Mandelbrot Magic

Higher Order Exponents

Now that you understand a litlle bit about the functioning of the Mandelbrot Set, we can start to change the equation and explore what happens to the resulting fractal image. Below you will find the Mandlebrot Set and Julia Sets for Z2 + C followed by the Mandelbrot Set and Julia Sets that result from iterating Z4 + C. That is, Z * Z * Z * Z + C. Recall that in polar coordinates, squaring a number just means doubling the angle and squaring the distance of the point from the origin. Raising a complex number to the 4th power simply means quadrupling the angle and multiplying the distance of the point to the origin by itself four times. In both cases, the new point is rotated around the origin, and scaled.

Zn+1 = Zn2 + C




Zn+1 = Zn4 + C



Explore the Z4 Mandelbrot Set and Julia Sets above by <Alt>-clicking in the windows to zoom in. Observe that all the same relationships apply with the Z4 system as you've learned about with the Z2 equation.

For instance, the Bifurcation Rule applies, but in the Z4 Mandelbrot Set, the branches always quadruple in number, from 4 arms to 16, 64, 256, etc.

Symmetry

Explore the symmetry of the Mandelbrot Sets and Julia Sets above. The Z2 Mandelbrot Set has one line of symmetry: it is reflected around the horizontal X-axis. The Z2 Julia Set however, has two axes of symmetry: it is made of two copies of the same shape, rotated around the origin.

Questions:
How many lines of symmetry does the Z4 Mandelbrot Set have? [ ]

How many lines of symmetry do the Z4 Julia Sets have? [ ]

Assuming the same pattern holds for other exponents,
How many lines of symmetry would the Z3 Mandelbrot Set have? [ ]

How many lines of symmetry would the Z3 Julia Sets have? [ ]

Bonus:
You can explore the Z3 Mandelbrot Set in the standalone fractal zooming prgram XaoS to verify if this assumption about symmetries is correct.


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