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y!

***B* ! *

* ! ** ! ** ! *

--*--------O-----*--xC ! A

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

The Golden Spiral is a logarithmic spiral; that is, its radius growsexponentially. Because of this, it's easiest to put it in polarcoordinates. (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 relyheavily on them.)

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

r = a*e^(k*@)

where r is the distance from the origin (the radius), @ representsthe 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 whata and k are.

To derive the spiral formula, we also need to remember that the ratioof the sides of a golden rectangle is equal to (1 + sqrt(5))/2, oftenrepresented 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 theGolden 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 ourequation r = a*e^(k*@) and call r the length of OA to get thefollowing:

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 onthe 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.

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Thenk * (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 theexponents 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 offby the spiral fit the Golden Ratio), and the one we've just derivedusing the polar equation. Since these are both equal to the ratio ofOB 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 equationr = a*e^(k*@). We'll solve for k now by taking the naturallogarithm 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 GoldenSpiral had an equation of the form r = a * e^(k * @)? We now know whatk 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'tdetermined yet, phi is the golden ratio = 1+sqrt(5))/2, and @/thetais the angle that the radius has opened up to, measured from thepositive 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

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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 whatabout a? Well, this equation will always produce a golden spiral, nomatter what a is. (Can you see why? Look back at how a golden spiralis defined. How do different values of a affect the graph?)