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
(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:
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.