Fractals:
Powers of the Julia and Mandelbrot
sets
Julia Set analogues
If
we characterise the Julia Set
as being the result of writing "z →
z^{2}
+ 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 "z^{2}"
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:







z^{2}
(Julia) 
z^{3} 
z^{4} 
z^{5} 
z^{6} 
z^{7} 
z^{8} 
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 z^{2},
we find that they all repeat if we rotate them through exactly 180
degrees (rotational symmetry of "two").
For z^{3},
the shapes selfmap three times within a 360degree rotation,
for z^{4}
the shapes selfmap four times, and so on.
Since
the Julia Set is associated with a twodimensional range
of images, each of these powers generates its own family of
corresponding shapes.
Here are seven tables of Julialike
images generated using z^{2}
, z^{3}
, z^{4} ,
z^{5},
z^{6}
, z^{7}
, and z^{8}
.
You can click on them to see them fullsize.
Each of these maps only shows the upper half of
the (mirrorsymmetrical) 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 z^{2}
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
crosssection, does this mean that the higherpower Julia Set
counterparts also generate higherpower
counterparts of the Mandelbrot Set?
In a word: Yes.
Here's
the corresponding table of Mandelbrotlike slices, for z^{2}
to z^{8}.
The "bordered" images can be clicked to launch hires 1800 ×
1800
versions.







z^{2}
(Mandelbrot) 
z^{3} 
z^{4} 
z^{5} 
z^{6} 
z^{7} 
z^{8} 







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 z^{2},
we have a rotational symmetry of "one" (the thing requires a full
360degree rotation to map onto itself). For z^{3}
we get a twofold
rotational symmetry, for z^{4},
a threefold symmetry, for z^{5},
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 z^{3}
tends to
have twopronged features, and the bulbs that we get with z^{4},
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" z^{2}
Mandelbar, we get a threecornered shape, for the z^{3}
variant we get four corners, for z^{4}
we get a fivepointed shape. and so on:







z^{2}
(Mandelbar) 
z^{3} 
z^{4} 
z^{5} 
z^{6} 
z^{7} 
z^{8} 
Higherpower fractal
solids
The
odd threepronged 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
threepronged beastie.
The same
trick also works with
the higherpower 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 threedimensional
fractal solids.
External Links:

"Alt.Fractals: A visual guide to fractal geometry and design" (2011) ISBN 0955706831, sections 1721, "HigherPowered ..."
all original material
copyright © Eric Baird 2007/2008
