We've
seen on the previous page
that the Julia Set has four independent
parameters (the real and imaginary components of the start-value of z,
and the real and imaginary components of c, the
added constant), so, technically the full Julia Set is a
four-dimensional shape.

While
it's traditional to use slices taken through the AB
plane, we
can generate cross-sections through it at any angle we like
through these
four dimensions, and with any offset.

The Mandelbrot Set is
a section of the 4D Julia Set

The Mandelbrot Set exists as a cross-section
through the 4D Julia Set, in
the plane CD, with A and B
set
to zero. Since it cuts through
the shape at right angles to our usualAB-plane Julia Set
images, the intersection isn't obvious when we compare the images.

In
a three-dimensional space, two angled planes are
guaranteed to intersect somewhere to form a line, but in
four-dimensional
space they're only guaranteed to intersect at a point. So in order to
be
able to see the tell-tale signs of intersection, we need to start
assembling collections of AB
Julia Set images to be able to see the
larger pattern emerging.

Let's take
our earlier table of Julia Set images, and calculate it again, to a
higher resolution.

With just a couple of hundred Julia Set images,
the
shadow of the Mandelbrot Set is starting to emerge.

The
next image is built from a rectangular array of several
thousand tiny Julia images.

The pattern of Julia Sets that have a
dark centre builds up to form a
noticeable Mandelbrot Set outline. More precisely, the
exact centre point in a given Julia Set
image
selected using (0, 0) (C, D) will be identical to
the
correspondong point on the Mandelbrot Set (0,0), (x,y)
... so if we'd only plotted the single
centre-pixel
of each Julia Set in our 2D array,
we'd actually be plotting the Mandelbrot Set. Since this map is a
two-dimensional array
of two-dimensional images, it's technically a four-dimensional
image (with very
limited
resolution).

One
thing that we notice from these images is that although the Mandelbrot
outline is visible, we can also see what appears to be
additional overlaid detail. The
interior of our shape isn't entirely black, because
we've been plotting
more than just the centre pixel of each Julia image. Pixels away from
the centre of the individual Julia tiles are also playing a part in
the final
picture, and are showing us hints of further detail that exists
away from the
Mandelbrot plane. When we look at the overall image (above), we're not
just
seeing the shadow of the Mandelbrot Set, we're also seeing
glimmerings of additional
four-dimensional
structure,
compacted into 2D.

Maps and Subsets

The
images above illustrate how the Mandelbrot Set can be used to as a map
of the usual Julia Set images. A "zoom in" on a particular point on
the Mandelbrot Set reveals characteristics reminiscent of the
character of the Julia Set image that would have been called
up by using those same two coordinates as "selection numbers"
– "spiky"
regions of the Mandelbrot Set tend to correspond to "spiky" Julia set
images, "twisty" regions on the Mandelbrot tend to correspond to
"twisty" julia images, and so on. This
correspondence
appears because the larger 4D Julia Set shows local
self-similarity
in four dimensions – local
patterns and
themes that appear
in the CD
(Mandelbrot) plane also tend to "infect" small intersecting regions of
the AB
(standard Julia) plane, and vice versa.

Sometimes
the effect is very striking. If we zoom in on the Mandelbrot, we find
smaller shapes that are noticeably "Mandelbrot-like", and it's said
that
if we zoom in far enough, we can also find
points on the Mandelbrot Set that seem to correspond to full Julia Set
images. This might suggest to us that the Julia Sets are somehow contained
within the Mandelbrot set.

In
reality, it's the other way around.

Although mathematicians
sometimes
get carried away and say that the Mandelbrot Set contains perfect
copies of itself, it doesn't.

The Mandelbrot Set has the
unusual
property (for a cross section of the full 4D shape) that all points on
its
boundary are connected within the plane. This means that if we placed
the point of an infinitely-sharp pencil onto one part of the Mandelbrot
boundary, and traced out an infinite length of line, we'd end up back
at our starting position having traced out the entire set. The shape
has no separated "islands" (unlike most of the standard Julia
Set
images).
So when we find a "mini-Mandelbrot" within
the larger Mandelbrot Set, by definition, it can't be a perfect copy,
because the
condition of "connectedness" means that the smaller offspring must be
connected to the parent by threads and tendrils that the original
parent doesn't have. Similarly, although some people have claimed to
have found things that look like Julia Set
islands floating within the Mandelbrot Set, they must have
internal interconnections that don't necessarily exist in the
corresponding "standard" Julia Set image, and must have external
connections that definitely won't exist in the
original.

So,
although the Mandelbrot Set can be considered as a "map" of how certain
aspects of the full Julia Set change with location across two of its
four dimensions, technically, the Mandelbrot Set is a subset
of the Julia Set, rather than the other way around.
We can't
zoom in on the Mandelbrot and obtain a perfect standard
Julia image.
However, we can slice the
full 4D Julia Set and obtain a perfect Mandelbrot.

Into the Fourth Dimension

Given that the full Julia Set is
four-dimensional, how can we visualise it?

If
we want to slice the Julia Set to produce 2D images, we have six main
ways of doing it. We're familiar with the "Mandelbrot" slice (0,
0), (x,
y), and with the usual Julia Set image slices [(x,
y), (n,
n)], but we also have four other
major
planes that we can use for taking cross-sections through the solid's
centre.

Here they all are:

Primary
Planes

AB

CD

AD

BC

AC

BD

Each
of these six planes can then be extended in one other
dimension to create
a three-dimensional solid. Stripping away the duplicates, we then end
up
with four different major solids: (ABC)
[(x, y), (z, 0)], ABD [(x,
y), (0, z)],
ACD [(z, 0), (x,
y), and BCD [(0, z), (x, y)].
Each of these incorporates three of the six primary
cross-sections.

Primary
Solids (click images to enlarge)

1: A×B×C

2: A×B×D

3: A×C×D

4: B×C×D

We're
free to slice the four-dimensional shape at any angle and
offset we like,
but these four
images show the four primary solids.

Solids # 1 &
2

The first two of
these solids have the central "circle" AB
cross-section

To
someone familiar with the usual Julia Set images, these two are the
most easily visualisable – they can be assembled by
"stacking" Julia
Set images from
our earlier arrays. The first solid represents a "stack" of
mirror-symmetrical
images taken along the central "spine" of the map, and shows a
"Mandelbrot-like evolution from the bottom of the diagram to the top,
starting with thin "spike-like" features, developing into a
middle-sized
bulb, which is then followed by a large void terminated in a
cleft.

However, if we cut the ABC
solid in the hopes of finding a Mandelbrot, we fail. We can get a sort
of triangular wedge that looks a little like a
Mandelbrot, but the
"side detail" is all wrong. That detail is contained in parameter D,
which isn't used in this solid.

If
we now look at the ABD solid, we can produce
cross-sections that are
reminiscent of the Mandelbrot's side-bulbs, but the earlier "spine"
evolution is missing (because of the lack of parameter C).

If
we're trying to visualise the full four-dimensional shape, it's
probably easiest to think of ABC as the
mirror-symmetrical "core set"
of the full Julia Set, and to visualise D as a
parameter that makes the
shape break up and twist to the left or the right, depending
on
whether D is positive or negative.

Solids
# 3 & 4

With different
combinations of three parameters, the contained Mandelbrot is more
explicit. In the second two images, we're using ACD
and BCD ,where C
and D describe the Mandelbrot plane. These two
images have been coloured
according to the magnitude of A or B,
and as a result they're showing
a central horizontal red band which, if we cut along it, would give the
Mandelbrot
set as a cross-section.

The first image
(left) shows the effect of varying A, the initial
"real" component of z.

The last image
shows the effect of varying B, the initial
"imaginary" component of z.

If
we now look back at our Julia Set mosaic image, with its hints of
additional four-dimensional detail, those details correspond to the
additional detail that
appears in these last two images.

Mandelbrot
Echoes

With these two last shapes, we see something odd:
as well as the central Mandelbrot cross-section, there seem to be some
suspiciously similar-looking shapes intersecting these solids at 45
degrees. How come?

Well,
the Mandelbrot also "echoes" through the main Julia Set. If
the "official"
Mandelbrot is given by 0,0, x,y, and the first two
parameters are the
starting
values of z, and the second two (x
and y) are the constant offset, c,
that gets added at each stage,
then we can see that after one iteration,
squaring 0,0 again gives 0,0
(no change) and adding x,y
to
this gives us (x,y),
(x,y).

So, if we start
at (x,y), (x,y), we'd seem to be entering the same
Mandelbrot sequence
as before,
with a "one iteration" head start. So as well as a Mandelbrot
cross-section cutting through the 4D Julia hypersolid in the CD
plane (0,0), (x,y), we might also expect a
Mandelbrot section at (x,y),
(x,y), and perhaps another at (-x,-y), (x,y).

So there ought to be some 45-degree Mandelbrot
resonances within the larger four-dimensional
shape.

"Fluffy Mandelbrot"

Julia Set plot,
(A&B), C, D
(click to enlarge))

Working out an
exact expression
for the number of perfect Mandelbrots that we can obtain from
the Julia Set, using various types of flat and curved cross-sections is
trickier problem, but perhaps mathematicians will have a crack
at
it one day.

Video

Here are some animated renderings of parts of the
full four-dimensional Julia solid: