Cool Stuff: Fractals Alt.Fractals paperback - out January 2011


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2008

Fractals: Powers of the Julia and Mandelbrot sets

Julia Set analogues

If we characterise the Julia Set as being the result of writing "z → z2 + c" (and plotting the dependency of the result against the initial value of z), then it's reasonable to ask: What happens when we change the "z2" part of the equation for a different power of z?

Some things happen that are quite fun.

Here are a few corresponding counterparts of a single frame from the Julia Set, all taken at exactly the same coordinates, using different powers of z:

sample Julia set image, z^2 equivalent analogue of the Julia set fractal image, for z cubed (z^3) equivalent analogue of the Julia set fractal image, for z raised to the fourth power (z^4) equivalent analogue of the Julia set fractal image, for z raised to the fifth power (z^5) equivalent analogue of the Julia set fractal image, for z raised to the sixth power (z^6) equivalent analogue of the Julia set fractal image, for z raised to the seventh power (z^6) equivalent analogue of the Julia set fractal image, for z raised to the eighth power (z^8)
z2 (Julia) z3 z4 z5 z6 z7 z8

What we find is that the rotational symmetry of each shape corresponds to the power of z that we use. So for the "proper" Julia Set images, which use z2, we find that they all repeat if we rotate them through exactly 180 degrees (rotational symmetry of "two").
For z3, the shapes self-map three times within a 360-degree rotation, for z4 the shapes self-map four times, and so on.

Since the Julia Set is associated with a two-dimensional range of images, each of these powers generates its own family of corresponding shapes.

Here are seven tables of Julia-like images generated using z2 , z3 , z4 , z5, z6 , z7 , and z8 .
You can click on them to see them full-size.

table-array of Julia Set fractal images, for z-squared, in redtable-array of Julia Set fractal images, for z-cubed, in orangetable-array of Julia Set fractal images, for z raised to the fourth power, in yellowtable-array of Julia Set fractal images, for z raised to the fifth power, in greentable-array of Julia Set fractal images, for z raised to the sixth power, in bluetable-array of Julia Set fractal images, for z raised to the seventh power, in deep bluetable-array of Julia Set fractal images, for z raised to the eighth power, in purple

Each of these maps only shows the upper half of the (mirror-symmetrical) table – the missing lower half is just a mirrored reflection. The reason for the unusual proportions and colour scheme is that I originally did these as wraparound designs for a matching set of coffee mugs. (!)

Note the shape of the shadows. In the red z2 example the shadow corresponds to (half of a) Mandelbrot Set, the other shadows correspond to ...



Mandelbrot Set analogues

Since the usual 4D Julia Set generates the 2D Mandelbrot Set as a cross-section, does this mean that the higher-power Julia Set counterparts also generate higher-power counterparts of the Mandelbrot Set?

In a word: Yes.

Here's the corresponding table of Mandelbrot-like slices, for z2 to z8. The "bordered" images can be clicked to launch hi-res 1800 × 1800 versions.


analogue of the Mandelbrot Set fractal, for z cubed (z^3) analogue of the Mandelbrot set fractal, for z raised to the fourth power (z^4) analogue of the Mandelbrot set fractal, for z raised to the fifth power (z^5) analogue of the Mandelbrot set fractal, for z raised to the sixth power (z^6) analogue of the Mandelbrot set fractal, for z raised to the seventh power (z^7) analogue of the Mandelbrot set fractal, for z raised to the eighth power (z^8)
z2 (Mandelbrot) z3 z4 z5 z6 z7 z8

Again, we see a change in rotational symmetry, but this time, the rotational symmetry is one less than the power of z being used.

So for the conventional Mandelbrot set, using z2, we have a rotational symmetry of "one" (the thing requires a full 360-degree rotation to map onto itself). For z3 we get a twofold rotational symmetry, for z4, a threefold symmetry, for z5, a fourfold symmetry, and so on. in each case, the overall character of the set is reflected in the shape of the smaller features, so we find that z3 tends to have two-pronged features, and the bulbs that we get with z4, tend to cascade down to smaller scales in groups of three.


"Mandelbar Set" analogues

Since we can switch the Mandelbrot Set into becoming a "Mandelbar Set" ("Tricorn fractal") by flipping the sign on its "imaginary" component, does this trick work for higher powers too?

Yep!

But with the "Mandelbar" analogues, the rotational symmetry of the resulting shape is now one higher than the power of z.
For the "proper" z2 Mandelbar, we get a three-cornered shape, for the z3 variant we get four corners, for z4 we get a five-pointed shape. and so on:

analogue of the 'Mandelbar' fractal, for z cubed (z^3) analogue of the 'Mandelbar' fractal, for z raised to the fifth power (z^5) analogue of the 'Mandelbar' fractal, for z raised to the sixth power (z^6) analogue of the 'Mandelbar' fractal, for z raised to the seventh power (z^7) analogue of the 'Mandelbar' fractal, for z raised to the eighth power (z^8)
z2 (Mandelbar) z3 z4 z5 z6 z7 z8

Higher-power fractal solids

The odd three-pronged fractal solid that we met earlier was an amalgam of the Mandelbrot and Mandelbar shapes. Since the two shapes had identical "real" components, they were guaranteed to mesh together to create a larger shape. When we tried to calculate that shape we got our three-pronged beastie.

The same trick also works with the higher-power counterparts of the Mandelbrot and Mandelbar sets: again, the formulae for each pair of shapes share a common "real" axis component, and can be combined to create a family of three-dimensional fractal solids.


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