Powers of the Julia and Mandelbrot
Julia Set analogues
we characterise the Julia Set
as being the result of writing "z →
+ 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:
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").
the shapes self-map three times within a 360-degree rotation,
the shapes self-map four times, and so on.
the Julia Set is associated with a two-dimensional range
of images, each of these powers generates its own family of
Here are seven tables of Julia-like
images generated using z2
, z4 ,
, and z8
You can click on them to see them full-size.
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
example the shadow corresponds to (half of a) Mandelbrot Set, the
other shadows correspond to ...
the usual 4D Julia Set generates the 2D Mandelbrot Set
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.
the corresponding table of Mandelbrot-like slices, for z2
The "bordered" images can be clicked to launch hi-res 1800 ×
we see a change in rotational symmetry, but this time, the rotational
symmetry is one less than the power of z
So for the conventional Mandelbrot set,
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,
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
have two-pronged features, and the bulbs that we get with z4,
tend to cascade down to smaller scales in groups of three.
we can switch the Mandelbrot Set into becoming a "Mandelbar Set"
fractal") by flipping the sign on its "imaginary" component,
trick work for higher powers too?
with the "Mandelbar" analogues, the rotational symmetry of the
resulting shape is now one higher than the power of
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:
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
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
"Alt.Fractals: A visual guide to fractal geometry and design" (2011) ISBN 0955706831, sections 17-21, "Higher-Powered ..."
all original material
copyright © Eric Baird 2007/2008