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).

fibonacci

The function is just the standard Binet formula, interpreted over the complex plane:
z - -φ-z)/√5
or equivalently:
z - 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:

spiral2
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:

spiral1

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 z - 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:

cosspiral

Again, the red parallelogram illustrates the recurrence.

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