Converging on the Eye of God

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Converging on the Eye of God

• D.N. Seppala-Holtzman
• St. Josephs College
• faculty.sjcny.edu/holtzman

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Announcements

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

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Several Mathematical Objects Play Central Roles

• F, the Golden Ratio
• Golden Rectangles
• Golden Spirals
• The Eye of God
• The Fibonacci numbers

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The Divine Proportion

• The Golden Ratio, sometimes called the Divine
  Proportion, is usually denoted by the Greek
  letter Phi F
• F 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

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A Line Segment in Golden Ratio
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F The Quadratic Equation

• The definition of F leads to the following
  equation, if the line is divided into segments of
  lengths a and b

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The Golden Quadratic II

• Cross multiplication yields

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The Golden Quadratic III

• Dividing by b2 and setting F equal to the
  quotient a/b we find that F satisfies the
  quadratic equation

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The Golden Quadratic IV

• Applying the quadratic formula to this simple
  equation and taking F to be the positive solution
  yields

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Two Important Properties of F

• 1/ F F - 1
• F2 F 1
• These both follow directly from our quadratic
  equation

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F Is an Infinite Square Root
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F as a Continued Fraction
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Constructing F

• 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 F to 1.

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Constructing F
B
ABAC
C
A
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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 F, 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 F, a Golden ancestor
• Both constructions can go on forever

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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.

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The Golden Spiral
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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

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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 gt2)
• Thus, the sequence is
• 1 1 2 3 5 8 13 21 34 55

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The Binet Formula

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

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The Binet Formula II

• The Binet Formula can be expressed in terms of F
  

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The Fibonacci F Connection

• Taking the limit of the quotient of
• sequential Fibonacci numbers in their
• Binet form yields

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Fibonacci and F in Nature

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

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Sunflowers
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Pineapples
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The Chambered Nautilus
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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 F and the
  Fibonacci numbers
• He read Livios book and was deeply impressed
  with the many remarkable relations, connections
  and properties he found there

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Francisco Rangel

• Having observed that the limiting ratio of
  Fibonacci numbers yielded F, 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

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Francisco Rangel II

• Knowing that a Golden Rectangle has sides in the
  ratio of F to 1, and knowing the relationship of
  F to the Fibonacci numbers, he considered
  aspiring Golden Rectangles, rectangles with
  sides equal to sequential Fibonacci numbers
• These would have areas Fn1Fn

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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
• Fn1Fn / F2n-1 and Fn1Fn / F2n2

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The Spreadsheet
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Stable Quotients as Limits

• In more mathematical terminology, he found
  several convergent limits. The results from the
  previous slide were

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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!

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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

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The Investigation Begins

• First off, we computed the coordinates of the Eye
  of God in closed form in terms of F. We got

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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

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The Golden Spiral
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Two Theorems

• We proved that the limits that we had found
  earlier corresponded precisely to these two
  expressions involving F. That is

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Sketch of Proof

• Use Binet formula and observe that

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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

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Lemma

• For any integer k, we have

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Proof of Lemma

• Observe

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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

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The Sequences

• The first several terms of these sequences are
• x sequence F1F1 F0F2 F-1F3 F-2F4
• y sequence F-2F1 F-3F2 F-4F3 F-5F4

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Related by Powers of F

• Note that the y-coordinate can be obtained from
  the x-coordinate by dividing the latter by F3.

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Recall

• the x-coordinate of the Eye
• the y-coordinate of the Eye

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Hold That Thought!

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

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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

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Eye of God 1
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Eye of God 2
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Eye of God 3
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Eye of God 4
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The 4 Eyes
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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

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The 4 Eyes
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Retrieve that Thought

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

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Golden Offspring
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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

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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 F

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The Eyes of the Offspring
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The Eyes of the Ancestors

• A similar result holds for Golden Ancestors

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Tying it All Together

• This leads us to our unifying theorem
• We proved the following

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Unifying Theorem

• The limit of (Fn1 Fn )/ Fsnk behaves as
  follows
• It diverges to infinity if s lt 2
• It converges to zero if s gt 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.

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Sketch of Proof

• The Binet formula yields

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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

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Conclusion

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

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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.