# Questions tagged [hausdorff-dimension]

Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension. To be used with [tag:fractals] or [tag:dimension-theory].

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### How can I measure or approximate the fractal dimension of the Barnsley Fern?

I would like to calculate a fractal dimension of the Barnsley Fern, but I am not sure what method may I use, nor even what fractal dimension I should use for this fractal. I know in this post it's ...
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### Fractal dimension of a fractal that is comprised of discrete objects

I'm trying to understand fractal dimension in the context of colloidal gels. But more on that later. I'm confused about a more fundamental thing, which I think relates to the discreteness of the ...
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### Holomorphic maps preserve Hausdorff dimension.

In a paper I read there is the following claim: Let $f:\mathbb{C}\to \mathbb{C}$ be a non-constant entire transcendental function(essential singularity at infinity) and $A\subset \mathbb{C}$ a set in ...
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### Sets of infinite Hausdorff dimension in a second countable metric space

I am wondering if there exists an example of a second countable metric space $X$ containing a set $A$ with infinite Hausdorff dimension.
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### How do you find the Hausdorff dimension of a fractal curve given by an equation?

For example I was playing around in Desmos and came across the following fractal:$$y=\sum_{i=0}^{\infty}\cos(b^ix)b^{-i}$$So I'm curious to know what the fractal dimension is as a function of $b$ ( ...
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### The Hausdorff measure of the unit interval

I am trying to calculate the Hausdorff measure of the unit interval. Here's my attempt: Fix $\epsilon>0$. consider the open balls $B(x,\epsilon)$ with $x\in[0,1]$. How many can cover the unit ...
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### Are there two sets $X$ and $Y$ such that the following inequality for box dimension holds

I am trying to find sets $X$ and $Y$ s.t. $\dim_B(X\bigcup{}Y)>\max\{\dim_B(X),\dim_B(Y)\}$. At first I thought taking $X=[0,1]$ and $Y=\{10+1/n^2:n\ge{}1\}$ but I don't think that works. Is this ...
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### Finding the dimension of the sphere cube

If you take an $2r\times 2r\times 2r$ cube, and divide it to 27 equal cubes, and then remove all the "axis" cubes (all the cubes which are straight left, straight right, straight up, etc. from the ...