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?*

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