# Complex Fibonacci

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:
`(φz - -φ-z)/√5`
or equivalently:
`(φz - eiπzφ-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 `(φz - cos(πz)φ-z)/√5` 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 `0.1 <= im(z) <= 0.2`:

Again, the red parallelogram illustrates the recurrence.