# 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 ...
1 vote
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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 ...
1 vote
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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 ...
261 views

### 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 ...
1 vote
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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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### 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 ...
155 views

### 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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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] 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. ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 $x_k\in E$ ...
1 vote