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Golden Spiral

Date: 03/23/98 at 11:46:10
From: Erica Dobney
Subject: Golden Spiral

What is the equation of the Golden Spiral?

Date: 03/31/98 at 14:53:20
From: Doctor Barrus
Subject: Re: Golden Spiral

Hi, Erica!

If I understand the Golden Spiral the same way you do, then here's 
what we're thinking:

             *    ! * 
           *      !   *
          *       !    *
         *        !     *
         C        !     A

This is a view of part of one turn of the spiral. If you were to 
construct a line segment from A to B, the triangle OAB would form 
half a golden rectangle. The same thing would be true if you connected 
B and C for triangle OBC.

The Golden Spiral is a logarithmic spiral; that is, its radius grows 
exponentially. Because of this, it's easiest to put it in polar 
coordinates. (If you haven't learned about polar coordinates yet, 
don't worry - they're pretty simple to learn. You might want to talk 
to your teacher, though, because this explanation is going to rely 
heavily on them.)

As with every logarithmic spiral, the Golden Spiral's equation can be 
written in the form of the polar equation 

     r = a*e^(k*@)

where r is the distance from the origin (the radius), @ represents 
the Greek symbol theta, which is the angle the graph is open up to, 
and a and k are constants. To get the equation for the Golden Spiral 
(and not just any logarithmic spiral), we'll need to find out what 
a and k are.

To derive the spiral formula, we also need to remember that the ratio 
of the sides of a golden rectangle is equal to (1 + sqrt(5))/2, often 
represented by the Greek symbol phi.

If the spiral above is a Golden spiral, then we know the following:

   The lengths of the parts of the axes cut off by the spiral fit the 
   Golden ratio. That is,

     (length of OB)/(length of OA) = phi = (1 + sqrt(5))/2

Also, since we're dealing with polar coordinates, we'll use our 
equation r = a*e^(k*@) and call r the length of OA to get the 

  length of OA = a*e^(k*n*2*pi),   where n is an integer. 
                                   (We put n*2*pi in for @/theta,  
               = a*e^(k*2n*pi)     because we want to start on   
                                   the positive x-axis.)

Now call r the length of OB:

  length of OB = a*e^[k*(n*2*pi + pi/2)]  Here we add pi/2 to the       
                                          angle we used for point A  
               = a*e^[k*(2n+1/2)*pi]      to indicate that B is on the    
                                          positive y-axis.

                                         k * (2n+1/2] * pi
                                    a * e
  (length of OB)/(length of OA) =  ---------------------------
                                             k * 2n * pi
                                        a * e

The a's will divide out, and to divide the e-terms we subtract the 
exponents to get

                                   [k * (2n+1/2) * pi] - (k * 2n * pi)
  (length of OB)/(length of OA) = e

                                   [k * (1/2) * pi]
                                = e

So now we have two expressions for the ratio of OB to OA - the one 
from the definition of a Golden Spiral (that the axes' lengths cut off 
by the spiral fit the Golden Ratio), and the one we've just derived 
using the polar equation. Since these are both equal to the ratio of 
OB to OA, we can set them equal to each other to get:

    phi = (1 + sqrt(5))/2 = e^[k * (1/2) * pi]

    \________  _________/   \_______  _______/
             \/                     \/
      from Golden Ratio     from polar equation

Now, remember that in order to get the equation for the spiral, 
we need to know a and k, the constants in the polar equation 
r = a*e^(k*@). We'll solve for k now by taking the natural 
logarithm of both sides:

    ln(phi) = ln(e^[k * (1/2) * pi])
            = k * (1/2) * pi

Now multiply both sides by (2/pi):

(2/pi) * ln(phi) = (2/pi) * k * (1/2) * pi
                 = k

Now, remember way back at the beginning when we said that the Golden 
Spiral had an equation of the form r = a * e^(k * @)? We now know what 
k is. So we can now say that the equation for the Golden Spiral is

                   (2/pi) * ln(phi) * @
          r = a * e

where r is the radius of the spiral, a is another constant we haven't 
determined yet, phi is the golden ratio = 1+sqrt(5))/2,  and @/theta 
is the angle that the radius has opened up to, measured from the 
positive x-axis.

We can write this another way. Using the rules of exponents, the 
exponential function, and natural logs, we can say that

    k = (2/pi) * ln(phi)            moving the coefficent of the  
                                    natural log to the  exponent of   
      = ln[phi^(2/pi)]              its argument - a natural log rule 

             ln[phi^(2/pi)] * @  
    r = a * e                       substituting k back into 
                                    r = a*e^(k*@)

         /    ln[phi^(2/pi)] \ @
      = a * (e              )  )    a rule of exponents
         \                    /
             / (2/pi) \ @
      = a * (phi       )            simplifying the inside using a  
             \        /             natural log rule

               [(2/pi) * @]

    r = a * phi                     a rule of exponents

Here, finally, is our equation, given in polar coordinates. But what 
about a? Well, this equation will always produce a golden spiral, no 
matter what a is. (Can you see why? Look back at how a golden spiral 
is defined. How do different values of a affect the graph?)

Well, this has been a long answer, but I hope it's helped. Good luck!

-Doctor Barrus,  The Math Forum
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Associated Topics:
College Euclidean Geometry
High School Euclidean/Plane Geometry

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