Questions tagged [fractals]
For questions on fractals, which are irregular, rough, or "fractured" sets that often possess self-similar structure.
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Why is Assouad dimension the largest and Lower dimension the lowest?
I understand both definitions, but I can't see why Assouad is maximal and Lower is minimal.. Could someone detail these "why's" in a by absurd? I failed writing this proof. If someone just ...
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Generating fractal outlines
I am looking for an algorithm that will generate natural-looking (as in created by nature) polygonal shapes. The goal is to create 2D colorful art. This might be via parameterized fractals (I found ...
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Julia Set programming in Wolfeam Mathematica [closed]
How Julia set is generated in wolfram Mathematica coding language?
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Does an infinite line segment make any sense? [closed]
In school, I was taught that a line segment must be finite. But a fractal has an infinite perimeter. If you just took two points along that perimeter, and "stretched it out" wouldn't you ...
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Local dimension of stationary measures for iterated function systems with an expanding map
This question was previously posted on MathOverflow.
Consider the iterated function system (IFS) $X_n$ on $I = [0,1] $generated by the functions $\Phi = \{f_1,f_2,f_3\}$ and the probability vector $P =...
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About Frostman Lemma [closed]
Let $H^g$ denote the Hausdorff measure with respect to a gauge function $g$. Suppose $K \subset \mathbb{R}^d$ is compact. Is it true that $H^g(K)>0$ if and only if there exists a (regular Borel) ...
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Non-square Moore-curve-like space filling curve in three dimensions
I wonder whether there is a space filling curve that looks like the following
if the width of the domain has an odd number of cells and looks like the following
if the width of the domain has an ...
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Dragon curves, fractional dimensions and whatever could be deduced
I put out a question about a thesis on Dragon Curves which was deleted. I did not know it is against the guidelines to advise on academia. Please don't consider my question as seeking advice on that ...
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Novel approach to the box counting calculation of fractal dimension
The box counting method of measuring the fractal dimension of an object is
$$D = \lim_{\epsilon \rightarrow 0}{ {\log N( \epsilon)} \over {\log { {1}\over{ \epsilon }}}},$$
where the classic example ...
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Periodic Points in Julia Set under Rational Map
Let $A_1$ be the finite set of periodic points in the Julia set for a rational map, $R$, which has lowest possible period.
Inductively, define $A_{n+1} = R^{-1} (A_n) \cup A_n$.
Why is it true that $...
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This fractal isn't a Sierpiński carpet so what is this variant?
While building fractals in minecraft I built this fractal with the intent of making a Sierpiński carpet but I made a mistake and created this (I also built this in 3d).
The procedure I used to create ...
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Is there a Hamiltonian path though a Menger sponge of level-n?
This is a thought that I was having while building a model of a level 4 Menger sponge in minecraft.
Imagine a Menger sponge to be built of cubic voxels the same size as the smallest void. You can ...
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Markov chains: Hitting time of random walk on Sierpinski triangle
Given a Sierpinski triangle $G_n$ and a random walk on $G_n$ denoted as $(X_i)_{i\in\mathbb{N}}$, I'm attempting to prove that the hitting time $T_n$ to go from one corner to any of the other two ...
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The fractal dimension of the complementary space
I'm a physicist who specializes in studying the behavior of granular materials. Research has confirmed that certain distributions of granular sizes exhibit fractal characteristics. In simpler terms, ...
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Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency
let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$
I'm studying fractal geometry and ...
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Why does the Mandelbrot fractal appear when plotting $\underbrace{x\cos(x\cos( \cdots x\cos}_9(x))))$?
while plotting the function $x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x)))))))))$
using matplotlib in python I found the mandelbrot fractal.
What is the reason that the mandelbrot fractal ...
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How is category theory relevant to fractal geometry?
In this YouTube video by Richard Southwell at the 2:30 timestamp, it is said that:
Fractal geometry can be studied quite profitably using category theory.
The fields seem very different to me but I ...
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Is a non-fractal, continuous curve made of tiny line segments?
EDIT: This question was in-part made unclear by my misconception of the hyperreals not being dense, since I thought there were no numbers between zero and consecutive infinitesimals
($0, 1/\infty, 2/\...
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False formula of Galois and Abel fractal and the quintic
6Well all start with the well-know sentence due to Abel : "At 16 years old i found a false formula for the general quintic "
After many attempts I found by myself this :
Fractal formula :
It'...
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Mapping a Line onto a Plane
I was recently thinking about space filling curves and the following question popped into my head.
Clearly $\mathbb{R} \times \mathbb{R}$ is equinumerous to $\mathbb{R}$ so that there is a bijection ...
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Is there a minibrot in the end of the "endless" spirals in the Mandelbrot Set?
I'm new to fractals and especially the Mandelbrot Set. I've noticed these never ending and self-similar spirals all around the Mandelbrot set, just like the one below:
At lower max-iterations, there ...
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Example of a geodesic metric space with non-integer Hausdorff dimension
If one googles metric spaces with a finite non-integer Hausdorff dimension, one usually finds fractals like the Cantor set or the Koch snowflake.
These two examples are not so well-behaved in the ...
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Does Multi-Fractal analysis on time series signal help extracting local properties?
Fractal based measures help quantify the self-similar geometrical properties of set (in this case a time-series signal). It mostly represents the global characteristics of geometry of the signal.
But ...
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Newton fractal formula conversion to coordinate plain
starting to do some shader with Newton fractal I can't understand examples.
If I have $$z^{2}+c$$ fractal,
it could be presented as a $$(a +b*i)^{2}+c$$ $$a^{2}+2*a*b*i - b^{2}$$ and I will get point ...
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Are there "continuously generated" space-filling curives (or other fractals) that allow "differentiable approximation"?
I think all fractals I am aware of are based directly or indirectly on the iterated application of some function or substitution rule.
Typically, e.g. space-filling curves are presented as ...
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Does the Box-Counting Dimension always exist after applying a Lipschitz mapping?
When you have a bi-Lipschitz mapping and the Box-Counting Dimension of $A$ is defined, then the Box-Counting Dimension of $f(A)$ ist also defined (and the same as the B.-Dim. of $A$).
We can also ...
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Generalisation of Fatou-Julia theorem
In an already answered question regarding the Fatou-Julia theorem (every basin of attraction contains a critical point), I saw that a user stated:
There are some other situations where an analogous ...
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Marstrand's slicing theorem - lower bound and equality
Consider a fractal $\mathcal{S}\subset\mathbb{R}^2$ (e.g., a Koch snowflake) and a 1-dim line $\ell$. Marstrand's slicing theorem states that
$$dim_H(\mathcal{S}\cap\ell)\leqslant dim_H(\mathcal{S})-1$...
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Caputo Fractional Derivative of $t^-\alpha$
There is a theorem in fractional derivative that states that the Caputo Derivative of order $\alpha >0$ with $n-1<\alpha<n$ of the power function $f(t) = t^p$ for $p≥0$ satisfies:
$$\...
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Are there hilbert space filling surfaces (2D) / volumes (3D) / (N-D)?
If I understand correctly Hilbert space filling curves only map N dimensions to 1.
However, I'm looking for a similar concept, that maps N dimensions to X dimensions. (My choice of how many).
And ...
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What is the Hausdorff measure of the Vicsek Fractal?
If we consider a vicsek fractal of width 1 and height 1 (in other words constructed from the unit square) its easy to show that the hausdorff dimension of this is $\frac{\ln(5)}{\ln(3)}$ through the ...
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Verify that the attractor of an IFS is the graph of an interpolation
I am going through exercises in Fractals Everywhere by Michael Barnsley. I have a confusion about Exercise 2.1 in Chapter 7:
The function $f(x) = 1 + x$ is an interpolation function for the set of ...
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Counting fractals via "stagewise complements"
This is motivated by this older question. Now posted at MO.
Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to ...
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Naïve definition of a measure on a fractal
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use the so-called Hausdorff measure $\mathcal H^...
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Why do these patterns appear in custom fractals? [closed]
Last year I programmed a fractal renderer and played around a lot with fractals such as the Julia set and Mandelbrot set. Eventually I got curious and inputted my own algorithms for the fractal, and ...
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Is the interior of the mandelbrot set connected?
I know that the Mandelbrot set is connected, but what about its interior? It doesn't seem intuitively like it should be, but I can't find any information online confirming this. I can think of an ...
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How is the Lorenz attractor a fractal? [closed]
Based on my intuition of what a fractal is, the Lorenz attractor doesn't fit that category for me. A fractal should have some self similarity, but the attractor seems just like two two-dimensional ...
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Drawing the Apollonian gasket
Note: I've already asked a question in a similar spirit to this, but I don't feel like I got my point across so here I am.
I am working on a program that draws the Apollonian gasket fractal. I need ...
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Finding radii of three tangent circles given their centers [closed]
I'm interested in making a program that draws the [Apollonian gasket][1] fractal. For this, I need a way to find the radii of three mutually tangent Soddy circles given their centers, for example ...
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Algebraic solutions to the problem of Apollonius for Soddy circles
I'm interested in making a program that draws the Apollonian gasket fractal. To do this, I need a method to find the center of a circle internally or externally tangent to three Soddy circles. ...
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How do I algebriacally analyze these results from the mandlebrot set?
The equation for the mandelbrot set is z = z^2 + c,
in my exploration, I changed the values of c and analyzed the iteration in the table below
What algebriac patterns emerge in this table?
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Hausdorff dimension of the Koch curve
It is well known that the Hausdorff dimension of the Koch curve is $\log_34\approx1.26>1$. Thus by the property of Hausdorff measure, $\forall1<p<1.2$, the $p$-Hausdorff measure of the Koch ...
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Confusion over dense level sets of the Takagi function
I was told that the Takagi function, $T:[0,1]\rightarrow \mathbb{R}$, is continuous and has uncountable dense level sets in $[0,1]$. This has confused me for the following reason:
Suppose $L$ is a ...
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Oscillations in Newton's fractal
I'm working on a program that draws Newton's fractal for a given polynomial. Newton's fractal is a fractal derived from Newton's root-finding method, which given some initial guess $x_0$ and function $...
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Why is the Sierpinski carpet connected and locally connected?
The Wikipedia article on the Sierpinski carpet fractal says that it is compact, connected and locally connected.
It is clear from the construction that the Sierpinski carpet is closed and bounded in $\...
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How to find the limit to infinity of the iterative function used in the Mandelbrot set with a fixed C.
I'm trying to do my math IA on the effect of changing complex constant c in the mandelbrot set on the series's convergence, but I don't know how to algebraically solve the limits for iterative ...
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Intrinsic dimension calculated via 2_NN method vs Hausdroff dimension
I followed the following video:
https://www.youtube.com/watch?v=zZ_NmaMeblU&ab_channel=YoavFreund
to understand the intrinsic dimension. He took a shape that looks like a blob ($2$ dimensions) and ...
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How to calculation the Hausdorff dimension of this set?
It's about Exercise 1.19, Fractals in Probability and Analysis?
Suppose $S\subset\mathbb{N}$, and we are given $E,F\subset\{0,1,2\}$. Define $B_S=\{x=\sum_{k=1}^\infty x_k2^{-k}\}$ where $x_k\in E$ ...
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Box dimension of a perturbed set in the interval $[0,1]$
I've asked a couple questions about box dimensions (also called the Minkowski-Bouligand dimensions) recently. My main goal is to better understand the box dimension of discrete sets of points within ...
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Does any small enough circle around a point of a curve, intersects it at exactly two points?
If we choose any point of a line, a parabola, a circle, a sinusoidal curve etc, as a center of a circle of radius r, then there is a distance d such that, if r<d, the circle intersects the curve at ...
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