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 well-known shape of the standard two-dimensional Mandelbrot set.

But what happens if we want to be able to produce a three-dimensional 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 three-dimensional shape ( r, i, - i ) that will generate a "Mandelbrot set" cross-section when the Tricorn-specific part of the formula gives zero, and a "Tricorn set" when the Mandelbrot contribution is zero.

We then have a three-dimensional fractal surface enclosing a solid that can be "cut" to produce either set.

Here's the full three-dimensional shape:

In this image, the shape was colour-coded 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" cross-sections more obvious.

Here's a couple of animations of the solid rotating in three dimensions:

Self-similarity

Like the Mandelbrot and Tricorn fractals, this solid shows some self-similarity at different scales.

If we work leftwards along the central axis of the Mandelbrot set, we come across the smaller "Mandelbrot-like" shape (leftmost image):

Cross-sections of a smaller "island" in the combined Mandelbrot-Tricorn solid, taken at 0, 18, 36, 54, 66, 72 and 90 degrees

, while the "Tricorn-like" 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 X-axis at different angles, this location generate a series of shapes that echo how that larger solid appears for that cross-section. Ignoring the little offshoot strands, the smaller shape, in three dimensions, is basically an echo of our larger solid. The sort of self-similarity that shows up in the Mandelbrot and Tricorn fractals appears (to some extent) in our new three-dimensional shape, too.

Additional features

The left-hand "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 X-axis. 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 2-D shapes.

Taking a close-up 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 cross-connections 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 centre-right 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 2-D fractal shapes, it's a proper fractal solid in its own right.

"High-z" 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.

z3 fractal solid

all original material copyright © Eric Baird 2007/2008