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Questions tagged [fractals]

For questions on fractals, which are irregular, rough, or "fractured" sets that often possess self-similar structure.

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Show that lipeomorphisms are homeomorphisms

If S and T are metric spaces with $\varrho$ as a metric, a function $f:S \to T$ is a lipeomorphism iff there exist positive constants A and B such that $A \varrho(x,y) \leq \varrho(f(x),f(y)) \leq B\...
Rubén Sales Castellar's user avatar
1 vote
2 answers
59 views

Sierpinski Gasket coordinate description

I was reading Gerald Edgar's "Measure, Topology, and Fractal Geometry" when I came across this exercise Let coordinates $(u,v)$ be defined in the plane with origin at one corner of the ...
Rubén Sales Castellar's user avatar
3 votes
0 answers
74 views

What are the order-3 non-fractal irreptiles?

An irreptile is a shape that can be dissected into smaller copies of the same shape. An order-3 irreptile would divide into three similar copies of the original shape. What are the order-3 non-fractal ...
Ed Pegg's user avatar
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Fractal Subdivision for Procedural Generation

This is my first attempt at creating a thread here, thus I'm not sure how much context I need to provide you. Please leave a commend if you need additional info. With that disclaimer out of the way, ...
James Sarantidis's user avatar
2 votes
1 answer
34 views

What does it mean to be $(M, s)$-homogeneous?

I'm trying to get a handle on the Assouad dimension and Wikipedia says that for a metric space $(X, \zeta)$, The Assouad dimension of $X$, $d_A(X)$, is the infimum of all $s$ such that $(X, \zeta)$ ...
roundsquare's user avatar
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-2 votes
2 answers
43 views

Existence of generalization of space-filling curves [closed]

The concept of space-filling curves is well-known: There exist continuous maps from $[0,1]$ that fill a box in $n$ dimensions. Does there exist a more general concept of filling $n$-dimensional space ...
Jas Ter's user avatar
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2 votes
1 answer
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Hausdorff dimension of the Sierpiński triangle calculated from definition

I already know how to calculate the Hausdorff dimension of the Sierpiński triangle the way it is presented on Hausdorff dimension of Sierpinski triangle less than log3/log2. However, my complaint is ...
Dave the Sid's user avatar
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Is there an L-system for coral sea fans?

I've been looking at several kinds of sea fan such as: https://en.wikipedia.org/wiki/Gorgonia_ventalina It occurred to me that this could likely be modelled with an L-system, but I couldn't see anyone ...
thenapking's user avatar
3 votes
1 answer
40 views

Fractal Geometry: Show that the Hausdorff dimension of a set and its image under $f(x)=x^2$ are the same.

Let $f:\mathbb{R} \to \mathbb{R}$ be the function $f(x) = x^2$, and let $F$ be any subset of $\mathbb{R}$. Show that $\text{dim}_Hf(F) = \text{dim}_\mathrm{H} F$. Here $\text{dim}_\mathrm{H}$ refers ...
wiishopwednesday's user avatar
2 votes
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Is there a notion of a "smooth approximation" of a fractal?

The motivating example that comes to mind is the koch snowflake fractal. Given that the recursive structure of the fractal comes from injecting triangles into triangles, it seems like one could ...
Makogan's user avatar
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Cantor set Self-similarity

I’m defining the Cantor set in the following way $C:= \bigcap_{n\ge 0} C_n$ with $C_0:=[0,1]$ and $C_n:= \frac{1}{3} ( C_{n-1} \cup (C_{n-1}+2))$ for all $n\ge 1$ I should prove that $T(C)=C$ with $...
Shiva's user avatar
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Hutchinson's Notation in Fractals and Self-Similarity

I have been examining https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf for my thesis but couldn't find an explanation for two notations in the paper. ...
Dave the Sid's user avatar
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Hypersphere of Fractal Dimensions

Is the concept of a fractal dimension when applied to questions like a fractal dimension hypersphere volume nonsense? For example: Can a 4.5 dimensional hypersphere exist and have a 4.5D volume. And ...
lucasbachmann's user avatar
1 vote
1 answer
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Reflecting a fractal

Assume you have a certain IFS (iterated function system - a finite set of contractions) given by affine transformations. As reflections are affine transformations, any reflection of an IFS fractal ...
Psaro's user avatar
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Hölder regularity and Hölder condition

Is there any textbooks or papers introducing Hölder regularity and Hölder condition? When I read the paper(https://www.ams.org/journals/tran/2021-374-11/S0002-9947-2021-08489-5/home.html), I found ...
Pei-Zhi Liu's user avatar
3 votes
0 answers
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Cover the set $\{0,1,1/2,1/3,1/4,\dots\}$ with the least amount of closed intervals of length $\delta>0$.

Let $\delta>0$ and define $N(\delta)$ to be the least number of closed intervals of length $\delta$ necessary to cover the set $S:=\{0,1,1/2,1/3,\dots\}$. For example, it is clear that $N(1/2)=2$. ...
Croqueta's user avatar
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Contraction mapping theorem for statistical self-similarity

The notion of self-similarity that I was given is the following: Let $(X, d)$ be a metric space, and for each $1 \leq i \leq N$, let $f_i: X \rightarrow X$ be a contraction. Then there exists a unique ...
parity's user avatar
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-1 votes
1 answer
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Do all >1D box fractals really have lines in them?

Consider an $n \times n$ square split up into $n^2$ cells the natural way. Now suppose we fill in $1 \le k \le n^2$ of these cells s.t. $\log_n(k) > 1$. Is it always the case that there will be a ...
Sidharth Ghoshal's user avatar
1 vote
1 answer
60 views

Iterated function system with a fractional number of contractions

Iterated function systems can be used to generate fractals. One starts off with a simple geometric figure and applies the IFS infinitely many times to obtain a fractal. For example, in the case of the ...
Artur Wiadrowski's user avatar
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Almost all projections of the four corner Cantor set have $0$ Lebesgue measure. A mistake in the proof and how to fix it

Let $E$ be the $1/4$ Cantor set, which I define briefly here $$E := \{ 3\sum_{n=1}^{\infty}{\frac{\epsilon_n}{4^n} } \, : \, \epsilon_n = 0 \text{ or } 1\}$$ Or it can be defined in this way Let $\...
Paul's user avatar
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1 vote
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Convergence in Hausdorff distance $\iff$ Lebesgue measure converges

There are numerous sources (e.g. here) mentioning that Lebesgue measure is not continuous with respect to the Hausdorff distance $d_H$. This means that for a sequence of sets $K_n$ converging to $K$, $...
JayP's user avatar
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Alternative proof Box-counting dimension Cantor set

From the Book of Falconer (Fractal Geometry: Mathematical Foundations and Applications, 2nd ed.), we have the following proposition: Proposition 3.2 Let $F \subset \mathbb{R}^n$, then $$ \begin{...
Mikeys00's user avatar
1 vote
1 answer
109 views

Evaluate $\int_{-\pi}^\pi1+\lim\limits_{n\to\infty}\min(\cos(x),\dots,\cos(nx))dx$ to find the fractal’s area

We want to find $$I=\int_{-\pi}^\pi1+\lim_{n\to\infty}\min(\cos(x),\dots,\cos(nx))dx$$ Here is a picture of $1+\min(\cos(x),\dots,\cos(nx))$ for $n=50$: $\displaystyle I_n= \int_{-\pi}^\pi1+\min(\cos(...
Тyma Gaidash's user avatar
2 votes
1 answer
48 views

Countable, self-similar total orders

A total order $I$ is said to be weakly self-similar if there exists a proper subset $J \subsetneq I$ together with a bijective, order-preserving function $f:I \to J$ (that is, $J$ is isomorphic to $I$)...
Andrea Marino's user avatar
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0 answers
28 views

Symmetry of King's dream fractal (orbit of a 2-dimensional iterated function system is a parallelogram?)

For fixed real numbers $a,b,c,d$ define the map $f : \mathbb{R}^2 \to \mathbb{R}^2$ by $$f : (x, y) \longmapsto (\phi(ax) + b \cdot \phi(ay),\hspace{1em} \phi(cx) + d \cdot \...
Watson's user avatar
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2 answers
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Need help naming a fractal/image generated by exponential

I'm taking an introductory comp-sci course this semester, and my professor pulled up this graphic while explaining mergesort (the graphic shows blocks of size $2^{-n}$). For some reason, the vertical ...
gingerale_pianoman's user avatar
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0 answers
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Correctness implementation perturbation theory on Mandelbrotset

I'm trying to wrap my head around the perturbation theory paper by K.I. Martin. I've made a toy Python script to try it out using float64 as 'high precision' for the reference and calculating the rest ...
Matige KunstIntelligentie's user avatar
1 vote
1 answer
54 views

Easy to understand proof of Self-similar IFS fractal dimension

I'm preparing for a seminar talk on fractals, the topic is Self-similar IFS fractal dimension, proving the main theorem used: Given $IFS=\left\{ \mathbb{R}^{n};w_{1},...,w_{N}\right\} $ with ...
Nadav's user avatar
  • 479
2 votes
2 answers
94 views

3d fractal helix modeling

I'm trying to build a 3d visual to illustrate a concept. Imagine a circular helix. We could define a cylinder that contains that helix. But now imagine this cylinder takes the helicoïd shape too ! We ...
nnuuurrrrcc's user avatar
2 votes
1 answer
115 views

Is the Mandelbrot set sufficiently self-similar to non-rigorously visually recreate procedurally from warped, nested structures?

While not strictly self similar, the Mandelbrot set shows significant similarity in its pixelated visual samplings. Are the overall shapes approximating each region well understood enough, not in a ...
Lucent's user avatar
  • 121
0 votes
1 answer
98 views

List of open conjectures on fractals [closed]

I want a list of open conjectures on fractals. Preferably easy to understand and low dimensions. Before you say Eremenko's conjecture ( https://en.wikipedia.org/wiki/Escaping_set ): https://arxiv.org/...
mick's user avatar
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0 answers
24 views

Best way to fit a huge balanced binary tree into a rectangle

Does anyone know if there is a known algorithm/fractal to draw a very large balanced binary tree nicely, with minimal empty space when you put in on a piece of paper? Obviously, simplicity is also ...
solasky's user avatar
  • 210
1 vote
1 answer
73 views

deriving/explaining the Distance Estimation Method (by gradient) for rendering Julia & Mandelbrot sets

This question is about the derivation / explanation of the Distance Estimation Method (DEM) for rendering Julia and Mandelbrot fractals. I have not succeeded in finding an explanation online so I ...
Penelope's user avatar
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3 votes
1 answer
74 views

Connecting two points inside the Koch snowflake without getting too close to the boundary

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. We say $\Omega$ is a uniform domain with constant $c \geq 1$ if for any $x,y \in \Omega$ there is a rectifiable curve $\gamma : [0, l_\gamma] \to ...
Tobi's user avatar
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1 vote
0 answers
39 views

Fractals by Fractional Integration

Since the nth order integral finds the volume enclosed by an n dimensional function, does that imply that fractional order integrals can be used to find the volume enclosed by rational (or irrational) ...
Anirudh Yamunan Govindarajan's user avatar
0 votes
0 answers
70 views

Properties of a 3D Julia set from squaring a 3D number?

Consider a commutative 3D algebra $T$ where the nonreal units $x,y$ satisfy $$x^2 = A_1 + A_3 y $$ $$xy = 1 + B_2 x + B_3 y$$ $$y^2 = C_1 + C_3 y$$ where all the parameters $A_1,A_3,B_2,..$ are real. ...
mick's user avatar
  • 16.4k
-1 votes
2 answers
95 views

Can a fractal have an infinite area if it's bounded by a box? [closed]

Can a fractal/or any 2d shape have an infinite area if it's bounded by a box with finite area? And oppositely, if it can be bounded by a circle/line, how can it's "arc length"/perimeter be ...
PHV's user avatar
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3 votes
0 answers
102 views

Polar Coordinate Calculation of Hypercomplex Fractals

I've been doing some reading on how hypercomplex fractals are calculated using cartesian to n-spherical coordinate conversion. I've specifically been looking at these two sources: https://archive....
squirem's user avatar
  • 81
2 votes
0 answers
83 views

Number of Koch snowflakes when tiling Koch snowflakes next to each other

Imagine starting out with a Koch snowflake. You can stack 6 smaller snowflakes with 1/3 the area around the larger snowflake. After that, you can stack 18 even smaller snowflakes around those. This ...
jv1's user avatar
  • 21
1 vote
2 answers
64 views

Why does this distance estimator method render The Mandelbrot set incorrectly (non-divergent regions as divergent)?

I am using the following algorithm to render the Mandelbrot set and the exterior: • for each test point, calculate $c$ • initialise $z_{0}=(0+0i)$ • also initialise the gradient $dz_{0}=(0+0i)$ • ...
Penelope's user avatar
  • 3,325
15 votes
3 answers
2k views

Is there any sequence of closed shapes whose limit tends to the unit circle while the limit of the perimeter goes to infinity?

I know there is a simple method that generates a sequence of closed shapes whose limit is the unit circle but the limit of the perimeter is not $2\pi$ , but in all of the cases that I know , the ...
Hesam's user avatar
  • 161
0 votes
0 answers
40 views

Multiply connected Fatou component of an entire function.

This question may be trivial but still I want to know the answer. Question: Is there any necessary condition (except boundedness) for the existence of a multiply connected Fatou component of a ...
Factorial_zero's user avatar
0 votes
1 answer
76 views

Why more iterations benefit deeper Mandelbrot zooms over shallow zooms?

When rendering the Mandelbrot set fractal, we set a maximum number of iterations to test each point. If the escape criteria are met within the maximum iterations, we can stop further iterations ...
Penelope's user avatar
  • 3,325
3 votes
1 answer
144 views

The number of triangles in the fractal, which is formed from a square by the additional construction of isosceles right triangles

There is an algorithm for constructing a fractal: Take a square with a side of size 1; An isosceles right triangle is completed on each side; GOTO step 2. There is an example of that shape: How can ...
Alexander D's user avatar
2 votes
0 answers
67 views

Example of a buried Julia component of a transcendental meromorphic function.

We know examples of buried Julia components (Definition: A Julia component is called buried if it is not contained in the boundary of any Fatou component) for rational functions. In 1998, McMullen ...
Factorial_zero's user avatar
0 votes
0 answers
53 views

Points that are in a Julia set, but not repelling periodic points / connection between binary expansions and trajectories

I'm recreationally learning about fractals, specifically about functions of the "Multibrot" type $f(z) = z^d + c$. Just recently I became aware of the fact that repelling periodic points are ...
apirogov's user avatar
  • 203
2 votes
0 answers
89 views

Why do fractal-like patterns appear in this sequence?

I came across this sequence called Digital River, where the next number in the sequence is defined as the sum of the digits of the previous number, plus, the previous number itself. It caught my ...
Kristada673's user avatar
0 votes
0 answers
30 views

Are co-ordinate systems within a fractal inhertly compressible due to them having non integer dimensions?

This is a thought that I came up with while I was building a giant fractal in minecraft. If you have a space of some sort you can also create a coordinate system for it. At first I defined that as ...
Q the Platypus's user avatar
0 votes
0 answers
72 views

Is there a Julia fractal that contains uncountable many copies of itself?

We know that the Mandelbrot fractal contains a countable number of copies of itself. See : Does the Mandelbrot fractal contain countably or uncountably many copies of itself? Where that is explained. ...
mick's user avatar
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1 vote
1 answer
76 views

"Necks" of the Mandelbrot set

I'm looking at some "neck" points of the Mandelbrot set where there is just one point connecting a part of the set to another, such as the point (-0.75,0). I'm currently interested in ...
Suzumisc's user avatar

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