Logarithmic Spirals

In my free time, if I'm really bored I actually draw the logarithmic spiral.  The logarithmic spiral is based off Phi (1.618).  The width of each spiral section is equal to the next by multiplying it by Phi.  It's kind of cool to do if you're bored.  You need a pencil, a calculator, a ruler, and at least one compass, but I find that a couple compasses of different sizes work best. 

Firstly, you must create a baseline for your spiral by bisecting your paper.  If you look down at my spiral you can see the pencil marks where my bisection occurred. 

Secondly, make a dot in the middle of your baseline.  Set a compass to an inch and make a semicircle keeping your compass on the dot you drew.

Next take your calculator and multiply the diameter of your semicircle (1 inch) by the rounded off Phi (1.62).  (You should be able to do that in your head, too, LOL.)  The answer of course is 1.62.  That's the width of your next spiral section.  I would dot where that is.

Now find the width of all the spiral so far (2.62).  Divide that by two to find where you should put your compass.  Measure that out, and dot it as well.  Now place your compass on the new dot and draw another semicircle, continuing from the end of the last one.

Recall the width of this last spiral section (1.62) and multiply that by Phi rounded to the hundredths.  The answer is 2.6244.  I would round that off to the hundredths.  That makes 2.62 the width for your next spiral section.  Dot that on the opposite side of the middle semicircle.

Now measure the width of the whole spiral to find the new place to put your compass by dividing that by two.  (2.62 + 2.62 = 5.24 ~ 5.25 / 2 = 2.125)  Dot that, and put your compass there.  Continue the spiral by drawing another semicircle from the tail of your last one.

You should now have three semicircles.

Continue the process of multiplying the last spiral section by rounded Phi, measuring it out on the bisection and dotting it, dividing the whole spiral by two, placing the dot to put your compass on, and drawing another semicircle.  You can do this as long as you want until you get bored with it, run out of paper, or your compass(es) is too small to continue.

Below my drawing is also a picture of a chambered nautilus, which is one of natures many examples of the logarithmic spirals.

Tags: Scholarship, PHI