Questions tagged [fractals]
For questions on fractals, which are irregular, rough, or "fractured" sets that often possess self-similar structure.
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What is the name of the property of $x\mapsto 3f(x)$ that its orbits are wellordered by $f$?
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 $\alpha$, but now I assume I’m incorrect.
Here is ...
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Rectifiable Subsets of Sets with Large Hausdorff Dimension
A set $E \subset \mathbb{R}^d$ is called $1$-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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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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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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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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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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What is the fractal dimension of tetration fractal?
What is the fractal dimension of tetration fractal?
1. What is Tetration Fractal?
Tetration Fractal is a fractal expressed on the complex plane, which indicates the area of complex numbers that its ...
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What mathematical principles underlie our perception of beauty in geometric forms? [closed]
From the golden ratio in art and architecture to symmetry in Islamic tiling and fractals in nature, it seems that certain mathematical structures are consistently associated with aesthetic appeal.
Are ...
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Can you have a nested recursively deepening hyperbolic fractal structure?
I am staring at this basically, from Margenstern's work on Cellular Automata in Hyperbolic Spaces, like this paper About the embedding of one dimensional cellular automata into hyperbolic cellular ...
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IFS and Hausdorff dimension
Let $(X,d)$ be a complete metric space, $\{ T_1, \dots, T_m \}$ an iterated function system of similarities defined on the set of compact nonempty subsets of $X$ and $F$ the corresponding fixed point (...
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Show that $\dim_H(F)=\dim_B(F)$,where $F$ is the attactor of a finite collection of similarities with ratios less than 1
I learnt form Fractal Geometry_Mathematical Foundations and Applications of Falconer that $\dim_H(F)=\dim_B(F)=s$,where $s$ is the similarity dimension of the IFS which produces $F$ if the open sets ...
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Open set condition (IFS)
Let $(X,d)$ be a complete metric space, $\{ T_1, \dots, T_m \}$ an iterated function system of similarities defined on the set of compact nonempty subsets of $\mathbb{R}^n$ and let $F$ be the ...
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Moment of inertia of a part of a fractal
A physics problem that I was solving recently went as follows:
Take a square plate of side l , and remove the “middle” square (1/9 of the area). Then
remove the “middle” square from each of the ...
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Hausdorff-Dimension and Ahflors-Regularity
I have come across the following result in fractal geometry
Given a non-empty bounded subset $A\subseteq \mathbb{R^n}$ and a Borel-regular measure $\mu$ on $\mathbb{R^n}$ with $0 < \mu(A) \leq\mu(\...
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Proof that The Mandelbrot Set is bounded
Lemma 3.2.1: Let $n \geq 1 \in \mathbb{N}$. Then
\begin{align*}
2^{n+2}-4 \geq 2^{n+1}, \hspace{10pt} 2^{2n} \geq 2^{n+1}.
\end{align*}
Consider the sequence $(z_{c_{n}})_{n \in \mathbb{Z}^+}$, ...
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Generating a uniformly random point on the Sierpinski Triangle?
Is there a math procedure that will theoretically generate a point on the Sierpinski Triangle uniformly at random? There are a bunch of numerical approximate methods I can think of (e.g. just ...
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How do these concepts involving tuples and sets give these patterns?
Let $r$ be a fixed integer ($r \geq 2$), and define
$C_0 = \{ (x_1, x_2, \dots, x_r) : 0 \leq x_i \leq 9 \text{ for } i = 1, 2, \dots, r \}$.
For each digit $d \in \{0, 1, \dots, 9\}$ and any tuple $(...
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Uniform partitioning of a subset of the square lattice into two connected graphs
Suppose that I have a subset of the square lattice.
Think of this box as having one face.
Now, I consider all possible divisions of the box in such a way that there are two faces.
On the picture I ...
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Formula to Determine the Value of a Specific Part of a Term in a Fractal Sequence
I am working with a fractal sequence where each term introduces additional intermediate values in a structured pattern where between terms the amount added between integers is divided by 2. The first ...
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If $a^3+b^3=1$, is there a set that can be divided into two similar copies of itself, scaled by $a$ and $b$ respectively?
Let $a$ and $b$ be positive reals.
If $a+b=1$, any line segment can be divided into two smaller line segments, whose ratio the the original is $a$ and $b$ respectively.
If $a^2+b^2=1$, then there ...
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Find the area of any Pythagorean tree fractal in terms of the side lengths a and b.
For those of you who don't know, this is how you generate a Pythagorean tree fractal:
Draw any right triangle.
Turn each of its sides into a square on the exterior of the triangle. You should now ...
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Is this novel fractal dimension related to the conventional ball counting dimension?
According to Wikipedia, the Minkowski ball counting fractal dimension of a bounded set $S$ is defined by
$$\dim_\text{b}(S) := \lim_{\varepsilon \to 0} \frac {\log N(S,\varepsilon)}{\log(\varepsilon_o/...
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Where are the units of the box counting fractal dimension?
The box counting dimension of a bounded set $E$, as defined on Wikipedia is defined as:
$$D=\lim_{\epsilon \to 0}\frac{\log{N(\epsilon)}}{\log{(1/\epsilon)}},$$
where $N$ is the number of boxes of ...
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True or False of the Cubic Newton Fractal ?: Each point of the Julia Set is a common boundary point of six disjoint "basin" components
Concerning the Cubic Newton Fractal, i.e. the Julia set, in the complex plane, for iterations of $z \mapsto z - (z^3 - 1) / (3 z^2)$,
I'd like to check a particular assertion which I have (perhaps ...
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