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Results tagged with fractals
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user 11919
Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
7
votes
Accepted
Hausdorff dimension of inverse images.
The boundary of $A = f^{-1}((-\infty, t))$ and $B= f^{-1}((t,\infty))$ is $C = f^{-1} (t)$. Therefore $C$ has Hausdorff dimension at least $d-1$, using this MO entry. I recommend Sergei Ivanov's respo …
- 105k
7
votes
How might M.C. Escher have designed his patterns?
Adding to Valerio Talamanca's answer, there was a lecture by Hendrik Lenstra on Escher's "Print gallery" at Leiden University (December 12, 2012) which can be viewed here.
Also, my memory says that el …
- 105k
18
votes
Accepted
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prim...
The sum is less than $1$. First of all, Mathematica and SAGE independently tell me that
$$\sum_{n=1}^{10000} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}=0.950344\dots.\tag{1}\label{1}$$
We estimate the tail sum …
- 105k