There are a couple of ways to extend the Fibonacci series to a continuous function on real or complex numbers. Here’s a nice one: the grid shows the mapping of the unit strip, `0 <= re(z)`

, `0 <= im(z) <= 1`

, grid points are 0.1 apart. The real axis is picked out in red, notice that it crosses itself exactly at the points of the Fibonacci series – twice at 1 after executing a loop-the-loop (clicking on the image should give you a larger version).

The function is just the standard Binet formula, interpreted over the complex plane:`(φ`

^{z} - -φ^{-z})/√5

or equivalently:`(φ`

^{z} - e^{iπz}φ^{-z})/√5

For negative values, the function turns into a nice spiral (the `exp`

component dominates). Here is the `0 <= im(z) <= 0.1`

strip:

The red parallelogram illustrates `F(z+2) = F(z+1)+F(z)`

still holds, even for our complex quantities.

Most of the action takes place around the origin, so let’s zoom in:

The image of the real axis passing through the points -1,1,0,1,1,2,… is clearly visible (the real axis is on the outside, the origin is the leftmost point on the red parallelogram).

Sometimes the series is extended as `(φ`

which produces a mapping that takes the real line to itself, ie. taking the real part of the exponential, which is also nice, here we see the mapping of the strip ^{z} - cos(πz)φ^{-z})/√5`0.1 <= im(z) <= 0.2`

:

Again, the red parallelogram illustrates the recurrence.