Understanding the Symmetry of Julia Sets: A Brief Explanation

[chaoseverlasting]

I have a question regarding Julia sets. As far as I know, they are made by graphing functions on the imaginary plane, so when you get the final graph, you have an image that seems to have a plane of symmetry; i.e, as if one part of the graph was reflected about that plane to get the whole graph.

Is this because complex numbers occur in pairs, so if one part of the graph is the root of the equation, its mirror is the conjugate root? If anyone out there has the answer, I would appreciate it if they could enlighten me.

 

[HallsofIvy]

I have a question regarding Julia sets. As far as I know, they are made by graphing functions on the imaginary plane, so when you get the final graph, you have an image that seems to have a plane of symmetry; i.e, as if one part of the graph was reflected about that plane to get the whole graph.

Is this because complex numbers occur in pairs, so if one part of the graph is the root of the equation, its mirror is the conjugate root? If anyone out there has the answer, I would appreciate it if they could enlighten me.

It's not clear to me what you mean by a "plane" of symmetry. Julia sets are, as you say, in the complex plane and are 2 dimensional, not 3 dimensional as they would have to be to have a "plane" of symmetry. Perhaps you meant "line of symmetry" but that is also not true. Julia sets are symmetric about the origin. That is, if (x,y) is in the Julia set, corresponding to x+ iy, then so is (-x,-y), corresponding to -(x+ iy).
That's obvious from the definition of Julia sets: J_c is the set of complex numbers z₀ such that the sequence defined by z_n+1= z_n²+ c, starting with z₀, converges.

 

[tacman]

Yes, you are correct! Julia sets are created by graphing the behavior of complex numbers under iteration of a function. The symmetry you see in the final graph is a result of the fact that complex numbers occur in pairs, with one being the conjugate of the other. This means that if one part of the graph represents a root of the equation, its mirror image will represent the conjugate root. This symmetry is a fundamental property of complex numbers and plays a crucial role in creating the beautiful and intricate patterns of Julia sets. I hope this helps answer your question!