[04 Sep 2011] Fern [image source] Here’s a recent question from
reader Pehr in Sweden: First of all, wonderful site. Love it.
I’ve been studying the polar coordinates section hoping to
expand my knowledge on the equiangular spiral. The interactive
tools are great, though I’m having a hard time to derivate the
exact mathematical solution to why the resulting function for
the golden spiral is r = ae^(b(theta)) For some background on
Pehr’s question, see: Spirals are common in nature and have
inspired mathematicians for centuries. Aloe spiral [Image
source] Spiral galaxy NGC 5194 [Image courtesy NASA] The Golden
Spiral that Pehr is asking about is a special case of the
logarithmic spiral. Logarithmic spirals grow such that the angle
of a line from the center of the spiral to the tangent to the
curve at that point is constant. This is why they are also known
as “equi-angular” spirals. To see what this means, the 3 acute
angles marked in the following fern image are approximately 80°
We normally use functions in P