# Math: Fibonaci Sequence Video Fallacy 📺

By Xah Lee. Date:

Beware of Math Fallacies Spreading on The Net

Recently this video is spreading fast on the net.

A beautiful video indeed, but filled with fallacies.

It would be good for math teachers to use this vid as a intro, then proceed to go into detail of every piece of math shown in the video, and point out the parts that are false.

First, there's the Fibonacci sequence: {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …}. This is formed by adding the previous 2 terms, starting with 0 and 1.

[0:32] The Fibonacci spiral, constructed by fitting quarter circle arcs on stacks of squares with sides of Fibonacci numbers. The spiral formed is a approximation of a spiral called the Equiangular Spiral (aka logarithmic spiral).

The Fibonacci spiral isn't a true spiral. That is, it has kinks at every place the square connects (to see the kinks, think of a circle. A circle is a curve that has constant curvature at any point on the curve. Different sized circle have different curvatures. So, by connecting arcs of 2 circles from different sizes, the curvature will suddenly jump at the connection point. Although, such connection will be superficially smooth, but on a second level it is not smooth (this is the idea of derivative). You cannot design roads, train tracks, or roller coasters by just connecting circles of different sizes.).

Here, you should remember that the stacking of circular arcs as done by Fibonacci spiral is only a approximation to a particular equiangular spiral, called Golden Spiral.

A easy way to understand equiangular spiral is that it's a spiral such that its growth rate is constant. (For much more and many pictures, see: Equiangular Spiral.)

The equiangular spiral is a family of spirals, with a constant angle as parameter. The Fibonacci spiral is a approximation of just one of them, called the Golden spiral .

then, they fit this spiral to the nautilus shell. Note again that it's the equiangular spiral the shell is based on, not the Fibonacci stacks of circular arcs, and nor is the shell shaped like the particular equiangular spiral that Fibonacci spiral is approximating.

Here is a quote from Wikipedia Nautilus:

The nautilus shell presents one of the finest natural examples of a logarithmic spiral, although it is not a golden spiral.

[1:30] then, they draw a rectangle outline of the shell, then cut it in half, then take a diagonal and sweep it down... as if to show that a rectangular hull of the shell has the right proportions to construct the golden rectangle. This is pure garbage.

[1:41] then, they sweep a circle and spit out speeds at every angle of (2*π - 2*π/φ) radians to show how it forms the sunflower pattern. This is also borderline of pseudoscience. You can spit out seeds in such a regular way but using just about any other constant angle and still be able to show the result being similar to the patterns of sunflower seed's arrangement.

[2:53] then, they pick out certain dots of the sunflower seeds, connect lines, to form the pattern on dragonfly's wings. This is again purely fairy tale. Am not quite sure what rule they used to pick out the dots. But it doesn't matter. The closest real math this part of vid is trying to show the idea of Voronoi diagram and Delaunay triangulation .