The animation shows the image in the complex plane of circles of
increasing radius, centre at 0, after applying a certain polynomial
function *f* of degree 4. It can be seen that the image curve
swells from a point to a curve that wraps four times round the origin.
It would eventually become indistinguishable from four laps of a
circle.

The fact that at some stage the image curve must pass through the
origin shows that there is some value of *z* for which
*f*(*z*) is zero. (In fact, it happens four times.)

The function in this example is

*f*(*z*) = (1+*i*)*z*4 +
3*z*3 – *iz* + (2+*i*).

The radius of the circle on which *z* lies goes from
zero to 2.8 in steps of 0.02, then from 2.85 to 4.2 in steps of
0.05, then from 4.4 to 8.0 in steps of 0.2, then from 8.5 to 13.0
in steps of 0.5.