Julia set facts for kids

A Julia set

A Julia set is a special type of picture called a fractal. Fractals are amazing shapes that look the same no matter how much you zoom in or out. A Julia set is created using a mathematical rule, or formula, that is repeated many, many times.

The Julia set is closely related to something called the Fatou set. Imagine you have a mathematical rule. When you repeat this rule over and over, some starting numbers will behave in a similar way, staying calm or moving towards a certain point. These numbers form the Fatou set. The Julia set is made up of all the other numbers – the ones that behave in a chaotic or unpredictable way when you repeat the rule. It's like the boundary between the calm and the chaotic parts!

What are Julia Sets?

Julia sets are beautiful and complex shapes that come from complex numbers. These numbers are a bit like regular numbers but have an extra part. To make a Julia set, you pick a starting complex number and then apply a simple mathematical formula to it. You take the answer, and then you apply the formula to that new answer, and you keep doing this over and over again. This process is called iteration.

How are Julia Sets Formed?

When you iterate the formula, some numbers will stay close to the starting point, or they might move towards a specific value. Other numbers will quickly fly off to infinity, getting bigger and bigger without limit. The Julia set is made up of all the starting numbers that do NOT fly off to infinity. It's the boundary line between the numbers that stay "trapped" and the numbers that "escape."

The exact shape of a Julia set changes depending on the specific mathematical formula you use. Even a tiny change in the formula can create a completely different and unique fractal pattern. This is why there are so many different kinds of Julia sets, each with its own amazing design.

Julia Sets and the Mandelbrot Set

The Mandelbrot set is another famous fractal that is very closely linked to Julia sets. You can think of the Mandelbrot set as a kind of "map" that shows you what different Julia sets look like. Each point in the Mandelbrot set corresponds to a specific Julia set. If a point is inside the Mandelbrot set, its corresponding Julia set will be "connected" (all in one piece). If a point is outside the Mandelbrot set, its Julia set will be made of many separate, scattered pieces, like dust.

Studying Julia sets helps scientists and mathematicians understand how simple rules can lead to incredibly complex and beautiful patterns in nature and in mathematics.

Explore More

• Fractal
• Mandelbrot set

Images for kids

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  Julia set (in white) for the rational function associated to Newton's method for f : z → z3−1. Coloring of Fatou set in red, green and blue tones according to the three attractors (the three roots of f).

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  Pictures in the field lines for an iteration of the form z^2 + c

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  A Julia set plot, generated using random IIM

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  A Julia set plot, generated using MIIM

See also

In Spanish: Conjunto de Julia para niños