Self-similar fractals scale equally in each direction, self-affine fractals can scale differently across directions, and multifractals accommodate differing local scaling behavior within one object.

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All can be analyzed as irregular fractal constructs that have more regularity to them than was first assumed. The variety of fractals is immense, but all have a few common traits. First, they scale up or down by a specific amount. That is, the parts echo the whole in accordance with a precise, measurable formula. The simplest fractals scale the same way in all directions, hence, are called self-similar. They are like high-quality zoom lenses that expand or shrink everything in the frame by the same degree. What they see at one focal length will be similar to what they see at another. But the Bachelier cartoon scales more in one direction than another, and the same will be the case with other cartoons of price variation to be introduced in later chapters. Such fractals are called self-affine. They are like an office laser photocopier, set to shrink a document's image more crosswise than lengthwise.

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If the fractals scale in many different ways at different points, they are multifractal, and their mathematical properties become intricate and powerful. Indeed, the mathematics of fractals in full glory is difficult in its detail, but in its broad strokes, thousands of 16- and 17-year-olds are now learning it as part of their basic math courses. Fractals are supremely visual, hence, supremely intuitive. Fractals can look haphazard. They often defy conventional geometry or analysis to categorize. They are usually irregular and perplexing, rather than nice and predictable like the parabolas and circles of the old geometers but the key point all fractals start simply simplistically some might say in its first stages any scientific investigation had better be simplistic, otherwise it will never fly. Every fractal is the logical expression of a few straightforward ideas, rules, or mathematical relations.