An Imaginary Tale: De Moivre’s Formula in a Gif

I’ve been reading the book, An Imaginary Tale: The Story of i [the square root of minus one] by Paul J. Nahin and it’s really interesting!  I’m almost to chapter 4 and so far it’s been introducing some nifty ideas about complex numbers.  Among the content, we find a very convenient way of calculating the value of a complex number when multiplied by itself an integer amount of times.  That is, z^n=(a+bi)^n=r^n \cdot \left[ \cos\left(n\theta\right)+i\sin(n\theta)\right]\mid n\in \mathbb{Z}^+.  This is De Moivre’s Formula.

In particular, this is when the complex number z = a+bi \mid a,b\in\mathbb{R} is expressed in polar form (r,\theta) where the radius r = \sqrt{a^2 + b^2} is length (magnitude) between the complex number and the origin on a 2D plane while the angle \theta = \tan^{-1}\left(\frac{b}{a} \right) is the measure between the positive side of the real axis and the complex number.  The tricky part about calculating the value of θ depends on which quadrant the complex number is located.  If θ is in quadrant 2 or 3, you may have to add or subtract integer amounts of π.

So what’s the big deal?  While experimenting with this formula on GeoGebra, I found some awesome patterns that form when you graph the positive integer powers of a complex number.  The gif below shows an animated mapping of when some complex numbers with a fixed radius less than one are multiplied by themselves from 1 to 100.  I can’t help but watch it over and over again.

De Moivre's Formula

[latex]z^n\text{ such that }|z|<1\text{ and } n\in[1,100][/latex]

What shapes do you see?

Does this remind you of anything you’ve seen in nature?

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