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

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: …

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 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
0.2727 5
0.2777 6 8
0.2758 7 13
0.2765 21
0.2763 9 34
0.2764

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 ≈ and y ≈
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-2F1 Φ-3F2 Φ-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. 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

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

51. Many Golden Relationships

52. Tying it All Together
At this point, we went in search of a unifying theorem
We wanted some general rule that tied this all together
We proved the following:

53. 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.

54. Some Observations
Note that 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

55. Overview We began with a suspicion This led to a discovery
Empirical experimentation gave it credence
Rigorous proof removed all doubt
Geometry elucidated the truth
All of this led to a deeper understanding

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