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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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Is the Mandelbrot set sufficiently self-similar to non-rigorously visually recreate procedurally from warped, nested structures?

While not strictly self similar, the Mandelbrot set shows significant similarity in its pixelated visual samplings. Are the overall shapes approximating each region well understood enough, not in a mathematically rigorous way, but visual way, to…
Lucent
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Is there an analogue of the Gosper flowsnake on the Cartesian lattice?

The Gosper curve has a sort of dual flowsnake which is a space-filling curve on a hex lattice. The unit motif traces a 7 hex 'super-hexagon' comprised of the origin and its six neighbors in the three +/- axis directions, which get connected into a…
patricksurry
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What's the topological dimension of Sierpiński's Triangle?

I am aware that Sierpiński's Triangle is a fractal, with Hausdorff dimension $1.5850$. Therefore my intuition leads me to believe it's topological dimension is 1 (as the topological dimension must be less than the Hausdorff dimension). However,…
M.Hatton
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Calculating the fractal dimension of a Romanesco Broccoli

It has been quite a while since I've had some introductions to fractal geometry, but only on special cases, e.g., the Sierpinski triangle and other self similar fractals. Now I have challenged myself to calculate the fractal dimension of the…
Fractal Admirer
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Name of this "cut 'n slide" fractal?

Can you identify this fractal--if in fact is has a name--based either upon its look or on the method of its generation? It's created in this short video. It looks similar to a dragon fractal, but I don't think they are the same. Help, please?
Justin Lanier
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Can different fractals have the same fractal dimensoins?

I first noticed that fractals can be defined by real numbers from this youtube video: https://youtu.be/gB9n2gHsHN4 My question is that: Are these fractal dimensions unique to the fractals, Can two different fractals have the same fractal…
Rounak Sarkar
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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?
Spiral
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Is a fractal per definition mise en abyme?

Mise en abyme is per definition a formal technique of placing a copy of an image within itself, often in a way that suggests an infinitely recurring sequence. The image below represents the technique: And we know from mathematics that a fractal is…
Rakozay
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Twin dragon behaviour under different scaling factors

I observed a weird behaviour plotting the twin dragon for different scaling factors $\alpha$. I used this mathematica code to generate them M = {{1/2, -1/2}, {1/2, 1/2}}; twindragon = { {M, {-1, 0}}, {M, {1, 0}} }; This corresponds…
user1868607
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Show that Hausdorff measure is semifinite

I am currently reading a book about fractals and the author states the result that Hausdorff measure is semifinite. Can someone tell me how to prove or provide a hint for me?
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What type of pattern is generated by mod(x * y, w)? Is this an interference pattern, a simple fractal or some other class of pattern?

I've stumbled across this pattern a number of times when working with shaders. The visuals created are very interesting - similar to Moiré patterns, but more intricate. I don't fully understand how it works and I'd like to know more about it, but I…
goulding
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Self Similarity in fractals

I'm looking into strict self similarity in fractals as part of a project for first year an from my understanding of the topic the Julia set is strictly self similar. However I've not been able to find anywhere this is firmly stated and wanted to…
Katie
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Is there a name for this figure with alternating colors in the complementary set of Sierpinski's triangle?

Start constructing Sierpinski triangle. On odd step - remove paint center white. On even step paint center blue. This figure has areas with arbitrary small patches of color, yet, area of both white and blue color is greater than zero. Lebesgue…
Stepan
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Fractals that have never been discovered before

I have a maths competition and I've cracked till round 2. This round mainly states that I require to make a fractal that's not on the internet (yet) and explain it. To be honest, fractals is an alien area for me and would love some help In…
The Dead Legend
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Fractal dimension of set in $\mathbb{R}^2$

I have a set of points in $\mathbb{R}^2$, of the form: $\Bigg\{\left(\frac{a}{\ell^2},\frac{b}{\ell^3}\right): \ell \in \mathbb{N}^+\Bigg\}$ where $a$ and $b$ are some real positive numbers. I am interested to know the box dimension of this set. Is…
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