Transcript Converging on the Eye of God D.N. Seppala-Holtzman and

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
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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
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Φ, the Devine Proportion
Golden Rectangles
Golden Spirals
The Fibonacci numbers
The Divine Proportion
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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
A Line Segment in Golden Ratio
Φ: The Quadratic Equation
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The definition of Φ leads to the
following equation, if the line is divided
into segments of lengths a and b:
ab
a

a
b
The Golden Quadratic II
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Cross multiplication yields:
a  ab  b
2
2
The Golden Quadratic III
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Setting Φ equal to the quotient a/b and
manipulating this equation shows that
Φ satisfies the quadratic equation:
   1  0
2
The Golden Quadratic IV
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Applying the quadratic formula to this
simple equation and taking Φ to be the
positive solution yields:
1 5

 1.618
2
Two Important Properties of Φ
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1/ Φ = Φ - 1
Φ2 = Φ +1
These both follow
directly from our
quadratic equation:
   1  0
2
Constructing Φ
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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
A
C
Properties of a Golden
Rectangle
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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
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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 Golden Spiral
The Eye of God
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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
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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
Sunflowers
Pinecones
Pineapples
The Chambered Nautilus
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 …
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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/F
ibonnaci
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Some Examples
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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
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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
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These two remarkable mathematical
structures are closely interconnected
The ratio of sequential Fibonacci
numbers approaches Φ as the index
increases:
Fn 1
lim

n  F
n
All of This is Background; Now
Our Story Begins
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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
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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
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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
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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
3
4
5
6
7
8
9
F(n)
2
3
5
8
13
21
34
F(n+1)F(n)/F(2n-1)
1.2
1.15384
1.176471
1.168539
1.171674
1.170492
1.17095
F(n+1)F(n)/F(2n+2)
0.2857
0.2727
0.2777
0.2758
0.2765
0.2763
0.2764
Stable Quotients as Limits
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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:
Fn 1 Fn
lim
 1.1708
n  F
2 n 1
Fn 1 Fn
lim
 0.27639
n  F
2n2
A Surprise
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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??!!
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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
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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:
Fn 1 Fn
 1
Fn 1 Fn
 1
lim

lim

n  F
n  F
1
1
2 n 1
2
n

2


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The Search for “Why”
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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
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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
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This lemma allowed us to rewrite the x
and y coordinates of the Eye of God as:
the x-coordinate of the E1 =
Fn 1 Fn
lim
 lim  2 n Fn
n  F
n 
2 n 1
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the y-coordinate of the E1 =
Fn 1 Fn
lim
 lim  1 n Fn
n  F
n 
2n2
Geometric Interpretation I
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Now that we had the x and y coordinates 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:
Geometric Interpretation II
Geometric Interpretation III
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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
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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
Every Golden Rectangle Has 4
Eyes of God, Not Just 1
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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 of God
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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:
Fn 1 Fn
lim
 lim   n 1 k Fn
n  F
n 
2nk
Striking Gold
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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
Many Golden Relationships
E3
E4
E2
E1
Tying it All Together
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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:
Unifying Theorem
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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.
Some Observations
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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
Overview
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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
Conclusion
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There are infinitely many paths that
converge upon the Eye of God