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  1. Converging on the Eye of God D.N. Seppala-Holtzman and Francisco Rangel St. Joseph’s College

  2. 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

  3. Several Mathematical Objects Play Central Roles • Φ, the Devine Proportion • Golden Rectangles • Golden Spirals • The Fibonacci numbers

  4. 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

  5. A Line Segment in Golden Ratio

  6. 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:

  7. The Golden Quadratic II • Cross multiplication yields:

  8. The Golden Quadratic III • Setting Φ equal to the quotient a/b and manipulating this equation shows that Φ satisfies the quadratic equation:

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

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

  11. 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.

  12. Constructing Φ B AB=AC C A

  13. 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

  14. 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.

  15. The Golden Spiral

  16. 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

  17. 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

  18. Sunflowers

  19. Pinecones

  20. Pineapples

  21. The Chambered Nautilus

  22. 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 …

  23. 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:

  24. 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

  25. 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

  26. The Fibonacci – Φ Connection • These two remarkable mathematical structures are closely interconnected • The ratio of sequential Fibonacci numbers approaches Φ as the index increases:

  27. 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

  28. 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

  29. 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

  30. 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

  31. The Spreadsheet

  32. 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:

  33. 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!

  34. 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

  35. 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]

  36. The Next Step • Next we proved that the limits that we had found earlier corresponded precisely to these two expressions involving Φ. That is:

  37. 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?

  38. 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”

  39. 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 =

  40. 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:

  41. Geometric Interpretation II

  42. 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

  43. 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

  44. 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

  45. Eye of God 1

  46. Eye of God 2

  47. Eye of God 3

  48. Eye of God 4

  49. 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:

  50. 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