# Squaring

Robert P. Munafo, 2002 Apr 18.

Refers to the act of multiplying complex numbers by themselves, and the analogous transformation seen in images such as Julia sets, that are generated by functions that have a quadratic behavior.

When you square a complex number, you can look at the operation two ways. One way is like this:

C^{2} = C × C = (a+bi) × (a+bi) = a^{2} - b^{2} - 2abi

and the other is to square the point's radius and double the angle. The two are equivalent.

You can also "square" any set of points in a 2-d space, such as an image or a part of an image. In this case you need to specify a center from which the angles and radii are to be measured. The squaring operation transforms the image in a way that makes it larger, stretches the outer parts more than the inner parts, and (if the original image completely surrounds the center point) wraps the image around twice, overlapping itself everywhere so that there are two parts of the squared image in any one area of the plane.

Image squaring shows up in the embedded Julia sets. For example, look at figures 1 and 3 on the paramecia page. The image in figure 3, when squared, gives an image very similar to that in figure 1.

The opposite operation is the square root mapping

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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