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A fractal is usually defined to be a self-similar shape (this is the informal definition). But, the formal definition is:

A fractal is a set for which the Hausdorff dimension strictly exceeds the topological dimension (formal definition).

Before proceeding to the question, I will tell what this means. The topological dimension is the "normal" dimension; a line is 0 dimensional, a surface is 2 dimensional, etc. The Hausdorff dimension is defined as $\log s/\log z$. What are $s$ and $z$? Suppose we zoom in to the shape. Here, $z$ is the zoom change, and $s$ is the size change in the shape.
So, my question is

In what cases and why are the formal definition and the informal definition equivalent?

I don't know much about topological and Hausdorff dimensions; I just know their intuitive meaning. So please pardon me if the definitions' equivalence is obvious from some elementary property of those dimensions.

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    How do you know they are equivalent? –  Dec 02 '20 at 10:24
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    I don't think these are equivalent. For instance, consider the Sierpinski triangle, but instead of always removing regular triangles, we remove regular $n+2$-gons in each step. So in the first step we remove a triangle, then squares, then pentagons, then hexagons, etc. I doubt that anyone would call this thing self-similar, but I'd bet that its Hausdorff-dimension is larger than its topological one. – Vercassivelaunos Dec 02 '20 at 10:24
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    The name fractal comes from the fractured shape they have, they're not necessarily self similar. Self similar fractals are like the regular polygons of the fractal world. They're nice because we can easily put our hands on them to compute things about them to gain some intuition of what fractals are like. – Merosity Dec 02 '20 at 10:28
  • @merosity: Sorry, I didn't know that. I have made a minor edit. –  Dec 02 '20 at 10:30
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    There are also random fractals which are not self-similar (eg Brownian motion can have a fractal character), and things like the problem of measuring the coastline. – Mark Bennet Dec 02 '20 at 10:33
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    First and foremost, it should be noted that the term "fractal" is not well-defined in mathematical literature. The "formal" definition you give is a definition which was proposed by Mandelbrot in the first edition of his Fractal Geometry of Nature. In the second edition, he essentially retracted this proposed definition. See here: https://math.stackexchange.com/a/2677204/468350 for further discussion. – Xander Henderson Dec 02 '20 at 12:06
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    Thus your question really seems to be "Is a set self-similar if and only if its Hausdorff dimension exceeds its topological definition?" Here, the answer is "no". For example, the unit interval $[0,1]$ has topological and Hausdorff dimension $1$, but is self-similar (it is, for example, composed of the two half-unit-intervals $[0,1/2]$ and $[1/2,1]$). – Xander Henderson Dec 02 '20 at 12:08

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