The taffy-like quaternion Julia sets are similar to complex Julia sets, except that they exist in four dimensions rather than just two. They aren’t really useful or significant; but, like other fractals, they tend to look pretty.

A Quick Review

Before getting into quaternion Julia sets, here is a short review of how complex Julia sets work:

Complex numbers have two parts, a real part and an imaginary part. Since they have two parts, they can be graphed in two dimensions. A complex Julia set is created by running each point on the plane through an iterative function z_n+1 = z_n² + c, where c is a constant (different values for the constant yield different Julia sets). If the result of the function goes to infinity, the point is not in the set. If the result stays bounded, the point is in the set. Coloring points in the set gives us a pretty picture.

The Quaternions

Quaternions, an extension of the complex numbers, are numbers of the form a + bi + cj + dk, where i² = j² = k² = ijk = -1. The first number a is the “real” part. The last three numbers b, c, and d are the “pure” or “imaginary” parts. Since quaternions have four parts, we can graph them in four dimensions.

Quaternion Julia Sets

Quaternion Julia sets are constructed just like complex Julia sets. Each point in four-space can be represented by a quaternion. That quaternion is then run through the function z_n+1 = z_n² + c many times. If the result goes to infinity, the point is not in the set. If the result does not go to infinity, that point is in the set.

Dimension Reduction

The resulting Julia set is a four-dimensional object. In order to be viewed, the object must be reduced by one or two dimensions. This is usually done by taking a three-dimensional cross-section of the four-dimensional object (that is, the set is intersected with a hyperplane). Practically what this means is one part of each quaternion is set to be a constant number, and the remaining three parts of the quaternion are graphed in three dimensions.

Stacking Up the Slices

However, we can perhaps get a better idea of what the 4D object looks like by arranging slices along a time axis. The following animation shows a series of cross-sections as the slicing plane varies from -1 to 1 and the object is rotated 360 degrees about the y-axis.

See Also

For more information about quaternion Julia set fractals, Paul Bourke has a good webpage on the subject, which also includes some pov-ray code.

If you would like to experiment with making your own quaternion Julia sets, Julia Shapes is a good program to try out. It has sliders that let you manipulate all the parameters.

Complex Julia Sets

A Julia set, named after French mathematician Gaston Julia, is a type of fractal defined by an iterative function over the complex numbers. The study of fractals has applications in complex dynamics, partial differential equations, statistics, etc, but most people like them because they produce pretty images. The following is an overview of how two-dimensional Julia sets work.

The Complex Numbers

Complex numbers are numbers of the form a+bi, where i² = -1. As these numbers have two parts (a real part and an imaginary part) they can be graphed in two dimensions. Usually, the real part (a) is plotted on the x-axis and the imaginary part (b) is plotted on the y-axis. Thus we can create a mapping to go back and forth between a point on a plane and a complex number.

Defining a Julia Set

A Julia set is the set of points on the complex plane which do not approach infinity after the function z_n+1 = z_n² + c is repeatedly applied. The initial value of z₀ is the point on the complex plane, and c is a constant complex number (different values of c create different Julia sets).

Here is perhaps a more simple explanation: Choose a point on the plane. That point has a corresponding complex number, which we will call z₀. Take z₀ and run it through the function several times - that is, square it, add c, repeat ad infinitum. If the resulting number goes towards infinity, the point z₀ is not in the set (color it white). If the result does not go to infinity, z₀ is in the set (color it black).

Strictly speaking, Julia sets are usually defined as either the set of divergent points (the ones that go to infinity, see wikipedia) or as the boundary of the convergent points (see mathworld). But for the purposes of creating pretty pictures, the above explanations suffice. Also, functions other than z_n+1 = z_n² + c can be used. More on that later.

Interesting Properties

Julia sets are not smooth. No matter how much you zoom in, the edges will always appear jagged (except when c = 0, which gives a circle).

When c is a real number, the corresponding Julia set is symmetrical about the real axis. Otherwise the set has 180-degree rotational symmetry.

If the value of c is inside the Mandlebrot set, the Julia set will be connected and is called a Fatou set. Values of c outside the Mandlebrot set form a disconnected Julia set, also called a Cantor set or Fatou dust.

Coloring Julia Sets

Julia sets can be made even prettier by adding color. The color of each point is usually determined by how fast that point diverges to infinity.

Other Functions for Julia Sets

While z_n+1 = z_n² + c is the most commonly used function, a Julia set can be defined by any function over the complex plane. Commonly used functions include:

• z_n+1 = z_n³ + c
• z_n+1 = c sin(z_n)
• z_n+1 = c i cos(z_n)
• z_n+1 = c exp(z_n)
• z_n+1 = c z_n (1 - z_n)

Rendering Julia Sets

Rendering a Julia set using a computer is rather simple using a brute-force method. Every pixel is mapped to a point on the complex plane and then run through an iterative process. If, after a certain number of iterations, the magnitude of the result is greater than a pre-defined threshold, the point is not in the set. Otherwise, the point is in the set. A greater number of iterations and a larger escape threshold value will result in a more accurate depiction of the Julia set, but also longer computation times.

Downloads

cJulia.cpp (4.42kb) - This is some simple C++ source code for creating a Julia set and writing the result to a bitmap file.

cJuliaEx.exe (80kb) - A small Windows program that renders quadratic Julia sets. Use the sliders to adjust parameters. If you want, you can take a look at the Visual C++ source code: cJuliaExSrc.zip

cJuliaExColor.exe (96kb) - Similar to the program above, except that it has more functions and is green. You can get the source code here: cJuliaExColorSrc.zip

I have to warn you about the last two programs, though. I don’t know very much about WIN32 API, so even though they work fine on my machine, the programs might be a bit wonky. If anybody wants to clean it up, make it faster (setPixel() is slow), or expand its functionality, go right ahead. But hopefully, they should work fine for you and are nice way to start exploring Julia sets.

Further Reading

The Wikipedia article on Julia sets is actually rather uninteresting. However, Paul Bourke has a good website on the topic. If you would like to read a book on the subject, Amazon has a few about fractals. I haven’t read any of them, but I’ve heard that Fractal Geometry by Kenneth Falconer is a good book (though expensive).

I began to show the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 3^2.

“… I suppose 3^3 must mean something in Geometry; what does it mean?”

“Nothing at all,” replied I, “not at least in Geometry; for Geometry has only Two Dimensions.”

- Edwin Abbott, Flatland

A lot of mathematical art concerns itself with objects that exist in four or more dimensions. Even physicists have been telling us that we live in a ten dimensional universe. How are we, as three-dimensional beings, able to understand this?

Starting Out Simple

Let’s begin by reviewing the lower dimensions:

0 - Point: A point has zero dimensions, and is usually represented graphically by a dot. Some people get confused about how it is possible for something to have zero dimensions. Try holding up two fingers in front of you, and imagine the point in space exactly halfway between your two fingers. This point exists; it has a definite location. However, measuring its length, width, or height would be impossible. It is zero-dimensional.

1 - Line: Now, suppose we move the point and look at the trail it leaves behind. The path of this point is a line. It is one-dimensional - it has length, but no height or depth.

2 - Square: If a line moves parallel to itself (or as mathematicians would say, “is extruded along an orthogonal axis”), we end up with a two-dimensional square.

3 - Cube: A square that is extruded along an orthogonal axis (in this case, the axis would be coming out of the computer screen) becomes a three-dimensional cube.

But we have to pause here, because the last figure in the diagram above is misleading. Borrowing from an idea of Rene Magritte:

The image on the left is indeed not a pipe. Pipes are solid objects that are used for smoking. What’s shown above is a painting of a pipe (or, rather, a computer image of a painting of a pipe). Likewise, what is shown above is not a cube. Cubes are solid objects. Unfortunately, it is impossible to display a solid object on a flat computer screen, so we are forced to use a two-dimensional representation of a cube.

4 - Hypercube: So what happens if we extrude a cube along an orthogonal axis? Well, naturally, we end up with a four-dimensional hypercube - also called a tesseract. We can attempt to make a two-dimensional drawing of a tesseract like so:

One question, however, still remains for us three-dimensional beings: where is that orthogonal axis?

But isn’t time the fourth dimension?

Well, that depends. Physicists, especially those interested in relativity, usually consider time to be the fourth dimension (or rather, one of the four dimensions). Mathematicians, however, can construct as many spatial dimensions as they want simply by adding on more variables. Mathematicians say that calling the fourth dimension “time” is an application of four-dimensional geometry.

Depicting 4D Objects in 3D Space

Because humans cannot see four dimensions, objects have to be reduced to three dimensions or fewer in order to be viewed. There are two basic ways of doing this.

The first is to slice the object and look at its cross section. For example, taking the cross-section of a sphere results in a circle. Just as a cross-section of a 3D object is a 2D figure, a cross section of a 4D object is a 3D figure. This would be called “intersecting” the object with a plane. Of course, what the cross section looks like will depend on where and at what angle the object is sliced. To get a feel for the nature of a four-dimensional object, it is helpful to examine a series of slices along an axis perpendicular to the slicing plane.

The second method is to look at the shadow of the object. If you shine a light on a three-dimensional object, you will see a two-dimensional shadow projected on the wall. Likewise, the shadow of a 4D object is a 3D object. What the shadow looks like will depend on the location of the light source relative to the object, as well as the orientation of the projection plane.

Experiencing Four Dimensions

Will human beings ever be able to directly experience four spatial dimensions? Psychologist Frances Wang has been doing research at the Beckman Institute to see how humans interpret four-dimensional space. Early results are showing that after a period of trial and error, subjects can gain some intuitive knowledge of how 4D space works.

Further Reading

Flatland: A Romance of Many Dimensions - Flatland, a book by Edwin Abbott Abbott, tells the story of A Square, his life in two-dimensional Flatland, his visit to Lineland, and his voyage to Spaceland. You can order the book at Amazon (they even made a DVD out of it), or you can read it for free at Project Gutenberg.

Fourfield - Fourfield:Computers, Art & the Fourth Dimension, by Tony Robbin gives some explanation of four dimensions and how it has been applied to art. It even comes with a pair of red and blue anaglyph 3D glasses and a printout of hypercube tiles. He has another book, Shadows of Reality, but I haven’t read that one yet.