Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [fractals]

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

The term fractal, derived from the Latin fractus meaning "broken" or "fractured," was coined by Benoît Mandelbrot in 1975 in order to describe mathematical objects (shapes, sets, processes, etc.) which possess irregular or rough structure at all scales. While there is little consensus on the precise definition of the term, fractals are typically characterized by self-similarity. The Cantor set, Sierpinski carpet, Koch Snowflake, and Mandlebrot set are examples of fractal sets.

1220 questions
1
vote
0 answers

Hausdorf dimension of fractal iterates

For fractals defined iteratively (via subdivision) like the Koch curve or Sierpinsky triangle, what is the Hausdorf dimension of the intermediate iterates? Specifically, for a fractal S defined as \Lim_n S_n, where each Sn is constructed by finite…
firdaus
  • 703
1
vote
1 answer

Can you help me find a fractal drawing program?

In a previous course on chaos, the professor had us experiment with a program. The program allowed you to draw a base image (with microsoft paint like tools), then it would iterate that image under various popular fractals. Note: The program was…
Chuck Franklin
  • 550
1
vote
3 answers

How does one determine the containing boundary of a fractal?

In the Mandelbrot set, the fractal is said to be contained in the circle of radius 2. $$ z_{n+1} = {z_{n}}^{2} + c $$ I did read about a proof that showed values of 'c' beyond this circle are not bounded and hence the set is contained…
Bharath
  • 205
1
vote
0 answers

multifractal scaling exponent tau(q) - concave up or down?

I have read some conflicting information from two reliable sources regarding the scaling exponent in multifractal systems - tau. On the Yale website devoted to fractals, they say "Tau is a decreasing function of q and is concave…
bibbly bobbly
  • 59
1
vote
2 answers

$L$-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 $a \to ba$, $b \to ccb$, $c \to a$. The study guide gives the first five generations…
Ray
  • 159
  • 4
1
vote
2 answers

A zero-dimension set and self-referencial equation

Let $K$ be a compact set in $\mathbb{R}^2$. Let $f_1,..., f_n$ be contracting similarities of $\mathbb{R}^2$ to itself. Suposse $K$ satisfies the self-referencial equation $$K=\bigcup_{i=1}^{n}f_i[K]$$ and $f_i[K]\cap f_j[K]=\emptyset$. Prove…
EQJ
  • 4,369
1
vote
1 answer

Structure of Mandelbrot bug antennae

I have analysed the structure of of Mandelbrot set and I have understood something, but I still have some questions, mainly about antennae and little bugs. Mandelbrot set consists of the main cardioid, heads, that are deformed copies of the whole…
BartekChom
  • 200
0
votes
1 answer

How to calculate line-length for fixed width koch fractal?

I am playing with fractals, and drawing them with Python turtle I am using this rules to create l-string for my koch fractal: begin: f f -> f+f--f+f In here, f means go forward by a fixed length, - means turn right for 60 degrees, + means turn…
yasar
  • 237
0
votes
0 answers

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 of the pixels in float32. Unfortunately, the…
Matige KunstIntelligentie
  • 1
0
votes
0 answers

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 important—it’s easy to draw a binary tree that…
solasky
  • 115
0
votes
0 answers

Why are there baby Mandelbrots in a non-Mandebrot fractal?

I'm playing around with a couple of different fractals, and a recurring theme I'm seeing is the presence of baby Mandelbrots (small versions of the Mandelbrot set) in nearly all of these. Here's one between -0.183346 + 0.36017685i and -0.18229 +…
Vivaan Singhvi
  • 11
0
votes
0 answers

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 is calculating fractal dimension of the coast of…
Chris
  • 189
0
votes
2 answers

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 this fractal is as follows. Start with a square…
Q the Platypus
  • 4,626
0
votes
1 answer

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 is a black point in the middle of the spiral,…
Yılmaz Alpaslan
  • 103
0
votes
0 answers

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
Mic
  • 1
1 2 3
…
11 12