Fractals: 3D
Into
the Third Dimension
The
Mandelbrot Set can be thought of
as the
result of an
"unstable" function with two parameters, and two zones ... a zone where
the
function obviously goes to infinity,
and a zone where it doesn't. In between the two, there's a
chaotic
region where these two outcomes mix together, and where the result of
the function seems to jump about like crazy when we make tiny
adjustments to the input parameters. If we plot how the function reacts
to combinations of the two parameters, by using them as the "X"
and
"Y" coordinates for a graph, we get a visual
description of how the pattern's boundary shifts with
position. That intricate and chaotic boundary shape gives us the
wellknown shape of the standard twodimensional Mandelbrot
set.
But
what happens if we want to be able to produce a threedimensional
fractal with similar characteristics?
To manage that,
we'd need a similar a formula that has with three
independent parameters rather than two.
To achieve this, we can fuse the Mandelbrot
set with it's alternative sibling, the "Tricorn" fractal
(sometimes known as
the "Mandelbar" set).
These two
fractals are mathematically very similar: the key part of both
formulae has two different components, a "real
number" component that's usually plotted horizontally on
the "X"
axis, and an "imaginary number" component that's
usually plotted
vertically, on the "Y" axis, giving the
signature ( r, i ).
The only difference between the two formulae is that the "Tricorn" has
a minus sign tacked onto the front of its "imaginary"
component ( r,  i ).
Since the two functions have the same "horizontal"
formula, we can compact their four parameters into three, and plot a
threedimensional shape (
r, i,  i ) that will generate a "Mandelbrot
set" crosssection
when the Tricornspecific part of the formula gives zero, and a
"Tricorn set" when
the Mandelbrot contribution is zero.
We then have a threedimensional
fractal surface enclosing a solid that can be "cut" to produce either
set.
Here's
the full threedimensional shape:
In
this image, the shape was colourcoded according to how far
each point was from the "Mandelbrot" plane, so if we cut the solid in
half, along the middle of the central red band, we get the Mandelbrot
set.
We
might have instead chosen a different colour scheme that emphasises
the "Tricorn" plane. The version below does this, and has also
had one
quarter of the shape cut away, to make the two "standard"
crosssections more obvious.
Here's
a couple of animations of the solid rotating in three
dimensions:
Selfsimilarity
Like
the Mandelbrot and Tricorn fractals, this solid shows some
selfsimilarity at different scales.
If
we
work leftwards along the central axis of the
Mandelbrot set,
we come across the smaller "Mandelbrotlike" shape (leftmost
image):



Crosssections
of a smaller "island" in the combined MandelbrotTricorn
solid,
taken at 0, 18, 36, 54, 66, 72 and 90 degrees 
,
while the "Tricornlike" shape on the
far right appears at exactly the same location and magnification, for
the
"Tricorn" fractal.
If
we now take our combined 3D
shape, and slice through the Xaxis at different angles, this location
generate a
series of shapes that echo how that larger solid appears for
that
crosssection.
Ignoring the little offshoot strands, the smaller shape, in three
dimensions, is basically an echo of our larger solid. The sort of
selfsimilarity that shows up in the Mandelbrot and Tricorn
fractals appears (to some extent) in our new threedimensional shape,
too.
Additional features
The
lefthand "bulb" of the Mandelbrot and Tricorn fractals look rather
similar, and in three dimensions we see that this part of the solid
looks suspiciously like the result of simply rotating and
distorting a shape around the Xaxis.
However, when we look at the full shape, we see
other, new, features
that don't correspond to a simple morph, rotation or distortion of the
two earlier 2D shapes.
Taking a closeup view of
the piece of the solid that's closest to us
... corresponding to the larger lower lobe of the Mandelbrot set
... we see that it appears in three dimensions to be circled by
"orbits" and
crossconnections between features and lobes that don't have direct
counterparts in either the Mandelbrot or the Tricorn
sets.
Although
most of the shape's detail appears near the Mandelbrot and
Tricorn
planes, this detail doesn't simply fade in and out . The solid includes
complex strands that circle and link features of the Mandelbrot set,
and act outside the Mandelbrot plane. Although the shape
has two fairly smooth curved faces, these regions are circled
and
overhung by "perimeter fronds"  the blue "shoulder" in the
centreright of the image above represents a bunch of these fronds that
are folded back on themselves to connect to the rest of the shape.
This isn't just a simple "morph" of two 2D
fractal shapes, it's
a proper fractal solid in its own right.
"Highz" Siblings
When we repeat this exercise with the counterparts of the
Mandelbrot and Mandelbar sets built using higher powers of z,
we get a whole
family of fractal solids.
z^{3} fractal solid 

External Links:

"Alt.Fractals: A visual guide to fractal geometry and design" (2011)
ISBN 0955706831, section 16, "The Mandelbrot/Tricorn Hybrid"
all original material
copyright © Eric Baird 2007/2008
