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## Converging on the Eye of God

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**Converging on the Eye of God**D.N. Seppala-Holtzman and Francisco Rangel St. Joseph’s College faculty.sjcny.edu/~holtzman**A Tale of Mathematical Intrigue**• Our story begins with a vague suspicion • It develops into a series of experiments • These yield a surprising discovery • This leads to a conjecture • Which is followed by a rigorous proof • All of which leads to elucidation in terms which relate to the original suspicion**Several Mathematical Objects Play Central Roles**• Φ, the Devine Proportion • Golden Rectangles • Golden Spirals • The Fibonacci numbers**The Divine Proportion**• The Divine Proportion, better known as the Golden Ratio, 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**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**• Setting Φ equal to the quotient a/b and manipulating this equation shows 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:**Constructing Φ**• Begin with a 2 by 2 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.**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, one gets a 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**Φ In Nature**• There are physical reasons that Φ and all things golden frequently appear in nature • Golden Spirals are common in many plants and a few animals, as well**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 Fibonacci Numbers in Nature**• Just as with Golden Spirals, the Fibonacci numbers appear frequently in nature • A wonderful Website giving many examples of this is: www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonnaci**Some Examples**• The number of “growing points” on plants are often Fibonacci numbers • Likewise, the number of petals: • Buttercups: 5 • Lilies and Iris: 3 • Corn Marigold: 13**More Examples**• The number of left and right oriented spirals in sunflowers and pinecones are sequential Fibonacci numbers • The “family tree” of male drone honey bees yield Fibonacci numbers**The Fibonacci – Φ Connection**• These two remarkable mathematical structures are closely interconnected • The ratio of sequential Fibonacci numbers approaches Φ as the index increases:**All of This is Background; Now Our Story Begins**• In the fall term of 2006, Francisco Rangel, an undergraduate, was enrolled in my course: “History of Mathematics” • One of his papers for the course was on Φ and the Fibonacci numbers • He was deeply impressed with the many remarkable relations, connections and properties he found here**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 examined the areas of what he called “aspiring Golden Rectangles” • These areas would be products of sequential Fibonacci numbers: 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**Stable Quotients as Limits**• In mathematics, one says that the limit of these terms approaches a fixed value as n approaches infinity. In this notation, the previous findings 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: • x = (Φ + 1)/[Φ + 1/ Φ] • y = (Φ – 1)/[Φ + 1/ Φ]**The Next Step**• Next we proved that the limits that we had found earlier corresponded precisely to these two expressions involving Φ. That is:**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? • Was there a geometric interpretation?**A Helpful Lemma**• We suspected that geometric insight would come from relating our results to Φ. We proved: • Lemma: lim ( Fn+k/Fn ) = Φk • This allowed us to recast our expressions in terms of Φ • Surely this would yield something “Golden”**A Reformulation in Terms of Φ**• This lemma allowed us to rewrite the x and y coordinates of the Eye of God as: • the x-coordinate of the E1 = • the y-coordinate of the E1 =**Geometric Interpretation I**• Now that we had the x and y co-ordinates of the Eye of God in terms of Φ, we could give a geometric interpretation of these sequences • x sequence: Φ1F1 Φ0F2 Φ-1F3 Φ-2F4 … • y sequence: Φ-2F1Φ-3F2Φ-4F3 Φ-5F4 … • Graphing this gives the following:**Geometric Interpretation III**• The terms of the x-sequence give, alternately, the upper and lower x bounds of Golden offspring • Similarly, the terms of the y-sequence give, alternately, the upper and lower y bounds of Golden offspring • In each case, the over and under estimates get smaller with each iteration • Both of these patterns persist to infinity, converging on the Eye of God**The Big Picture**• Now that we understood why these two specific sequences did what they did, we went in search of a more general rule • To make sense of what we found, we need make a few important observations**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 Every Golden Rectangle Has 4 Eyes of God, Not Just 1**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:**Striking Gold**• There are many Golden relationships that these four Eyes generate. For example: • The four Eyes form a Golden Rectangle • The rectangle with upper right-hand corner at E2 and lower left at the origin is Golden • So is the rectangle with upper right-hand corner at E4 and lower left at the origin