . . . . . . .

18

3.3

A New Definition for Point of Inflection . . . . . . . . . . . . . . . . .

20

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viii

CONTENTS

4 Magic Square Theorem

23

4.1

What does it mean to be a Point of Inflection? . . . . . . . . . . . . .

23

4.2

The Point at Infinity is a Point of Inflection . . . . . . . . . . . . . .

24

4.3

Magic Square Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1 Magic Squares Mathematicians find beauty in mathematical results which establish connections between two areas of mathematics that appear to be totally unrelated. Once these connections and patterns are made, mathematics appears to be a more organized and whole science. One famous example of this is Euler’s identity which is eiπ + 1 = 0. Some mathematicians call this formula “the most remarkable formula in mathematics” because it incorporates five fundamental mathematical constants. Year after year there are more discoveries connecting different fields of mathematics. In this paper I hope to illustrate one of these discoveries. In a natural, clear way, I hope to display how the nine points of inflection on an elliptic curve can be organized to create a 3 x 3 magic square. Much of this developement follows [1].

1.1

Background and Origins

Magic squares can be traced back as far as 2200 BC when they were mentioned in a manuscript by China’s first emperor. From then on, magic squares have fascinated numerous civilizations. The Mayan Indians were drawn to their symmetry and conti1

2

CHAPTER 1. MAGIC SQUARES

nuity. The Hausau people of northwestern Nigeria and southern Nigeria used magic squares as a calculating device with mystical associations. In Islam, it symbolized the power of Allah’s omnipresence. Even in ancient Babylonian times, people considered these squares to have magical powers, and in the eighth century AD, some squares were considered useful for turning ordinary metal into gold.[2] They have been respected and admired in almost every period and on every continent. Even when the squares lost their mystical meanings, the general public continued to use them as fascinating puzzles, while experienced mathematicians studied them as problems in number theory.[2] People seem to be drawn to them and the many relationships there are between the sums of numbers filling the squares.

1.2

What is a Magic Square?

A magic square is a square array of integers in which the rows, columns, and main diagonals have the same sum. One example of a magic square is of the fourth-order (the order of each magic square is based on how many squares make up the length). (Figure 1.1) 1

2

16 15

13 14

4

3

12

7

9

6

8

11

5

10

Figure 1.1: 4 x 4 Magic Square As we can see the magic sum for this square is 24 because that is the total sum of the numbers in the horizontal rows, vertical columns and main diagonals. It has been established there are 880 different fourth-order magic squares. 880 might

3

1.3. 4 X 4 NASIK SQUARE

seem like a lot but the magic is quite rare when one considers that there are roughly 2,615,348,736,000 possible arrangements of consecutive integers in 4 x 4 array.[2] This takes into account that no integers are used more than once and there are no rotations or reflection variations.

1.3

4 x 4 Nasik Square

Even though there are so many variations of these magic squares, some of them are remarkably unique. Such as a 4 x 4 Nasik magic square. (Figure 1.2) 1

14

7

12

15

4

9

6

10

5

16

3

8

11

2

13

Figure 1.2: 4 x 4 Nasik Square Some consider this Nasik square to be the most elegant and “perfect” of all fourthorder squares because not only do the rows, columns and main diagonals sum to 34 but also the broken diagonals.[2] Broken diagonals leave one end of the square and return along another. For example, 15, 14, 2, 3 and 10, 4, 7, 13 and 15, 5, 2, 12 are all broken diagonals with the sum of 34. In general, a magic square is called Nasik, pandiagonal, or diabolic if all its broken diagonals add up to the magic constant. What makes this square so special is that if there was a tiling of the two dimensional plane with this magic square then any 4 x 4 square would also be a fourth order magic square. Also if you add up any 2 x 2 square in this tiling then it would equal the magic sum of 34. Additionally, the squares at the corners of the magic square also add up to 34. And along every diagonal, any two cells separated by one cell add up

4

CHAPTER 1. MAGIC SQUARES

to 17, which is half of 34. For example, one of the main diagonals has the numbers 1, 4, 16 and 13 and when 1 and 16 are added or 4 and 13 these pairs add up to 17. This characteristic also holds for the broken diagonals. A Nasik square remains Nasik under four different transformations: rotation, reflection, a transfer of a row from top to bottom or from bottom to top, or a transfer of a column from one side to the other. There are only 48 different Nasik magic squares of the forth-order compared to the 880 fourth-order magic squares there are in total. Such squares can be constructed of any odd order above 3 and of any order that is a multiple of 4. However, a Nasik square of a singly even order such as 6, 10, 14 and so on, have never been found and is considered to be impossible.

1.4

Benjamin Franklin’s Square

There is no limit to how large a magic square can become. A large square, such as Benjamin Franklin’s “Most Magically Magical” Square, (Figure 1.3) is 8 × 8. 52 61

4

14

62 51 46 35 30 19

3

13 20 29 36 45

53 60

5

11

59 54 43 38 27 22

6

12 21 28 37 44

55 58

7

9

57 56 41 40 25 24

8

10 23 26 39 42

50 63

2

16

64 49 48 33 32 17

1

15 18 31 34 47

Figure 1.3: Benjamin Franklin’s “Most Magically Magical” Square Even though Benjamin Franklin thought that magic squares were trivial time wasters, he couldn’t help but spend countless hours finding new, incredible squares.[2] What

1.5. THE SIMPLEST MAGIC SQUARE

5

makes this square so magical is that there are so many different ways to get the sum of 260. Not only does every straight row (horizontal or vertical) of 8 numbers added together make 260, half of each row is half of 260. The bent rows of eight numbers, ascending and descending diagonally all add up to 260. An example of a bent row is starting from the 16 we add the diagonal numbers to 10, then we go from 10 to 23 and add the diagonal numbers descending from 23 to 17 which adds up to 260. Also the four corner numbers with the four middle numbers make 260. Any 2 x 2 square on Benjaminn Franklin’s square has a sum of 130, which is half of 260. Even after Benjamin Franklin passed away, mathematicians found more and more ways to find the sum 260 or 130. Sadly, despite all the marvelous symmetries, Franklin’s “Most Magically Magical” Square doesn’t satisfy the main diagonal sums, so it cannot strictly qualify as a “magic squares” according to the common definition that includes diagonal sums. A plethora of different kinds of magic squares exist and are being discoverd. Just to name a few, there are bordered squares, composite squares, knights move magic squares and imperfect magic squares. There are also many relatives to magic squares such as magic stars, magic circles, magic cubes and magic labeling.[2]

1.5

The Simplest Magic Square

The simplest magic square and probably most well known magic square is one of the third order. (Figure 1.4) It has an arrangement of the numbers 1 through 9 in a 3 x 3 square grid so that the numbers in each of the three rows, three columns, and two diagonals have the magic sum of 15.

6

CHAPTER 1. MAGIC SQUARES 8 1 6 3 5 7 4 9 2 Figure 1.4: 3 x 3 Magic Square

All magic squares of order 3 are essentially the same because a rotation or a reflection of the square remains magic. To understand why the number 5 must be at the center of the third-order square, consider all the ways in which the magic constant, 15, can be divided into the addition of three positive integers from 1 to 9: 9+5+1, 9+4+2, 8+6+1, 8+5+2 8+4+3, 7+6+2, 7+5+2, 6+5+2 In a third-order magic square, each of the three rows, three columns, and two diagonals must sum to 15. In other words, there must be eight sets of numbers that sum to 15. Since the center number belongs to one row, one column, and two diagonals, it must appear in four of the eight equations. The only such digit is 5. This means that 5 must be in the center of the 3 x 3 magic square.[2]

1.6

Magic Squares Geometrically

If magic squares are explored geometrically, this third order magic square is actually an example of the nine-point plane. The rows and the columns can be seen as six lines where every line has three points. Then by allowing the diagonals to wrap around the edges of the grid, we then find there are a total of twelve lines. Another way to look at it is as if each square in a diagonal is under a unique column and row that isn’t shared with any of the other squares in that diagonal. So it is simple to see how

1.6. MAGIC SQUARES GEOMETRICALLY

7

the squares 8, 5 and 2 are a diagonal. But under this definition we can see that a the squares 1, 4 and 7 are also in unique columns and rows and therefore are also a diagonal. The twelve lines in this nine-point plane are: three rows: (1, 6, 8), (2, 4, 9), (3, 5, 7) three columns: (1, 5, 9), (2, 6, 7), (3, 4, 8) three main diagonals: (1, 4, 7), (2, 5, 8), (3, 6, 9) three off diagonals: (1, 2, 3), (4, 5, 6), (7, 8, 9) For this case a line is a set of points that is not necessarily connected, straight or infinite.Prasolov So maybe our nine-point looks like (Figure 1.5):

Figure 1.5: Nine-Point Plane Some of the rules that this nine-point plane has are: (1) Each pair of points lies on a unique line. (2) Each pair of lines intersect in at most one point. (3) Each point lies on the same number r of lines–in this case, r=4. (4) Each line contains the same number k of points–in this case, k=3. (5) There exist four points with no three on a line. Rules one and two rule follow the postulates of Euclidean geometry also an ancient part on mathematics. Rules 3 and 4 insures there is continuity in our plane, which is

8

CHAPTER 1. MAGIC SQUARES

that all points and lines have equal status. And the last rule states that the object at hand is nontrivial. Arrangements that satisfy (1-5) are called finite planes.[1] By looking at magic squares geometrically, it is possible to see the relationship between a 3 x 3 magic square and the nine points of a nine-point plane. In this relationship, three points being on the same line is equivalent to three values that make the magic sum.

Chapter 2 Elliptic Curves If beauty is linking two fields of mathematics that appear to be completely unconnected, then elliptic curves are the muses of the mathematical world. Elliptic curves have important connections in number theory, cryptography and integer factorization. Also simple Diophantine equations often lead to elliptic curves and through Riemann surfaces they have connections to topology. Not to mention elliptic curves have played a role in the recent resolution of Fermat’s Last Theorem who many consider the most important discovery in mathematics of the 20th century.

2.1

What is an Elliptic Curve?

An elliptic curves is a type of cubic curve with the form: y 2 = f (x) = x3 + px + q where f(x) has distinct roots We have to assume that our cubic curve is non-singular. There is this distinction between singular and non-singular cubics because they actually have completely different types of behavior. If we write the equation as F (x, y) = y 2 − f (x) = 0, we can use implicit differentiation to get: 9

10

CHAPTER 2. ELLIPTIC CURVES 2ydy dx

= 3x2 + p

Then by definition the curve is nonsingular, provided that there is no point on the curve at which the partial derivatives vanish simultaneously. This means that every point on the curve has a well defined tangent line. Now if these partial derivatives were to vanish simultaneously as a point (x0 , y0 ) on the curve, then y0 = 0, and hence f (x) and f ′ (x) have the common root x0 .[3] Thus x0 is a double root of f . Conversely, if f has a double root x0 , then (x0 , 0) will be a singular point on the curve. There are two possible pictures for the singularity. Which one occurs depends on whether f has a double root or a triple root. In the case that f has a double root, a typical equation is y 2 = x2 (x + 1) (Figure 2.1), where the curve has a singularity with distinct tangent directions.

2

y

–1

1

–0.5

0.5

1 x

–1

–2

Figure 2.1: x2 (x + 1)

If f (x) has a triple root, then after translating x we obtain the equation f (x) = y 2 = x3 (Figure 2.2) which is a semicubical parabola with a cusp at the origin.

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2.2. CHORD-AND-TANGENT METHOD

2

y

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

–1

–2

Figure 2.2: x3

2.2

Chord-and-Tangent Method

Since we have defined what an elliptic curve is, we now can find what points that lie on this curve. Given two points on the curve, we can define an addition that gives us a third point on the curve. This method of adding points on an elliptic curve is called the chord-and-tangent method. If we let E be the set of all rational or real solutions to the equation for the elliptic curve, Morrdell proved in 1921 that E was an abelian group with the cord-tangent method defining the group addition. Mordell’s Theorem: Let y 2 = f (x) = x3 + px + q where f (x) is a nonsingular elliptic curve then the solutions are a finitly generated abelian group. This is how the chord-and-tangent method works. If we have P and Q points on the elliptic curve E, we can join P and Q by the line l. Now l meets E on a third point we’ll call (P ∗ Q) which is unique according to Bezout’s Theorem. Bezout’s Theorem: Two algebraic curves of degree m and n intersect in exactly mn points (counting complex intersections, multiplicities, and points at infinity).

12

CHAPTER 2. ELLIPTIC CURVES

Provided they do not have any components. In our case the elliptic curve is of degree 3 and the line is of degree 1 so 3 times 1 equals 3 so there are 3 points of intersection. The sum (P + Q) is defined to be the reflection of (P ∗ Q) in the x-axis. From an algebraic standpoint the chord-tangentmethod looks like this: If P = (x1 , y1 ) and Q = (x2 , y2 ), then the line l has an equation of the form y = mx + b. With the equations y = mx + b, we can find the one-variable equation x3 −m2 x2 +(p−2mb)x+q−b2 = 0 for y 2 = x3 +px+q. This cubic polynomial has three roots, namely x1 , x2 , and the x3 coordinate of (P ∗ Q) = (x3 , y3 ). Reflecting (P ∗ Q) in the x-axis gives us (P + Q) = (x3 , −y3 ). This function which associates the pair P and Q to the point (P ∗ Q) is called the chord-tangent method.[1]

2

y

–1

1

–0.5

0.5

1

1.5

2

x

–1

–2

Figure 2.3: x3 − 2x with three real roots Take, for example, the elliptic curve y 2 = x3 − 2x (Figure 2.3) which we will call (E1 ). P = (0, 0) and Q = (2, −2) will be the two points on the elliptic curve. From these two points we can find the slope by the equation m =

y2 −y1 , x2 −x1

which is −1. Then

l is the line y = −x. Once we plug in all of our variables in x3 − m2 x2 + (p − 2mb)x + q − b2 = 0 and the equation becomes x3 − x2 − 2x = x(x + 1)(x − 2), whose roots are

13

2.3. COUNTING POINTS CORRECTLY 0, -1, and 2. Then we know that (P ∗ Q) = (2, −2) and so (P + Q) = (2, 2).

For another example we can look at is y 2 = x3 + 2 (Figure 2.4) which we will call E2 . P = (−1, 1) and in this case P=Q so we have to use implicit differentiation to find the slope of the tangent line, which is m = and our line l looks like y =

3x+5 . 2

3x21 +p . 2y1

The slope turns out to be

From this information we can get our elliptic curve

to be a one-variable equation which is 0 = (x + 1)2 (x − double root) and

17 . 4

3 2

17 ), 4

whose roots are −1 (a

Then (P ∗ P ) = ( 17 , 71 ), and so (P + P ) = ( 17 , − 71 ). 4 8 4 8

2

y

–1

1

–0.5

0.5

1 x

–1

–2

Figure 2.4: x3 + 2 with one real root

2.3

Counting Points Correctly

If a straight line intersects a cubic at two points, then the line intersects the cubic a precisely at one more point. This statement is always true provided you count correctly, which means: 1. This means we allow complex coordinates. If complex coordinates are allowed then we can verify the line y = 2 meets E1 in three points (2, 2),(−1 + i, 2) and (−1 − i, 2)

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CHAPTER 2. ELLIPTIC CURVES

2. That we count multiplicities correctly. Thus, for E2 , the point P = (−1, 1) meets doubly since -1 is the double root of y = (x + 1)2 (x −

17 ) 4

and singly at

17 . 4

Going back to E1 , (P ∗ Q) and (P + Q) are on line x = 2. There is actually a third point on this line that doesn’t appear on the graph. The equation of the line that goes between P ∗ Q and P + Q can’t be found because these two points share the same x coordinates, so instead of using (P + Q) = (2, 2) we will use a coordinate that is very close to it such as S = (1.999, −1.9975). The equation of the line for these two points is y = 39997.5x − 7993, which is almost vertical. And from this: (P + Q) ∗ S = 15980002.25, 63880049500.313 This third point turns out to be very far from (P ∗ Q) and S, but it is still on the curve. Moving S closer to (2, −2) moves (P ∗ Q) ∗ S farther away; passing to the limit, if S = (2, −2) = (P + Q), then (P + Q) ∗ S is “infinitely far away” This last point does not have finite coordinates. We call it the point at infinity and label it O. It is considered a point on every elliptic curve. The third and last rule is: 3. We count the point at infinity, if necessary. Thus x=2 meets E1 at (2, 2), (2, −2) and O. Now we can tell how to add points on an elliptic curve so as to include the counting rules and the point at infinity. It is clear that a vertical line meets the cubic at two points and also at the point of infinity. And a nonvertical line meets the cubic in three points when we allow complex numbers and multiplicities.

2.4

The Chord-and-Tangent Method Algebraically

If we look at the chord-and-tangent method algebraically then we can obtain some helpful formulas. If E : y 2 = x3 + px + q is an elliptic curve, then we can express

2.4. THE CHORD-AND-TANGENT METHOD ALGEBRAICALLY

15

the sum P1 + P2 of points P1 and P2 on E by means of the following formulas. Let P1 = (x1 , y1 ), P2 = (x2 , y2 ) and P1 + P2 = (x3 , y3 ). If x1 = x2 and either y1 6= y2 or y1 = y2 =0, then P1 + P2 = O, and we say that P2 = −P1 . Furthermore, we write 2P for P + P, 3P for P + P + P, etc. We also know that x3 is always rational because x3 − m2 x2 + (p − 2mb)x + q − b2 = (x − x1 )(x − x2 )(x − x3 ). With this information and knowing P1 ∗ P2 = (x3 , −y3 ) we can figure out that: x3 = m 2 − x1 − x2 , y3 = −(y1 + m(x3 − x1 ) These equations are important because they will help us to define what is a point of inflection on an elliptic curve.

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CHAPTER 2. ELLIPTIC CURVES

Chapter 3 Tangent Lines and Points of Inflection So far we have not shown a direct connection between magic squares and elliptic curves. One seems to have very old origins with mystic connections, while the other recently has become a hot topic in many circles of mathematics. But the beauty lies in the fact that there is a one to one correspondents between the squares of a third-order magic square and the points on inflection on an elliptic curve.

3.1

Points of Inflection

Before we can find the points of inflection on an elliptic curve we must define what a point of inflection is. If we have the graph f (x) then f ′ (x) is the slope of the graph at the point (x, f (x)). The line that passes through the point(x, f (x)) with slope f ′ (x) is called the tangent line at the point (x, f (x)). If the point (x, f (x)) is a point of inflection, then either f ′′ (x) = 0 or f ′′ (x) does not exist. This means a point of inflection is a point which the graph of y =f(x) has a tangent line and where the 17

18

CHAPTER 3. TANGENT LINES AND POINTS OF INFLECTION

concavity changes. So, if f (x) is twice-differentiable, the f (x) has a point of inflection where y ′′ changes sign. Theorem: If the point (x,y) is a point of inflection, then either f ′′ (x) = 0 or f ′′ (x) does not exist. Proof : Suppose that (x, f (x)) is a point of inflection. Let’s assume that the graph of f is concave up to the left of x and concave down to the right of x. The other case can be handled in a similar manner. If this situation f ′ increases on an interval (x − δ, x) and decreases on an interval (x, x + δ). Suppose now that f ′′ (x) exists. Then f ′ is continuous at x. It follows that f ′ increases on the half open interval (x − δ, x] and decreases on the half-open interval [x, x + δ). this says that f ′ has a local maximum at x. Since, by assumption, f ′′ (x) exists, f ′′ (x) = 0. This shows that if f ′′ (x) exists, then f ′′ (x) = 0. The only other possibility is that f ′′ (x) does not exist.[4]

3.2

Points of Inflection on an Elliptic Curve

If we look at E2 , for example, it looks like there is only two changes of concavity in an elliptic curve but there are actually more. First let’s calculate y ′′ , where y 2 = x3 + px + q. We will write g(x) = x3 + px + q and differentiate both sides of the equality y 2 = g(x) twice; after some algebra this shows:

y ′′ =

2g(x)g ′′ (x) − g ′ (x))2 3x4 + 6px2 + 12qx − p2 = 8yg(x) 8yg(x)

(3.1)

We can see that the numerator to this equation is x4 +6px2 +12qx−p2 . So if (x0 , y0 ) is a point of inflection on the elliptic curve y 2 = x3 +px+q then 3x4 +6px2 +12qx−p2 = 0. We can see this is a forth-degree polynomial, therefore has four complex zeros (xi

19

3.2. POINTS OF INFLECTION ON AN ELLIPTIC CURVE

for 1 ≤ xi ≤ 4). Each of these complex zeros correspond to two points on the curve, p p (xi , g(xi )) and (xi , − g(xi )). So our picture for E2 is very deceiving and instead of just having 2 points of inflection we actually have eight. The only way that we could see these eight complex points is if we could visualize four-dimensional space. In order to prove this, we first must show that the four zeros of I(x) are distinct that is we have to show that for none of the zeros xi is g(xi ) = 0. Key Lemma: If E: y 2 = g(x) = y 2 = x3 +px+q, and I(x) = 3x4 +6px2 +12qx−p2 , then I(x) has four distinct zeros, none of which are zeros of g(x). Proof : Now I(x) is a polynomial with real coefficients. Hence, I(x) has distinct zeros if and only if it has no nonconstant factors in common with its derivative I ′ (x). We have already found that I(x) = 2g(x)g(x) − (g ′ (x))2 , and so I ′ (x) = 2g(x)g ′′ (x) + 2g ′ (x)g ′′ (x) − 2g ′′′ (x) = 12g(x) since g ′′′ (x) = 6. Next, it is clear that any nonconstant divisor of I ′ (x) is a nonconstant divisor of g(x), so that any such divisor of I(x) also divides the difference I(x) − 2g(x)g ′′ (x) = (g ′ (x))2 . So, any such divisor is a nonconstant common factor of both g(x) and g ′ (x). However, since y 2 = g(x) is an elliptic curve, g(x) is guaranteed to have distinct zeros because it is non-singular, and so it has no nonconstant factors in common with g ′ (x). We conclude that I(x) and I ′ (x) have no common factors, and so the forth degree polynomial I(x) has four distinct zeros.[1] Thus, no zero of I(x) is a zero of g(x), and so we have indeed found eight points of p inflection of the curve y 2 = g(x): namely, (xi , ± g(xi )), where xi is one of the four zeros of I(x). For example in our curve E2 where y 2 = x3 +2, it happens that I(x) = 3x4 +24x = √

3x(x3 + 8) has the four zeros 0, −2, −2w and −2w2 , where w = −1 + i 2 3 . With some algebra and we find the eight points of inflection are:

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CHAPTER 3. TANGENT LINES AND POINTS OF INFLECTION

√ √ √ √ (0, ± 2), (−2, ±i 6), (−2w, ±i 6), (−2w2 , ±i 6)

3.3

A New Definition for Point of Inflection

Now that we have found our eight points of inflection there is just one more that has to be found to make our final nine. The final point of inflection is actually the point at infinity, O. But in order to show this we have to look at tangent lines in a new way. What works is to notice that if the line y = mx + b is tangent to the curve y = f (x) at (u, v), then the equation mx + b = f (x) has a double root at u = x. For example, the tangent line to y = x3 − 3x + 2 at (2, 4) has the equation y = 9x − 14, and we see that x3 − 3x + 2 − (9x − 14) = x3 − 12x + 16 = (x − 2)(x − 2)(x + 4) Thus, x = 2 is a double root of f (x) − (mx + b) = 0, since (x − 2)2 is a factor of f (x) − (mx + b). The point of inflection has a similar correlation. If (u, v) is a point of inflection of the curve y = f (x), then the equation mx + b − f (x) = 0 has a triple root at x = u.[1] Using y = x3 −3x+2, P = (0, 2) is a point of inflection for y = x3 −3x+2, since the tangent line at P has equation y = −3x + 2, f (x) − (mx + b) = x3 , and so f (x) − (mx + b) has a triple root at x = 0. For elliptic curves we adopt this broader view of tangents and point of inflections. Let l be a line that meets the curve C at a point P. We’ll say that l is a tangent line to C at P if l and C intersect doubly at P. That is, if l has equation y = mx + b and C has equation F (x, y) = 0, then l is a tangent line at P = (x0 , y0 ) provided the equation F (x, mx + b) = 0 has a double root at x0 . Similarly, we’ll say that P is a point of inflection of the curve C if l and C intersect triply at P–that is, if the equation F (x, mx + b) = 0 has a triple root at

3.3. A NEW DEFINITION FOR POINT OF INFLECTION

21

P.[1] Using this definition, it is possible to show that O is the ninth point of inflection on an elliptic curve.

22

CHAPTER 3. TANGENT LINES AND POINTS OF INFLECTION

Chapter 4 Magic Square Theorem With the tools and information we have gathered from the previous chapters, we can finally show that the nine points of inflection on an elliptic curve make a nine-point plane.

4.1

What does it mean to be a Point of Inflection?

Using what the information we have gathered about the chord-and-tangent theorem we can find a broader, more fitting definition for a point of inflection. Lemma on the Points of Inflection: Let P = (x, y) be a finite point on the elliptic curve y 2 = g(x)–that is, P 6= O. Then P is a point of inflection if and only if 3P = O Proof : We now know that (x,y) is a point of inflection of y 2 = g(x) when (g ′ (x))2 = 2g(x)g ′′ (x) from the key lemma of section 3.2. The formulas to find the third point is the chord-and-tangent method also include (g ′ (x))2 , which are x3 = m2 − x1 − x2 and y3 = −(y1 + m(x3 − x1 ). This is apparent when x1 = x2 = x, y1 = y, so our first two points are going to be on the same x axis. Then our third point on the elliptic 23

24

CHAPTER 4. MAGIC SQUARE THEOREM

curve will be 2P = (x3 , y3 ), then we know: m=

3x2 +p 2y

=

g ′ (x) , 2y

and from x3 = m2 − x1 − x2 and y3 = −(y1 + m(x3 − x1 ) we have x3 = m2 − 2x =

g ′ (x) 2 2y

− 2x =

(g ′ (x))2 −8xy 2 4y 2

=

(g ′ (x))2 −8xg(x) 4g(x)

This is often called the duplication formula.[3] From this we can see that x3 = x if and only if x · 4g(x) = (g ′ (x))2 − 8xg(x) or just when (g ′ (x))2 = 12xg(x). But g ′′ (x) = 6x, so the x-coordinates of P and 2P are the same just when (g ′ (x))2 = 2g(x)g ′′ (x). This is the exact same condition for P to be a point of inflection.[1] √ For example in the previous chapter we learned that (−2, ±i 6) was a point of inflection for y 2 = x3 + 2. So lets see if it works with our new definition for a point of inflection. (g ′ (x))2 for y 2 = x3 + 2 is (3x2 )2 which is 144. 2g(x)g ′′ (x) for y 2 = x3 + 2 is 2(x3 + 2)(6x) which is also 144. So our new definition works. Will will have proved our lemma once we figure out the y-coordinates. If P and 2P have same x-coordinate, then the only possibilities for the y-coordinate of 2P are y and -y because the reflection of any x coordinate will also be on the elliptic curve. Since we said (P + P ) = (x3 , y3 ) then either 2P = P or 2P =-P. If 2P = P, then P = O (just add -P to both sides) which is impossible because at the lamma states that P is a finite point. So 2P =-P. But adding P to both sides show that 3P = O. Hence, we have shown that if P is a finite point, then P is a point of inflection if and only if 3P = O.[1]

4.2

The Point at Infinity is a Point of Inflection

As a corollary, we can show that O is also a point of inflection. Suppose l is a tangent line to the elliptic cure E at O. By point-counting, there must be another point where

4.3. MAGIC SQUARE THEOREM

25

l meets E: call it R. Could R be a finite point? No, for if R and O are on the line l, we have seen that -R is also on that line. So, l contains R and -R–that makes two even if R = -R–and also O doubly because it is a tangent line, which makes four in all. This means there are too many points. Hence, R cannot be a finite point, which means that R = O. We conclude that a line tangent to E at O intersects the curve E triply, Hence, O is a point of inflection of E, and we are done. Since 3O = O, we can now say that the points of inflection on an elliptic curve are precisely those points P for which 3P = O.[1] We can now prove that the nine points of inflection of an elliptic curve can be arranged to form a nine-point plane.

4.3

Magic Square Theorem

The Magic Square Theorem: Every elliptic curve has nine points of inflection, and these points form an affine plane of order 3. That is, each point of inflection lies on exactly four line, each of which contains two other points of inflection making 12 lines in all and each pair of points of infection determines a unique line. Proof : Let us first show that a line through two points of inflection meets the curve in a third point of inflection. The reason for this is that if P, Q and R are points of E which lie on a line, then R = P*Q = -(P + Q). Then, since 3P = 3Q = O, we see that O = O + O = 3P +3Q = 3(P + Q) = 3(-R) = -3R. Hence, 3R = -O=O (there is only one point at infinity) and we conclude that R is also a point of inflection. Next, by point-counting, no more than three distinct points of inflection of E can lie on a line. Since each pair of points of inflection on E lies on a unique line, each line containing two points of inflection exactly one other, and that line accounts for three pairs of points of inflection. Now there are a total of 36 pairs of points of inflection,

26

CHAPTER 4. MAGIC SQUARE THEOREM

36 being the number of 2-element subsets of a 9 element set. Thus, there are twelve lines in all, each containing three points of inflection. Finally, a given point of inflection Q must pair up with each of the other eight points of inflection. Since there are three points of inflection on a line, that puts exactly four lines through Q, each containing two other points of inflection.[1] For example, For the curve y 2 = x3 + 2, it happens that I(x) = 3x4 + 24x = √ 3x(x3 + 8) has the four zeros 0, −2, −2w and −2w2 , where w = −1 + i 3/2. With some algebra and we find that the eight points of inflection are: √ ±A = (0, ± 2) √ ± B = (−2, ±i 6) √ ± C = (−2w, ±i 6) √ ± D = (−2w2 , ±i 6) √ and O where w = −1 + i 3/2 Once we arrange these points to form the nine-point plane then the correlation is obvious. It clear that four of the lines are (O,A,-A)(O,B, -B)(O,C,-C) and (O,D, -D). To find the other eight lines use the slope formula from analytic geometry.[1] When you are done with this step then the points should form into a 3 x 3 magic square, like this one:

4.4

B

-A

-D

C

O

-C

D

A

-B

Conclusion

Even if you do not see the beauty in this proof, there is no doubt that this result ties in many mathematical fields. Finite geometry, recreational mathematics, combinatorics,

4.4. CONCLUSION

27

calculus, algebra, and number theory were all used to find the correlation between elliptic curves and magic squares.[1] This is an example of how interconnected and whole mathematics can be.

28

CHAPTER 4. MAGIC SQUARE THEOREM

Bibliography [1] Brown, Ezra (2001) Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves. The College Mathematics Journal vol. 32, No. 4 September 2001. The Mathematical Association of America, Washington, DC [2] Pickover, Clifford A (2002) The Zen of Magic Squares, Circles, and Stars. Princeton University Press, Princeton, NJ [3] Silverman, Joseph H. and Tate, John (1992) Rational Points on Elliptic Curves. Springer-Verlag Inc., New York, NY [4] Salas, S.L. Hille, Einar and Etgen, Garret J. (2003) One Variable Calculus: Ninth Edition. John Wiley and Sons, Inc., Danvers, MA [5] Prasolov, Viktor and Solovyev, Yuri (1997) Elliptic Functions and Elliptic Integrals. American Mathematical Society, Providence, RI

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