But four dimensions?

It occurred to me that A, B, C, D are each an axis of the Set, which means we are dealing with a 4-dimensional beast. There are six 2-dimensional slices (perm any 2 from 4). The two you probably know are:

• The Mandelbrot Set, which is the Real (C) + Imaginary (D) axes with the other two axes, A and B, at zero.
• The Julia Sets, which are an interesting case of A-axis and B-axis plus non-zero values for C and D axes, which I'll discuss below.

Is this 4D construct real or an illusion? Well, let's see.

For the 2D slice (Real C) axis plus A -axis), the B and D axes are zero. The Real (C) and A axes are the horizontal and vertical axes of the screen. That gives the pure object. By having a non-zero value for one of the other axes, the Imaginary (D) for example, we get a slice of Real + A as it looks a little along the Imaginary (D) axis. Here's the example. The first picture is the Real + A axes moved to 0.25 along the Imaginary axis. The second picture shows where this is. If you could cut out the first picture, it would stand out at 90° from the screen at that white line.

There are four possible 3D slices through a 4D object (perm any 3 from 4), so we could build up four such models. We can never visualize the 4D object itself, being 3D creatures, but mathematically it is easy to handle multidimensional objects. To my eyes, I'm building what seems to be consistent 2D and 3D views of some kind of 4D object.

 

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