Today is one of the annual celebrations of my quixotic quest to have the “days” associated with particularly important numbers, like Phi (φ) and Pi (π), placed upon days that actually reflect the math behind the numbers. The number Phi (φ) is the ratio between a longer line segment and a shorter line segment in a variety of geometric shapes, including the famous golden rectangle, pentagrams, and the Fibonacci spiral. August 14th is the day in the calendar year that best creates this same ratio between the total length of the year and the date in question. Therefore, August 14th is, or rather should be, celebrated internationally as Phi Day.
Since I run this joint, it is officially Phi Day at The Finch & Pea. If we had merchandise, there would probably be a discount. I suspect this would not change the likelihood that you would buy The Finch & Pea merchandise.
— Russ Creech (@RussCreech) April 2, 2012
For some, words like “Fibonacci” and “fire” are enough to set the little reward bells jingling in your nerd/pyro synapses. I, however, am made of more quantitative stuff1. How “rough” was Katy’s rough Fibonacci spiral (of FIRE!)?
To answer that, we first need a quick tutorial on how to draw a Fibonacci, or Golden, spiral. To create a spiral of your own, you tile squares with side lengths in the Fibonacci sequence (giving you boxes 1×1, 1×1, 2×2, 3×3, etc.) and draw a continuous curve between opposite corners. Like so:
If we try to draw the tiled boxes that would create Katy’s spiral, it looks like this.
Now, Katy is drawing free hand with a flaming branch. So, her squares aren’t precisely square. I accommodated that by calculating the area of each box and using the square root of the area as the square equivalent side length for each box. I normalized these side lengths to their position in the Fibonacci sequence. For example, the largest box (1) should have a normalized side length of 34. I tried the normalization in both directions, to the largest box (large to small) and to the smallest box (small to large).
Katy does a really good job with the three largest boxes (1-3). For the smaller boxes starting with box 4 the issue is that the turns are actually too large. This may be the effect of the camera settings or the limited resolution of fire as a drawing medium.
If we create an ideal Fibonacci tiling pattern and visually see how well it fits, we find that the flame is too wide to resolve any boxes smaller than box 4. In this one case, it is permissible for the craftsman to blame her tools.
It does leave a bit of a dilemma that no numbers can address. The fire is what makes the spiral “rough”, but the fire also makes the spiral “awesome”. Which would you choose? No, no, that’s the wrong answer. Pick the other one. Good.
1. I am also made of star stuff. And, sterner stuff, too, called scar tissue.