Cool Stuff: Fractals

2008

# The Julia Set in Four Dimensions

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 usual AB-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:

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