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Results tagged with fractals
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user 1056
Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
16
votes
Why are the Julia sets so simple? (quadratic family)
To extend my comment and emphasize the self-similarity of Julia sets and the Dragon curve, here is an interpolation between the two.
Each frame is generated by two complex functions,
f1[z_, t_] := …
- 15.8k
4
votes
Proof of im/possibility of constructing any fractal by iterated function systems?
Then, there are the random midpoint displacement fractals
which are self-similar in a much weaker sense than above, since these fractals involve random choices. … Not all fractals are given by iterating a map (I prefer to call these types of fractals discrete); some fractals are constructed using continuous methods, such as the Lorenz attractor. …
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4
votes
function that is the average of affine transformations of itself
We can define a sequence of functions converging to the one above:
$$f_1(x) = \begin{cases} -1 \text{ if } x<0 \\ 0 \text{ if } x=0 \\ 1 \text{ if } x>0 \end{cases}$$
and then $f_k(x) = \frac12[f_{k- …
- 15.8k
3
votes
Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$
This is just a partial answer, but it looks like your figures are not "pure" fractals, but a union of different fractals (with different fractal dimensions). … If I am not mistaken, these are called multi-fractals.
It looks like part of your figure is the Dragon curve fractal. …
- 15.8k
3
votes
sequences with a fractal dimension
Taking a truncated integer sequence is essentially the same as defining the "heights" of the line endpoints of a 2-dimensional curve, similar to http://www.gameprogrammer.com/fractal.html
my guess is …
- 15.8k
3
votes
L-systems and Sierpinski Triangle
Once you get the grips around constructing the Hilbert curve: https://en.wikipedia.org/wiki/Hilbert_curve
then this is not really all that hard to construct.
Any fractal that consists of $n$ identic …
- 15.8k
3
votes
Fractal questions: Weierstraß-Mandelbrot
You can also create a continuous, non-differentiable function by restricting the height map produced by the midpoint displacement algorithm to a line:
http://en.wikipedia.org/wiki/Diamond-square_algor …
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2
votes
Accepted
What is "graph-directed iterated function"?
From the wikipedia article, you have the substition rules $1\to12$, $2\to13$, $3\to1$.
We can write this in matrix form
$$
\begin{pmatrix}
1 & 1 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix}
$$
where t …
- 15.8k
2
votes
Is this a quasi-crystal and/or a fractal?
You will need to provide some more information, but your pattern has a resemblance with the Penrose tiling, that is indeed a quasi-crystal.
A quasi-crystal is a non-periodic pattern, but any finite p …
- 15.8k
2
votes
The graph of the fractal sets
Sure, here is the IFS, but I am not sure you gave the correct bounds for it, as it seem to only occupy $0\leq x \leq 10$ but be much larger in the $y-$direction. I plotted it for $0\leq x \leq 10$, $0 …
- 15.8k
1
vote
Are there any exact results for Hausdorff Measure?
(This is mentioned on wikipedias page with fractals listed after Hausdorff dimension). …
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1
vote
How to plot this fractal
It looks like a tiling of some type of julia fractals, then post-distorted with a 'spiral' map. Hence, it seems to fall in the "flame fractal" category. … Side note: I have developed some software to make this type of fractals, so I am rather familiar with the process. …
- 15.8k
1
vote
Accepted
A one dimensional fractal like set with the same line width within a bounded area?
It should be a cantor-set. Also, by construction, the length is preserved in each step, so the Haussdorff dimension should be 1.
To prove that it is in $[0,1]$, depends on how you construct your sets …
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-1
votes
Visualizing the l-adic fractal in the partition function p(n)
The Collatz function may however be analytically extended to an entire function, and iterating this function as when constructing a Julia set, DO create pictures one would expect.
99% of all fractals …
- 15.8k
-1
votes
Reference for the iterated function system of the Koch snowflake
id=HgBFW9hsGZwC&lpg=PP1&dq=fractals%20everywhere&pg=PP1#v=onepage&q=koch&f=false
There seems to be a reference in that book, but googlebooks does not show that page... … EDIT: The link is to the book
"Chaos and fractals: new frontiers of science"
by AvHeinz-Otto Peitgen,Hartmut Jürgens,Dietmar Saupe …
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