# 2. Strange & Complex

## 2.2 Julia Sets

Julia sets rendered with Object Mandelbrot & Julia's Dream.

Let's return for a while to our original map

ƒ: x → x^{2} + c.

The graph of this function is a parabola when "x" and "c" are real numbers. The orbits of well-behaved seeds are bounded for parameter values in the interval [−2, ¼]. These orbits can settle on to attracting fixed points, be periodic, or ergodic. A small set of fixed points, the repelling fixed points, do not generate orbits in the traditional sense. They neither roam nor run off to infinity and one need not wait for them to exhibit "characteristic" behavior. They are permanently and immutably fixed and nearby points avoid them. They lie on the frontier between those seeds with bounded orbits and those with unbounded orbits. Such is the behavior in general for all points and all parameter values — or is it?

The discussion so far has been constrained by a prejudice for
real numbers. What happens when we admit that i^{2} = −1 has a solution?
How does our function behave when "z" and "c" are complex
numbers? The answer, of course, is the same but the results are much more
interesting than such a flip statement implies.

The map…

ƒ: z → z^{2} + c

is equivalent to the two-dimensional map…

ƒ: (x, y) → (x^{2} − y^{2} + a, 2xy + b)

where z = x + iy = (x, y) is a point to
be iterated and c =

Stretch points inside the unit circle towards the origin. Stretch points outside towards infinity. | Wrap the plane around itself once without cutting or tearing such that every every angle value has doubled. | Shift the plane over so the origin lies on (a, b). |

Actually, it's easier to discuss these transformations if we represent the complex numbers as points on a complex sphere. The origin would be one pole and infinity another pole with the unit sphere being the equator. Imagine placing a light bulb at the infinity pole. Points on the sphere would leave shadows in unique positions on the plane. The complex plane is thus a projection of the complex sphere. While this is easier to deal with because we have a single point called infinity, it is not easier to diagram. Let's give it a shot.

Stretch points below the unit equator towards the origin pole. Stretch points above the equator towards the infinity pole. | Wrap the sphere around itself once without cutting or tearing such that every longitude value has doubled. | Stretch the sphere so the origin pole shifts to the point (a, b) but the infinity pole stays fixed. |

This family of mappings is said to be conformal;
that is, it leaves angles unchanged. Despite all this stretching, twisting, and
shifting there is always a set of points that transforms into itself. Such sets
are called the Julia sets after the French mathematician
Gaston Julia who first conceived of them in the 1910s.
The special case of c =

c = |
c = |
c = |
c = |

c = |
c = |
c = |
c = |

c = |
c = |
c = |
c = |

Sets on the real axis are reflection symmetric while those with complex parameter
values show rotational symmetry. With the exception of the parameter value c =