Converging on the Eye of God D.N. Seppala-Holtzman St. Joseph’s College faculty.sjcny.edu/~holtzman

Announcements Joint work with Francisco Rangel Mathematics Teacher (NCTM) Vol. 103, Nr. 2 (Sept. 2009) faculty.sjcny.edu/~holtzman  downloads

Several Mathematical Objects Play Central Roles Φ, the Golden Ratio Golden Rectangles Golden Spirals The Eye of God The Fibonacci numbers

The Divine Proportion The Golden Ratio, sometimes called the Divine Proportion, is usually denoted by the Greek letter Phi: Φ Φ is defined to be the ratio obtained by dividing a line segment into two unequal pieces such that the entire segment is to the longer piece as the longer piece is to the shorter

A Line Segment in Golden Ratio

Φ: The Quadratic Equation The definition of Φ leads to the following equation, if the line is divided into segments of lengths a and b:

The Golden Quadratic II Cross multiplication yields:

The Golden Quadratic III Dividing by b2 and setting Φ equal to the quotient a/b we find that Φ satisfies the quadratic equation:

The Golden Quadratic IV Applying the quadratic formula to this simple equation and taking Φ to be the positive solution yields:

Two Important Properties of Φ 1/ Φ = Φ - 1 Φ2 = Φ +1 These both follow directly from our quadratic equation:

Φ Is an Infinite Square Root

Φ as a Continued Fraction

Constructing Φ Begin with a unit square. Connect the midpoint of one side of the square to a corner. Rotate this line segment until it provides an extension of the side of the square which was bisected. The result is called a Golden Rectangle. The ratio of its width to its height is Φ to 1.

Constructing Φ B AB=AC C A

Properties of a Golden Rectangle If one chops off the largest possible square from a Golden Rectangle, one gets a smaller Golden Rectangle, scaled down by Φ, a Golden offspring If one constructs a square on the longer side of a Golden Rectangle, one gets a larger Golden Rectangle, scaled up by Φ, a Golden ancestor Both constructions can go on forever

The Golden Spiral In this infinite process of chopping off squares to get smaller and smaller Golden Rectangles, if one were to connect alternate, non-adjacent vertices of the squares with arcs, one gets a Golden Spiral.

The Golden Spiral

The Eye of God In the previous slide, there is a point from which the Golden Spiral appears to emanate This point is called the Eye of God The Eye of God plays a starring role in our story

The Fibonacci Numbers The Fibonacci numbers are the numbers in the infinite sequence defined by the following recursive formula: F1 = 1 and F2 = 1 Fn = Fn-1 + Fn-2 (for n >2) Thus, the sequence is: 1 1 2 3 5 8 13 21 34 55 …

The Binet Formula A non-recursive, closed form for the Fibonacci numbers is given by the Binet Formula:

The Binet Formula II The Binet Formula can be expressed in terms of Φ:

The Fibonacci – Φ Connection Taking the limit of the quotient of sequential Fibonacci numbers in their Binet form yields:

Fibonacci and Φ in Nature As an aside, I recommend the book, “The Golden Ratio” by Mario Livio Many surprising appearances of the Fibonacci numbers and Φ in nature are given

Sunflowers

Pineapples

The Chambered Nautilus

That was the Preamble; Now for the “Amble” In the fall term of 2006, Francisco Rangel, an undergraduate at the time, was enrolled in my course: “History of Mathematics” One of his papers for the course was on Φ and the Fibonacci numbers He read Livio’s book and was deeply impressed with the many remarkable relations, connections and properties he found there

Francisco Rangel Having observed that the limiting ratio of Fibonacci numbers yielded Φ, he decided to go in search of other “stable quotients” He had a strong suspicion that there would be many proportions inherent in any Golden Rectangle He devised an Excel spreadsheet with which to experiment

Francisco Rangel II Knowing that a Golden Rectangle has sides in the ratio of Φ to 1, and knowing the relationship of Φ to the Fibonacci numbers, he considered “aspiring Golden Rectangles,” rectangles with sides equal to sequential Fibonacci numbers These would have areas: Fn+1Fn

Francisco Rangel III He knew that these products would quickly grow huge so he decided to “scale them down” He chose to scale by related Fibonacci numbers He considered many quotients and found several that stabilized. In particular, he found these two: Fn+1Fn / F2n-1 and Fn+1Fn / F2n+2

The Spreadsheet n F(n) F(n+1)F(n)/F(2n-1) F(n+1)F(n)/F(2n+2) 3 2 1.2 0.2857 4 1.15384 0.2727 5 1.176471 0.2777 6 8 1.168539 0.2758 7 13 1.171674 0.2765 21 1.170492 0.2763 9 34 1.17095 0.2764

Stable Quotients as Limits In more mathematical terminology, he found several convergent limits. The results from the previous slide were:

A Surprise These two stable quotients, along with several others, were duly recorded They had no obvious interpretations Francisco then computed the x and y coordinates of the Eye of God. He got: x ≈ 1.1708 and y ≈ 0.27639 These were the same two values!

A Coincidence??!! Not likely! To quote Sherlock Holmes: “The game is afoot!” Clearly something was going on here We were determined to find out just what it was

The Investigation Begins First off, we computed the coordinates of the Eye of God in “closed form” in terms of Φ. We got:

Eye of God Coordinates These coordinates were easily derived as the Eye is located at the intersection of the main diagonal of the original Golden Rectangle with the main diagonal of the 1st Golden offspring

The Golden Spiral

Two Theorems We proved that the limits that we had found earlier corresponded precisely to these two expressions involving Φ. That is:

Sketch of Proof Use Binet formula and observe that:

The Search for “Why” At this point, we had rigorously proved that these limits of quotients of Fibonacci numbers gave us the coordinates of the Eye of God The question was: Why? We proved the following helpful lemma

Lemma For any integer k, we have:

Proof of Lemma Observe:

A Reformulation This lemma allowed us to rewrite the x and y coordinates of the Eye of God as: the x-coordinate of the Eye = the y-coordinate of the Eye =

The Sequences The first several terms of these sequences are: x sequence: Φ1F1 Φ0F2 Φ-1F3 Φ-2F4 … y sequence: Φ-2F1 Φ-3F2 Φ-4F3 Φ-5F4 …

Related by Powers of Φ Note that the y-coordinate can be obtained from the x-coordinate by dividing the latter by Φ3.

Recall the x-coordinate of the Eye = the y-coordinate of the Eye =

Hold That Thought! Keep that thought in a safe place as we shall need it shortly Let us, now, reconsider our original Golden Rectangle

Every Golden Rectangle Has 4 Eyes of God, Not Just 1 When we generated Golden offspring from our original Golden Rectangle, we excised the largest possible square on the left-hand side. We followed this by chopping off squares on the top, right, bottom and so on. We could have proceeded otherwise There are 4 different ways to do this sequence of excisions: Start on the left or right and then go clockwise or anti-clockwise These give 4 distinct Eyes of God

Eye of God 1

Eye of God 2

Eye of God 3

Eye of God 4

The 4 Eyes

The 4 Eyes of God The point that we have been calling the Eye of God is E1 The remaining 3: E2 , E3 and E4 all have x and y coordinates that are of the same form (but with different values of k) as those of E1, namely:

The 4 Eyes

Retrieve that Thought Note that all of the coordinates of all 4 eyes are obtainable from one another by multiplying by a power of Φ Furthermore, this pattern persists in the golden offspring and ancestors

Golden Offspring

Golden Offspring Note that the eye that we called E4 was the South-West eye in the original Golden Rectangle Here it is as the North-East eye in a Golden Offspring

The Other Eyes The 3 other eyes in the Golden Offspring also have x and y coordinates that can be obtained from one another by multiplying by some power of Φ

The Eyes of the Offspring

The Eyes of the Ancestors A similar result holds for Golden Ancestors

Tying it All Together This leads us to our unifying theorem We proved the following:

Unifying Theorem The limit of (Fn+1 Fn )/ Fsn+k behaves as follows: It diverges to infinity if s < 2 It converges to zero if s > 2 It yields the x or y coordinate of some Eye of God in some Golden Rectangle (offspring or ancestor) when s = 2. Precisely which of these it converges to depends on the choice of k.

Sketch of Proof The Binet formula yields:

Some Observations All of the stable quotients that Francisco found were precisely of this form, just with different values of k His initial hunch that many stable proportions would be found hiding in any Golden Rectangle proved to be prescient

Conclusion There are infinitely many paths that converge upon the Eye of God

References Burton, David M. Elementary Number Theory, 6th ed. New York, McGraw Hill, 2007. Livio, Mario. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York, Random House, Broadway Books, 2002.