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Fractals (mathematical definition here) are essentially objects with infinite levels of detail, which can be self-similar or not. Self-similar means that you see smaller versions of the object as you zoom in, but that's not necessarily true for all of them.
Usually we think of fractals as 2-dimensional things, but I was able to create these 3-Dimensional fractals with an amazing program called Mandelbulber2. The results are below, enjoy!