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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.

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What sort of function is this? (Logistic map?)

What sort of function is this? This looks like a bifurcation diagram of a logistic map to me, but there seem to be extra parameters that are changed along with the different colours, what are these?
Sapiens
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Topological and Hausdorff dimension of Koch tetrahedron fractal

I am trying to find the topological and Hausdorff dimensions of the fractal on the attached picture (with proofs). It is obtained by similar approach as Koch snowflake, but applied to a tetrahedron instead of a triangle. Could you help me how to…
Marko Ruman
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Does a particle moving in a fractal have a non integer number of degrees of freedom?

I was watching this video by 3blue1brown and he discusses the idea of fractal dimensions. He notes that fractals can have non-integer number of dimensions. For example, the Sierpinski triangle has a dimension of 1.5849. +Itai Efrat asked the…
Byte11
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Is the "average slope" method an appropriate substitute for the boxcount method in computing the fractal dimension of a curve in the plane?

Financial practitioners commonly use the average slope of a price chart over a given time interval to compute the fractal dimension of a stock price series. they approximate this average slope by simply taking the difference between the highest and…
Constantin
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Construct a set $A$ for which the lower box dimension of $A$ is less than the upper box dimension of $A$

I'm revising for an exam and would appreciate any answers to this question. Thanks
Carpodacus
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Fractal dimension of a spring whose wire is actually a spring, etc.

Inspired by this video: https://www.youtube.com/watch?v=gB9n2gHsHN4 At a high level, he talks about how shapes can have different dimensionality depending on the scale they're observed from, an example being a spring. Far away, it looks like a line.…
allidoiswin
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Minkowski–Bouligand box count dimension

What are the problems with using the box counting method to determine a fractal dimension? I am currently looking at the fractal dimension of a Barnsley fern. By differing the number of points used to create the fern, different answers are obtained…
bgrantham
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Moran's open set condition.

The iterated function system $(f_1,f_2,\cdots, f_n)$ satisfies Moran's open set condition iff there exists a nonempty open set $U$ for which we hacve $f_i(U)\cap f_j(U)=\emptyset$ for $i\neq j$ and $U\supset f_i(U)$. Such an open set $U$ will be…
EQJ
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How would I calculate points on the Peano curve?

The Peano curve is often given as an example of a space filling curve which maps the unit line to the unit square. So, it is a function of the form $[0,1] \rightarrow [0,1]^2$? In which case can I evaluate it for given numbers like 0.2, 0.5 and 0.7?…
Molly Stewart-Gallus
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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_{ij}:=\Big(\frac{x-j-1}{2^{i}},…
user123043
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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?)
Kanu Kim
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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 tetrahedrons of half the side length you will end up…
k-l
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How many vertices are in the Koch Snowflake?

EDIT: The question was put on hold because I didn't specify what I meant by vertex. In a comment below by Mark McClure, by "vertex" I mean one of the vertices of the standard, polygonal approximations to the Koch curve. I have been trying to create…
user242594
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Is the boundary of the Mandelbrot set jagged or smooth?

As the title states, I am wondering if the boundary of the Mandelbrot set is jagged or smooth. If it is jagged, is there some algorithm to find the vertices of any one of them? Are there an infinite number of vertices? I am aware that we do not know…
user242594
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how do i prove that a collection of contractions does not satisfy the open set condition?

I am studying a fractal that is defined by 4 similarities, similar to the Von Koch curve, and I am trying to verify that it does not satisfy the open set condition. The fractal is heavily self-intersecting so it seems obvious to me that it does not…
user3450230
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