Questions tagged [fractals]
For questions on fractals, which are irregular, rough, or "fractured" sets that often possess self-similar structure.
1,287 questions
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There are sets $A \ne C$ also satisfying $A=f_1[A] \cup f_2[A]$. How many can you find?
Problem. There are sets $A \ne C$ also satisfying $A=f_1[A] \cup f_2[A]$. How many can you find?
I'm working on this exercise in Gerald Edgar's "Measure, Topology, and Fractal Geometry". I'...
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Show that $C_{k+1} = f_1[C_k] \cup f_2[C_k]$ for $k = 0, 1, 2, ...$ by induction.
Show that $C_{k+1} = f_1[C_k] \cup f_2[C_k]$ for $k = 0, 1, 2, ...$ by induction.
Define $f_1(x) = \frac {x}{3}$ and $f_2(x) = \frac {x+2}{3}$.
Let $C_0 = [0,1]$. Then $C_1$ is obtained by removing ...
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What is the name of the property of $x\mapsto 3f(x)$ that its orbits are wellordered by $f$? [closed]
Consider the dynamical map that terminates on all natural numbers:
$f_o:x\mapsto (x+1)/2$ if $x$ odd
$f_e:x\mapsto x/2$ if $x$ even
This is easily proven to terminate for all natural numbers.
Now ...
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What is the Hausdorff dimension of this Cantor-like set?
Suppose $\alpha=\ln(2)/\ln(3)$. (This is the Hausdorff dimension of the Cantor set.) I originally assumed the Cantor-like set has Hausdorff dimension $\alpha,ドル but now I assume I’m incorrect.
Here is ...
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How do we construct a subset of $[0,1]$ with Hausdorff dimension $\ln(2)/\ln(3)$ with infinite Hausdorff measure in its dimension?
This is different from the questions here and here: we want a subset of $[0,1]$ instead of $\mathbb{R}$.
Suppose $\dim_{\text{H}}(\cdot)$ is the Hausdorff dimension and $\mathcal{H}^{\dim_{\text{H}}(\...
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Rectifiable Subsets of Sets with Large Hausdorff Dimension
A set $E \subset \mathbb{R}^d$ is called 1ドル$-rectifiable if there exists a (countable) family of Lipschitz mappings $f_i : \mathbb{R} \rightarrow \mathbb{R}^d$ such that
$$
H^1 \bigg(E \setminus \...
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Chaotic fringe of a fractal within $x\in(0.551,0.56)$. Can this transition be quantified rigorously?
I was playing around on Desmos a few months ago, with a basic Mandelbrot Set visualizer I made, experimenting with what new shapes I could make by changing the function iterated on $\Bbb{R}^2$. This ...
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Reference request: Koch snowflake is the boundary of a bounded and simply connected open set of the plane
I have found the (intuitive) statement that this well-known snowflake fractal curve encloses a simply connected region both in this article and in this one, the latter also being cited in the ...
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Understanding the Hausdorff Dimension of the Sierpinski Triangle: Mathematical Proof and Geometric Interpretation [closed]
I'm working on understanding the mathematical foundations behind the Hausdorff dimension of the Sierpinski triangle, and I'm particularly interested in both the rigorous proof and the geometric ...
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Is the Mandelbrot set a projection of a 3D object?
Is the Mandelbrot set a projection of a 3D object? Some of the images I have seen look like there is depth if you allow yourself to look at it that way. See attached image for example. Some views look ...
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A continuous and injective map from the unit interval to a subset of the unit square with Lebesgue measure 1?
Description of the Problem:
Netto's Theorem tells us that a continuous bijection between smooth manifolds preserves dimension. Hence, a continuous and bijective map from the unit interval onto the ...
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Sampling theory for non-differentiable functions?
I am interested in sampling fractal like functions. In three dimensions, but for now let's focus on the real line case.
Smooth case
If I have a smooth function $f$ I can generate a sampling of $f$ by ...
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Confused about a quadratic Julia set, Jordan curve and "noncomputable"
I was reading here :
https://mathworld.wolfram.com/JuliaSet.html
And it said, if I am not mistaken :
Consider
$$z_{n+1} = z_n^2 + c$$
for small $c,ドル then the Julia set $J_c$ is also a Jordan curve, ...
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Is a digit-reversal function well-defined for all real numbers in [0,1)?
Motivation
Conway's base 13 function has the intriguing property of mapping any non-empty interval to every real number — yet almost every input is mapped to zero.
I’m interested in finding a function ...
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Fractal approximated by interference patterns [closed]
Question
Can we define a fractal resembling the given image?
See "enhanced" image of a binary matrix $\mathcal H_q$ below.
See "Answer attempt" at the end.
That is, define an ...