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, . This is De Moivre’s Formula.
In particular, this is when the complex number is expressed in polar form where the radius is length (magnitude) between the complex number and the origin on a 2D plane while the angle 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.What shapes do you see?
Does this remind you of anything you’ve seen in nature?