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

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1answer
39 views

Discrete systems with complicated basin boundaries?

I am trying to come up with the strategy to write my Master's thesis in mathematics. At the moment it is as follows: Finding a (preferably) discrete dynamical system that possesses at least 3 ...
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0answers
57 views

Area of 2D fractal?

Some fractals have a whole fractal dimension, can their measure be calculated? For example if you start with a tetrahedron of a given size and recursively remove the central octahedron leaving 4 ...
3
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112 views

What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
3
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1answer
53 views

Centroid of Mandelbrot Set

How to find the geometric centroid of Mandelbrot Set?
1
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1answer
49 views

Asymptotic rendering time for koch snowflakes

I posted a similar question on stack-overflow, but this may be a more proper forum since it is more math-related than programming related: I'm currently working through the online course material for ...
1
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1answer
34 views

Hausdorff dimension calculation related to Jarnik's theorem

Let $$F=\{x \in R:||qx||\le2q^{1-\alpha}\log q \text{ for infinitely many } q \in \mathbb{R}\}$$ Show for $\alpha>2$, $\dim_H F\le 2/\alpha$. Jarnik's theorem (By Falconer) says: Suppose ...
-1
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1answer
63 views

fractal curve and fractal set

Would it be correct to say that all fractal curves are fractal sets, but not all fractal sets are fractal curves? If that is correct, what would be an example of a fractal set that is not a fractal ...
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2answers
92 views

Revolution of fractal

How to find the volume and surface area of a shape which made from revolution of Koch Snowflake? (I think the surface area will be an infinity, because length of the Koch snowflake is infinity.) And ...
2
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1answer
53 views

Fractal signal analysis

What kind of results can be proven about continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which present some sort of fractal behaviour / self-similarity? Do you have some textbook ...
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1answer
56 views

Does this simple pattern show that all integers are part of a fractal set?

The simple pattern Count up through nine, then on ten you wrap to the next line and keep going until you've counted nine more integers - then wrap to the next line. Repeat ad infinitum. ...
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117 views

Properties of King's Dream fractal

My question is focused on the King's Dream fractal, which can be defined as follows (nice pictures can be found here) : $$ \Omega = \{f^n(0.1,0.1) \;\vert\; n \in \mathbb N \} \quad ...
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84 views

Is it known whether the boundary of the Mandelbrot set is not continuous?

I might be missing something obvious here, but my understanding is that nobody currently knows whether the boundary of the Mandelbrot set is a Jordan curve because otherwise we would know that the ...
0
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2answers
41 views

Adding Examples to Math Paper

I'm writing a paper on the Mandelbrot set and want to add some examples of iteration to it to show values that are members of the set and to show values that are not members of the set. What's the ...
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1answer
27 views

Hausdorff dimension of a countable set

I don't understand why the Hausdorff dimension of a countable set in $\mathbb{R}^n$ is $0$. Can someone please give me a hint? Thank you!
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0answers
26 views

Showing that points are in the Mandelbrot set

I am given ( a simplistic definition I think ) of Mandelbrot set: M- set of complex numbers $c \in \mathbb{C}$ s.t. the sequence $(z_n)$ is bounded where $z_0=0 , z_{n+1}=z_n^2+c $ Need to show ...
2
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3answers
71 views

Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
0
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1answer
27 views

Cantor set and triadic expansions

I'm trying to prove that the Cantor set is equal to a certain set of 'escape points' for a mathematical feedback system. In this proof I'm going to need the fact that every element of the Cantor set ...
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2answers
30 views

There exist fractal with similarity dimension between 0 an 1?

How to prove that there exist a fractal with similarity dimension D = x, where x is between 0 and 1?
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1answer
21 views

Finding the similarity dimension of a variation of the Cantor Set.

If we take the Cantor set and instead of removing the interval $[1/3, 2/3]$, we remove the open interval $[x,1-x]$, with $0<x<1/2$, will the similarity dimension change? What I think is that we ...
0
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1answer
82 views

Why is the Mandelbrot Set to the power of 2?

I've been looking at the Mandelbrot set and playing around with the powers on it. In the sense that I know the equation is $z^2 +c$, and I've been changing up the $2$. As it happens gradual increases ...
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2answers
17 views

Set characterization

I need to understand what are the elements of the set $\left[0,1\right]$ whose non-terminating decimal expansions contain only the digits 3, 5 and 7. I suppose it is an auto-similar set, up to an ...
1
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1answer
24 views

Interpreting a nonstandard definition of a tree

Definition: A tree is a triple $(T,\sigma,\pi)$ where $T$ is a set and $\sigma$ is a so-called successor function from $T$ to the set $T^*$ of all nonempty subsets of $T$, together with a surjective ...
2
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0answers
37 views

Showing a point is a limit point of a set

For sake a clarity, when I say a "limit point", the definition I am using is: $x$ is a limit point of a set $A$ if there exists a sequence $\{x_k\}$ such that $x_k \neq x$ and $x_k \rightarrow x$ ...
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2answers
83 views

Fractals with Moduli in Pascal's Triangle

I'm working through a problem for my graduate math class and am hitting a wall. Here's the problem: For the first 10 lines of Pascal's Triangle, replace the odd numbers by black squares and the even ...
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1answer
63 views

Box dimension of $\{\frac{ 1}{5^n} : n \in \mathbb{N}\}$

I am working through a first course in Fractal Geometry, and have encountered a problem that has asked me to calculate the box-counting dimension of $F=\{ \frac{1}{5^n} : n \in \mathbb{N}\}$ However, ...
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3answers
110 views

Union of two Cantor sets is also a Cantor set

To prove that a set is Cantor, we have to prove that it's closed and bounded (compact), contains no intervals of positive length, and is perfect. The union of two Cantor sets would also be compact ...
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1answer
37 views

Cantor set and ternary expansions

I'm trying to show that if $x \in [0,1]$ has a ternary expansion consisting only of $0$'s and $2$'s, then $x$ is in the Cantor ternary set. The proofs I've seen typically rely on induction. Is it ...
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2answers
45 views

Are there methods to recursively calculate the decimal expansion of real numbers?

Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence, $$(1) \quad C_0=0.1, \ C_1=0.101, \ ...
2
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1answer
73 views

Definition of Minkowski dimension

I'm trying to understand the definition of Minkowski dimension given by Wikipedia here. In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting ...
3
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1answer
67 views

Is there a rigorous mathematical definition of the Koch curve?

Is there a rigorous mathematical definition of the Koch curve? Wikipedia says that mathematics is not given a rigorous formal definition of a fractal in general. And also I have not found a strict ...
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0answers
48 views

Non-empty interior of a Mandelbox

I have a question which has been interesting me for some time, namely when a Mandelbox has a non-empty interior. A definition of the Mandelbox may be found here. Essentially, it is defined by a ...
8
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2answers
199 views

How to avoid overlap in circle fractals?

I had asked this on reddit and someone suggested that I try here: Assuming that the pattern in the image below continues infinitely, how much would each generation of circles have to decrease to ...
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0answers
26 views

On a curve every point of which is a point of ramification

The title of my post is the same as the title of a known article written by Sierpinski where he introduced its famous triangle. In the book Handbook of the history of general topology by Lowen said ...
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1answer
52 views

Proving that we have $A \subset K$ where $K$ is self-similar

Let $f_1, \ldots$, $f_N:X\to X$ be contractions in the complete metric space $X$, and $K$ the self-similar set with respect to the $f_i$. If $A\subset X$ is compact and $$A\subset ...
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0answers
50 views

Relations between the 2-disc operad and fractals?

As you can see, as of late I opened a thread on n-disc operads: Clarification regarding little n-discs operads The thing is, those drawings there could somehow be construed in the real world as ...
2
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1answer
43 views

Are fractal image generators one-way functions?

Is it hard to calculate the coordinates and zoom factor that was used to generate a fractal image of, say, the Mandelbrot set? If you know the rest of the parameters, like how many iterations where ...
3
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1answer
66 views

Finding parameter paths for beautiful fractal animations

So I just got renewed interest in fractals and especially animations with fractals. To make an image or a frame, we usually need to evaluate a fractal for a subset of it's parameters. However for many ...
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0answers
70 views

Basins of attraction for Newton-Raphson fractal colouring

What's the general strategy/approach for defining the basins of attraction within the Newton-Raphson(NR) function in the complex plane? I would like to understand where these are to colour-in a NR ...
2
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2answers
358 views

A question about a fractal like iteratively defined function

I am trying to figure out what the following function $f:\Bbb{R}^3-\{\mathbf{0}\}\to\Bbb{R}$ defined below (in pseudocode) does: function $f(\mathbf{v}\in \Bbb{R}^3-\{\mathbf{0}\})$ { ...
4
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1answer
69 views

Hausdorff metric and $\varepsilon$-thickenings

Let $h(A,B)$ be the Hausdorff metric defined by: $$ h(A,B)=\inf\{\varepsilon >0 \; | \; A \subseteq B_\varepsilon, B \subseteq A_\varepsilon \}, $$ where $A_\varepsilon$ is the ...
2
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2answers
65 views

How does one turn an object into fractal? [closed]

I've seen a lot of digital art made using fractals e.g landscapes, flowers, trees and the like, but I was wondering how does one turn an object into a fractal? Like say....if I wanted to make a ...
5
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1answer
156 views

Choosing an appropriate sequence of $\{1,2,3\}$

Let $f_1,f_2,f_3$ be the contracting maps $f_i:x\mapsto \frac{1}{2}(x+p_i)$ from $\mathbb{R^2}$ to itself and $p_i\in \mathbb{R}^2$. Denoted by $S$ the attractor Sierpinki gastek of the iterated ...
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2answers
330 views

Could this odd insight help explain part of the difficulty in proving the Collatz Conjecture?

Background: Here's a crash course on the Collatz Conjecture. Basically, you take a number and if it is even you divide it by two. If a number is odd, you multiply it by three and then add one. You ...
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1answer
117 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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1answer
70 views

The mandelbrot fractal and it's relations to algebras and groups

I have been fooling around with Mandelbrot fractal to and fro for many years. One of the latest years I learned some general algebra with some of the most basic groups, like cyclic groups, dihedral ...
2
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1answer
55 views

Are Square and equilateral triangle the only convex regular ngons that can be composed to smaller versions of themselves?

While I was trying to make an analogous question to Select $n^2 + 1$ points in the unit square. Show that at least two points are no more than a distance $\frac{\sqrt{2}}{n}$ apart, using equilateral ...
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1answer
56 views

The unit ball is not auto similar.

I want read a prove that the unit ball $B$ is not auto similar. I mean that there is not similarities $f_1,...,f_n$ with contracting constants <1, such that $$B=\bigcup_{i=1}^n f_i[B] $$ Anyone ...
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1answer
14 views

What can we do on $S$ in order that $H(S)$ be compact?

Let be $S$ a metric space. We define the hyperspace $H(S)$ as the metric spaces consisting of every no empty compact subset of $S$ and the Hausdorff metric. I want that $H(S)$ be compact imposing ...
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0answers
116 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
3
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1answer
144 views

Can a fractal be a manifold?

Here it is said that it is not possible: Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower? But I am confused about this. What about the invariant ...