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==Chaos and the complex squaring map==
The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but angles which differ by multiples of 2π ([[radians]]) are identical. Thus, when the angle exceeds 2π, it must ''wrap'' to the remainder on division by 2π. Therefore the angle is transformed according to the [[dyadic transformation]] (also known as the 2x mod 1 map). As the initial value ''z''<sub>0</sub> has been chosen so that its argument is not a rational multiple of π, the [[Orbit (dynamics)|forward orbit]] of ''z''<sub>''n''</sub> cannot repeat itself and become periodic.
More formally, the iteration can be written as:
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