Having an idea of what a fractal is, by example, etc., then seeking the actual definition is, at first, both obvious and imprecise. You'll see it defined as an object that is self-similar in some sense or other, but none of the definitions along these lines are at all rigorous.

Then looking back to Benoit Mandelbrot's original characterization of them: they are topological spaces wherein the Hausdorff dimension exceeds the topological dimension. Let's take those two dimension definitions one at a time.

The Hausdorff dimension of a space is the infimum of values for $d$, for which the $d$-dimensional Hausdorff content of the space is zero. The $d$-dimensional Hausdorff content of a space, in turn, is interpreted by considering all possible covers of the space by a collection of balls, then for each such covering, forming the sum of the radii of all the balls each raised to the power $d$. Then from among those possible sums, one selects the infimum. And so the Hausdorff dimension then is the smallest $d$ value for which that infimum is zero.

Now to the topoligical dimension: there are some choices here, so I opted with the one that said something about a fractal's boundary, because that is a very recognizable aspect. We must also assume our space is separable, which I don't mind. The definition is recursive. So we require that the topological dimension is always integral, and that the topological dimension of $\emptyset$ is $-1$. Then we consider all possibilities of choosing simultaneously a point and an open set from the space, and for each possibility we inductively require that there is a set containing the point, but contained in the set, whose boundary has it's topological dimension bounded by $n$. And so we insist that the dimension of the space must then be no more than $n+1$.

Well, I understand that stuff taken one step at a time, but I feel pretty in the dark about how that definition implies the self-similarity property we observe. Does raising the radius of the balls to the power of $d$, in the $d$-dimensional Hausdorff content, have the effect of measuring something like the dimension of the space it embeds into, when we take that infimum? That touches on conceptualizing fractional dimension, but still, why should the only way of achieving this fractional dimension be by self-similarity?

Surely there must be equivalent definitions of a fractal out there that are not so impenetrable as this, do you know of one? Especially one that elucidates where the self-similarity results come from?

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    $\begingroup$ Not every set of fractional dimension is self-similar, that condition has to be additionally imposed on "interesting" fractals. Converse is also not true, straight line, plane, etc. are self-similar but of whole dimension. $\endgroup$ – Conifold Jul 17 '14 at 19:43
  • $\begingroup$ Is there a precise rigorous condition one can impose to make them "interesting?" $\endgroup$ – j0equ1nn Jul 17 '14 at 19:45
  • $\begingroup$ I don't think there is ever a "precise" condition that makes a mathematical object interesting. But Wikipedia lists some conditions, self similarity, fine structure and irregularity among them en.wikipedia.org/wiki/Fractal#Characteristics. $\endgroup$ – Conifold Jul 17 '14 at 19:49
  • $\begingroup$ Of course not! Hence the quotes! But we do search for mathematical framework behind what we find interesting. In this case I refer to the self-similarity. As far as I can see Wikipedia classifies various forms of self-similarity but does not indicate a point-set theoretic condition that could be appended to Mandelbrot's definition, which would precisely capture the behavior. $\endgroup$ – j0equ1nn Jul 17 '14 at 20:19

how that definition implies the self-similarity property we observe.

It does not. Fractals, defined as sets with $\operatorname{H-dim}<\operatorname{T-dim}$, have no intrinsic reason to be self-similar. That we tend to think of fractals as self-similar is a kind of observational bias.

We can only imagine sets (and maps) with a finite (and pretty low at that) number of different features. Our visual field (either physical or mental) has limited resolution. A "generic" subset of real line is something we can never hope to comprehend in its entirety.

To augment our limited vision, we can zoom in on a set and see what it looks like on some scale. But there are infinitely many scales. So we give up and say: let's think of the sets that have the same pattern on every scale, or perhaps have a finite number of patterns that appear on different scales in some regular way. What else is there to do?

By the way, the smooth, non-fractal, objects also have such paucity of features: on all sufficiently small scales, a smooth surface looks just like a plane.

Conclusion: don't try to infer self-similarity from inequality of dimensions. If you want a definition that makes self-similarity transparent, look at the definition of self-similar set.

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    $\begingroup$ I see I had made false assumptions about what a fractal is. The distinction between self-similarity and fractal is certainly what I was missing - this helps a lot. I'm tempted to delete my question since it requests an explanation of something that's not even true. On the other hand I think it's a common ignorance about fractals, so perhaps your answer will be of use to others. $\endgroup$ – j0equ1nn Jul 18 '14 at 21:54

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