One of the reasons why mathematics is so amazing is that even a simple at first glance concepts have interesting behaviors and features. I think that the bifurcation diagram of the logistic map is a great example.
The logistic map is given by the equitation:
xn+1 = rxn(1 - xn), x0 < 1, r ∈ (0,4]
Below you may generate its bifurcation diagrams for different ranges and for different values of the x0. It's amazing that those diagrams have fractal nature - self-similarity is even visible at first glance.
X-axis represents values for different r values, Y-axis represents x0...n element value, n < 90 because in real world we can't show elements ad infinimum.
rstart>0: rstop≤4: x0<1: