Question regarding Julia sets

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In summary: If z0 is in Jc then z1= z02+ c is in Jc and so on. If z0 is in Jc then so is -z0, since (-z0)2= z02. That's what gives the symmetry. I'm not sure what you mean by "complex numbers occur in pairs". The equation zn+1= zn2+ c has only one root, zn. But the graph of the function f(z)= z2+ c, its Julia set, can be symmetric if c is symmetric.In summary, Julia sets are graphed on the complex plane and are symmetric about the origin. This is because if a complex number is in the Julia set
  • #1
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. :wink:
 
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  • #2
chaoseverlasting said:
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. :wink:

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: Jc is the set of complex numbers z0 such that the sequence defined by zn+1= zn2+ c, starting with z0, converges.
 
  • #3


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!
 

1. What are Julia sets?

Julia sets are fractal sets that are associated with complex quadratic polynomials in the complex plane. They are named after the French mathematician Gaston Julia who first studied them in the early 20th century.

2. How are Julia sets created?

Julia sets are created by iteratively applying a quadratic polynomial to a complex number. The resulting values are plotted on the complex plane, with different colors or shading indicating whether the value remains bounded or tends to infinity under repeated iterations.

3. What is the significance of Julia sets?

Julia sets have significant applications in mathematics and physics, particularly in the study of dynamical systems, chaos theory, and complex dynamics. They also have aesthetic appeal and have been used in art and design.

4. Can Julia sets be generalized to other types of polynomials?

Yes, Julia sets can be generalized to other types of complex polynomials, such as cubic or quartic polynomials. These sets exhibit more complex and intricate patterns compared to the simpler quadratic Julia sets.

5. How are Julia sets related to the Mandelbrot set?

The Mandelbrot set is closely linked to Julia sets, as each point on the boundary of the Mandelbrot set corresponds to a unique Julia set. In fact, the Mandelbrot set can be thought of as a "family" of Julia sets, with each point on the boundary representing a different parameter value for the quadratic polynomial used to generate the Julia set.

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