Questions on fractals, irregular-looking mathematical objects that display the property of self-similarity.

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Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
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1answer
355 views

Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
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1answer
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a conundrum regarding integrated Brownian motion and fractals

Let $X(t)$ be a Brownian motion. I know that the integral \begin{equation} Y(t) = \int_0^t d\tau ~ X(\tau) \end{equation} is well-defined, since Brownian motion $X(\tau)$ is a.s. continuous. Thinking ...
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1answer
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Hausdorff dimension of a Cantor Set: attaining a lower bound

I'm considering the problem of calculating the Hausdorff dimension of a Cantor set, according to the following lemmas: Lemma 1 Let $C: [0, 1] \rightarrow [0, 1]$ be a Cantor staircase function. Then ...
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109 views

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve $f_h:[0,1]\...
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Fractal identification

I was trying different algorithms out, and after a while, I found this fractal: The generation has similarities to Koch's curve, but instead of putting triangles on triangles, I put circles on top ...
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3answers
269 views

What is the moment of inertia of a Gosper island?

We know that regular hexagons can tile the plane but not in a self-similar fashion. However we can construct a fractal known as a Gosper island, that has the same area as the hexagon but has the ...
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Hausdorff metric between attractor sets of iterated functions systems

Let the set $X:=\{(x,y)\in \mathbb{R}^{2}:0\leq x\leq 1,0\leq y\leq 1-x\}$ and for each positive integer $i$ and $j\in\{1,\ldots,2^{i}-1\}$ define the contracttion $f_{ij}:X\longrightarrow X$ by $f_{...
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67 views

Can a “Julia set” fractal be described in a “closed form”?

What I mean by that is, consider, say, the "Koch snowflake" curve. It is formed by repeatedly applying a substitution to the lines of a triangle to get the final curve in the limit. What I am after ...
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Integral over Julia Set (Is my math correct?)

So I was answering this question about whether or not the Julia Set was self-similar in a known way. Of course it is, and that got me thinking. Even though the self similarity is nonlinear, what if ...
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Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
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Interpretation of $ \tau $ in the Stephen Astels paper '' Cantor set and numbers with restricted partial quotients?

I am trying to read Stephen Astels paper 'Cantor sets and numbers with restricted partial quotients'. Visit http://www.ams.org/journals/tran/2000-352-01/S0002-9947-99-02272-2 In this he directly ...
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2answers
139 views

formal definition of “fractal” or standardized categories?

fractals are many decades old and come up in a wide variety of contexts and can be generated in so many different ways. however, a formal definition of fractal seems really slippery/ difficult. are ...
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2answers
319 views

Area fractal pentagrams II

A simple fractal. How to find the area of it? (only the arms of the star) Working with pentagrams is quite complicated, I can not solve this.
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2answers
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Hausdorff dimension via ergodic theory

This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of ...
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0answers
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What algorithm can be used to reconstruct a self similar time series from a portion of it?

I am working on a process which produces time series similar to the one shown in the graph below: I calculated Fractal dimension $D$ and Generalised Hurst Exponent $H$, to confirm that equality $D=...
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192 views

Pythagoras tree bounding size

The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed ...
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2answers
138 views

Proof of x-intersection of the Mandelbrot Set?

I'm trying to prove that the Mandelbrot set intersects the X-axis on the interval [-2,.25]. I understand and have proven that the Mandelbrot set lies in a radius of 2. Mostly, I'm wondering how to ...
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what part of a m-set fractal showing spiral behaviour?

What part of a fractal Mandelbrot Set showing spiral behaviour like this one: what is it's direct equation?
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On the construction os a space-filling curve by interpolation fractals functions

I'am trying to construct a space-filling curve in the unit square $I^{2}:=[0,1]\times [01,]$ following the results shown in the Barnsley's book "Fractals Everywhere". Thus, let $\Delta:=\big\{ (0,0),(...
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How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
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What are the Geometric Properties of Non Integer Vector Spaces?

I found a paper from Princeton called "Axiomatic Basis for Spaces with Non Integer Dimension" that presents five axioms and then starts to create a framework similar to what I'd think the subject ...
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1answer
48 views

Geometric generation principle form constructing the Hilbert Curve

I have some questions on the generation of the Hilbert's space-filling curve. Any help to clarify doubts a-e would be very appreciated. The Hilbert's space-filling curve is a function $f_h:[0,1]\...
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1answer
74 views

Continuous or Differentiable but Nowhere Lipschitz Continuous Function

What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval? What is a real valued function that is differentiable on a close interval but not ...
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4answers
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Does this Fractal Have a Name?

I was curious whether this fractal(?) is named/famous, or is it just another fractal? I was playing with the idea of randomness with constraints and the fractal was generated as follows: Draw a ...
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1answer
115 views

Is a hypersphere of non-integer dimension a fractal?

Thanks to the gamma function the formula for the surface of a unit http://mathworld.wolfram.com/Hypersphere.html $$ S(n) = \frac{2 \pi^{n/2}}{\Gamma(n/2)} $$ allows to calculate the surface of a ...
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3answers
236 views

Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
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1answer
88 views

What is the shortest path to a “little Mandelbrot” from $i$?

As you all already know, the Mandelbrot set has little "copies" of itself strewn throughout the boundary region (some of them distorted somewhat), and these are all connected. The point $i$ (or $x = ...
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Is there a there a non intersecting mapping to unit square.

Is there a way to go from the fat cantor set to a half unit square in a non intersecting way using Hilberts curve? How would I go about constructing a non intersecting space filling curve of non zero ...
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Sierpinski gasket is the closure of a set of its vertices

Why Sierpinski gasket is the closure of a set of its vertices? Let $V_0:=\{0=p_0, p_1, p_2\}$ be vertices of an equilateral triangle, and let $\hat {\mathcal H} := \bigcup_{i=0,1,2}(- + p_i)/2$ (...
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1answer
40 views

IFS which construct this fractal and have affine transformation only

[Image updated] Is there an IFS which construct this fractal and have affine transformation only? (I think there must be a restriction, which is not an affine transformation. Can it be proved?)
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When the self-similar dimension and the Hausdorff dimension are different?

By the en.wikipedia, for the self-similar sets in a metric space, the self-similar dimension and the Hausdorff dimension are often the same, but not always. Is there a known sufficient-necessary ...
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2answers
58 views

$L$-Systems: Order of Substitution

I am working the a subject guide on involving $L$-Systems and have the alphabet $A = \{a, b, c\}$. The initiator is the string $a$ and the rules of substitution $a \to ba$, $b \to ccb$, $c \to a$. ...
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Continuous path inside the Mandelbrot set connecting i to zero?

This relates to another challenge Question about drawing Mandelbrot filaments. It is possible to compute a formula for a continuous path inside the Mandelbrot Set connecting $c=i$ to $c=0$? Obviously,...
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1answer
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Exact value of Hausdorff measure of middle-third Cantor set

Is there any result about the exact value of $\log_3 2$-dimensional Hausdorff measure of the middle-third Cantor set? And is there any fractal (in $\mathbb R^n$) which is not contained in a $p$-...
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What´s the dimension of a Sierpinski fractal?

I know the dimension of a Koch snowflake (log4/log3), but what numbers do I have to put in to obtain the dimension of a Sierpinski fractal?
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1answer
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What is the condensation set of a fractal?

Is there a definition of the condensation set of a fractal that is both clear and rigorous? I've been searching around to get a sense of what exactly the condensation set of a fractal is - I've ...
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Can the fractal dimension of a surface be less than 2?

I have two surfaces represented as raster images with heights as grayscale values. One is natural landscape elevations; the other is just distance from a line. I have computed Minkowsky D = 2 - H ...
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How was one derivied from the other?

In the geological paper entitled The power–law relationship between landslide occurrence and rainfall level by C. Li et al, a power-law cumulative probability distribution is derived. However, I don't ...
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2answers
244 views

How to correctly calculate the fractal dimension of a finite set of points?

The box-counting dimension is defined by: $\lim\limits_{\epsilon \to 0} \dfrac{N(\epsilon)}{1/ \epsilon}$ What works well if you are solving algebraically or if you can recursively generate more ...
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1answer
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How to determine constant $C$ in $p(x) = Cx^{-D}$?

Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L_{cf}(> X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given ...
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54 views

Scaling factor closest to 1 in an infinite sequential rectangle packing

The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $ k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest ...
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1answer
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Is this a valid definition of “self-similar fractal”?

I have always been fascinated by self-similarity, particularly in fractals. I was always wanted to find a simple definition of a self-similar fractal. Of course, saying "is self-similar, and is a ...
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2answers
54 views

hyperbolic spaces and fractals

Is there a relation between hyperbolic spaces and fractals? In group theory, if we take the Cayley graph of a free group on two generators, we get a fractal quaternary tree, which I'd like to think as ...
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What exactly are fractals

I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of ...
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1answer
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Why c>1/4 is not in Mandelbrot set

As title: $f_c(x)=x^2+c$ I got to the step: $f_c(x)>x$ (for all x) But what's next? How to show that after k iterations, $f^k_c \to \infty$ as $k \to \infty$ Thanks,
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Fractal fundamentals

I am a programmer by trade, and am very interested in fractals. To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and ...
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1answer
84 views

Is every basin of attraction completely invariant?

I can't seem to find a definitive answer in the literature. I believe the answer is yes, but my focus has been on the rational maps on the Riemann sphere. At the very least I'm confident that if the ...
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61 views

Is there a general metod to construct a fractal?

I would like to construct a fractal (traditional, self-affine, and fat fractal) with a given embedding and fractal dimension, but I don't know how to do it programmatically. The shape of the fractal ...
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How is this fractal produced?

It is stated here: Iterating the above optimized map $$f(z)=\frac{1}{4}(1 + 4z - (1 + 2z)\cos(\pi z))$$in the complex plane produces the Collatz fractal. The point of view of iteration on ...