# Square Root Mapping

Robert P. Munafo, 2023 Aug 9.

A transformation in which points are mapped onto new positions by the function

f(z) = C_{1} + C_{2} (z-C_{3})^{(1/2)}

where C_{1}, C_{2} and C_{3} are constants; usually C_{1}=C_{3} and
the absolute value of C_{2} is less than 1. Notice the symbol +
indicating that the square root has two possible values — each point
maps onto two new locations, positioned 180^{o} apart from one
another. Thus, if the mapping has been applied to an image, you get
two distorted copies of the original image.

A close approximation to the square root mapping can be seen in the
bifurcation in the filaments around islands. For example, when
zooming in towards an island, there is an area of 2-fold symmetry,
then further in you find an area of 4-fold symmetry. The inner area of
4-fold symmetry is very close to what you would get by taking the
2-fold symmetry features and applying a square root map with C_{1} =
C_{3} equal to the coordinates of the island at the center and a C_{2}
corresponding to the conbined rotation and size ratio between the
2-fold and the 4-fold features.

On his web page Quick Guide to the Mandelbrot Set, Paul Derbyshire sums it up pretty well with this description:

*The familiar double spiral is another example of this phenomenon.
Inside a seahorse spiral there is a mini Mandelbrot; near the mini
Mandelbrot there are two spirals joined exactly at the point where the
minibrot is. [...] [in these examples] the bottom image is invariably
two of the top image joined at the marked point, sometimes in a
convoluted fashion.*

The square root mapping can also be applied in reverse (particularly useful when you have a mystery Mandelbrot image and want to find its coordinates); this reverse mapping is squaring.

### Relationship to Leavitt Navigation

The acquired skill known popularly as "morphing" requires a good understanding of the squaring and square root maps; see Leavitt navigation and exponential map.

revisions: 20020517 oldest on record; 20230809 link to Leavitt navigation and exponential map

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

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Aug 09. s.27