Explore what happens when the power of the Mandelbrot zoom is a fractional value. In this case the exponent is 2.5. The standard Mandelbrot 2, and other integer values produce similar fractals. However things start to become a little wild when we use a fractional power. (z = z2.5 +c)
Raising a value to the power of 2.5 is actually the same as introducing a square root in the exponent:
As a result of introducing the exponential, the function is no longer continuous over the complex plane. So we say the function is non-analytic. The square root function is not continuous, it has areas where there are sharp branch points. In fact, there are multiple results to the square root function, so we are forced to choose one. The most obvious choice is the primary root. These sharp branch points propagate throughout the fractal as we iterate over the function again and again. This means that the fractal is non-continuous, and you can see break points and islands in this video.
Fractal : 2.5th Power Mandelbrot Set (z = z2.5 +c)
Imaginary (note: exponential e-12): -1.211002419848878844842099e-12