If you think about the shapes that a fractal (or coloring method) generates, and specifically about their size (or 'slope') with respect to the size of the pixels, you can start to form an idea about how many pixels the detail is covering, or how fast the actual function underneath is changing with respect to the pixels that you are trying to use to capture a picture of it.

If you think about a fractal as being made up of waves (like a sound), this aspect would be thought of as the 'frequency' (or pitch) of the waves. Many fractals have higher than pixel-frequency content, (that is, wavelengths that are /smaller than a pixel/) and unless you do something about this, it causes an artifact called 'aliasing' in the image, in other words, bad image quality.

If you look at things from the pixels side, you can start to look at how much detail you are trying to fit into each pixel, when you try to fit too much in, you have to use some extra sampling (called 'antialiasing') or else (if you only take one sample per pixel for instance) you get a bad estimate of the function under that pixel - this leads to nasty looking jagged edges, uncontrolled noise, "moire" interference patterns or (if its an animation) unwanted sparkling, crawling or strobing.

In short, aliasing in a render leads to a more inaccurate (than necessary) representation of what is actually going on under those pixels.

Most classical fractals (like mandelbrot, julia etc) have one specific line (or a 'dust' of points) around which the detail is smaller than the pixels that you're rendering, the rest of the fractal is usually a fairly smooth gradient, which doesn't contain any higher frequencies than the function it is trying to represent. This makes them easier (and faster) to render because you only have to do a lot of extra sampling work near the edge.

There is some more insight about this on wikipedia:

Returning to the explanation of my comment (from the ultraFractal mailing list), I really like it when the detail in a fractal makes the antialiasing do a lot of work everywhere, which is about having some high frequencies /everywhere in the frame/. I find that it makes the image feel very high quality, like it would have taken patience and care to render and it hints at the idea that it could even be infinitely detailed without even zooming. (to Marc Librescu) Some of your renderings are moving towards that kind of situation, and I like it :D.

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