# Fermat’s spiral

Whoa, holy crap. I sat down to try to understand Fermat’s spiral a little bit better last night, and I ended up with a pretty cool animation.

I started with the Wikipedia page, and there I saw this image, with the following caption:

The pattern of florets [

say, of a sunflower], produced by Vogel’s model, (central image). The other two images show the patterns for slightly different values of the angle.

Those “other two images” had me wondering… for *what* different values of the angle? I wanted to see what the curve would look like with different angles (see the equations provided on that page), so I wrote a little script with my new favorite programming package, pygame, creating one big loop from 0° to 360°. Some very interesting patterns result!

(Click on image above to view video. It may take a few minutes to load, depending on your connection speed. A much smaller video is available here.)

The only thing changing here is the angle in the second equation (top left of video). ** c** is just a constant to stretch things out whatever amount, and

**is all the integers from 1 to 140 (so we get 140 dots per changing angle = 140 dots in every video frame).**

*n***is the radius of each dot from the faint gray dot in the center of the window, and**

*r***Θ**is the angle of each dot’s location, with 0° straight

When do you think the pattern will repeat?

Addendum/Correction: Since I originally posted this, I rewrote the code in Processing, and discovered that I had an error that is evident in the movie in this post. I don’t know why I didn’t notice it before. You can see the error kick in around whenΘ=nx 37° … More and more dots show up in the wrong position, soon after resulting in some major weirdo non-spiral action (as is clearly evident at, for example, 44.4°). This error is the result of me treating radian values as if they are values in degrees. Dammit, if this doesn’t catch me again!Still looks neat, though. ^_^

## 3 Comments

**Anita replied:**

500 dots, sized proportional to radius, using golden angle of 137.5Â°…

June 26th, 2007 at 7:14 pm. Permalink.

**Brent Fitzgerald replied:**

lovely. great to see you using pygame for visual experiments like this.

June 27th, 2007 at 2:17 am. Permalink.

**karlo sostaric replied:**

my new vision double spirale (spheric helix, yin yang!

authorSeptember 24th, 2007 at 3:31 am. Permalink.