golden spiral

"My new hypothesis: If we're built from spirals while living in a giant spiral, then is it possible that everything we put our hands to is infused with the spiral?"¹ -Max Cohen in the motion picture pi.

The golden spiral is an absolute shape in mathematics and less exact in chaotic nature. Pythagoras first discovered the spiral in the 5^th century B.C.. He derived the shape from the golden rectangle which has a specific ratio, the golden ratio. When the ratio is squared, it leaves a smaller rectangle behind that has an identical golden ratio as the previous rectangle. Even if the squaring continues to infinity it will always have the same result, which is the property that separates the golden rectangle from the rest.

     Illustrating this ratio results in concentric rectangles illustrated in golden ratio by SpudTater. Drawing a curve connecting the opposite corners of these rectangles forms a golden spiral. The spiral is found in nature, in a ram's horns, in the nautilus shell, a fingerprint, the cochlea of an ear, DNA, a whirlpool, amd the shape of the Milky Way Galaxy. A representation of the golden spiral within concentric golden rectangles is below. (It will not display correctly in Netscape 4.7.)

                                                                                                                                      
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¹pi, http://www.pithemovie.com

How to plot a Golden Spiral

Recently my mom went to a conference of sorts. The speaker was talking about Fibonacci's mathematics, and how they relate to the world around us. I do not wish to restate what other people have already, so I will get on to how to draw a golden spiral.

First thing to know when drawing a golden spiral is the Fibonacci sequence which is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144….

To start graphing, it is best to use graphing paper, and pick on the upper right hand side of the paper (not too far), playing around with positions will help you find that perfect place. Move one block up from your starting point and draw a three block by three block square with the starting point + 1 box in the lower left hand corner. Find the exact middle of that 3x3 box, and mark it. Drawing the next size up (5), draw a box that follows the same line as the 3x3 box, count 5 boxes, and make the square; find the exact middle of the 5x5 box. This is your second point in the spiral. Next is the 8x8 box, and it shares a line with the 5x5, and its point is in the exact middle as well. After that would be the 13x13 box; and so on.

         ___________________
        |                   |
        |                   |
        |                   |
        |                   |
        |         0         |
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        |                   |
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         _____  ___ ________
        |     |    |        |
        |  0  |    |        |
        |     |D_0_|        |
        |_____ X__     0    |
        |          |        |
        |    0     |        |
        |          |        |
        |__________|________|

This picture is an example of a golden spiral, if you were to continue in the pattern, and connect the 'o' you'd have a spiral. Also, the 'X' is the starting point, and the 'D' is the starting point + 1. Try it sometime. It is fun, and algorithmic; what could be more fun than that, beside a barrel full of soy monkeys.

Other examples of Golden Spirals in nature are parrots beaks, galaxies, hurricanes, pine cones, pineapples scales, sunflowers, breaking waves, and filaries (a seed from a weed plant in California).