# Tagged Questions

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### Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
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### Drawing a nested epicycloid

I would like to learn how to draw this kind of pictures (possibly with Mathematica, as it is the only language I would be comfortable to code such a thing in): There is something similar on the ...
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### Fractals - when the number of seed shapes that can fit into the scaled-up copy is non-integer.

I've heard people say (for eg. here) that the dimension of fractal patterns (particularly, in this question, Lindenmayer fractals) can be formulated as follows: $$D=\frac{\ln N}{\ln S}$$ Where $N$ ...
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### Is the two-dimensional Koch curve space-filling?

Say, we'd like to make a Koch curve with self-similarity dimension of two. A Koch curve with the following generator seems to be two-dimensional, since if we double its size by scaling we'll find ...
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### Reference - Fractal Geometry

I am looking for textbooks or lecture notes about Fractal Geometry that reach an advance level on the topic and aren't just introductory.
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### Intersections of fractal sets with connected sets

Let $\beta \geq \alpha > 0$. Let $A\subset\mathbb R^n$ be a measurable set with Hausdorff dimension $\alpha$. Must there exist a closed connected set $B$ with Hausdorff dimension $\leq \beta$ ...
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### How to convert a right angled triangle into a equilateral triangle?

I want to use the Apophysis program to make a right angled sierpinski triangle into an equilateral triangle. But how can i do so? i have tried the second picture one but that is not correct.
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### Geometrical objects whose volumes are fractional powers of their sizes

While studying properties of foams (imagine bubbly soap or microscopical grids/networks), I started wondering on the relationship between the volume occupied by the matter $V_s$ itself and the overall ...
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### Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
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### Cantor Set and Fractals

I have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, but ...
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### Area fractal pentagrams III

how can I find the area of these two fractal? I've been try to solve some geometry exercises here but this and some other are so much difficult.
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### Area fractal pentagrams II

A simple fractal. How to find the area of it? (only the arms of the star) Working with pentagrams is quite complicated, I can not solve this.
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### Area fractal pentagrams I

When I saw this image I was a little curious. How can I find the area of this fractal?
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### What's the analogue of Sierpinski triangle to disk?

What's the (closest) analogue of Sierpinski triangle to disk?
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I read somewhere that we can not measure the length of the Adriatic Coast because it has fractal structure. I want some concrete explanation for the fractal structure
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### Defining distance in fractal dimensions.

Is it possible define a distance measure in fractal dimensions? namely, what the generalization of $$D(x,y)=\left(\sum_i(x_i-y_i)^2\right)^{\frac{1}{2}}$$ in fractal dimensions?
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### Fractal walking: well defined case or not?

Please consider the following recursive diagram: diagram Each triangle is connected at the midpoint of a side to the corner of an inner triangle which is 1/4 times the size. The total line length of ...
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### Snow Flake Problem: Limit of perimeter & area at $\infty$

I am supposed to find the limits as $n\rightarrow\infty$ of the perimeter & area of a snow flake. $$N_n = \text{Number of sides} = 3\cdot 4^n$$ $$L_n = \text{length of side} = \frac{1}{3^n}$$ ...
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### understanding if a point is inside or outside a Koch fractal curve

I continue this post because i have a problem: understand if a point is inside or outside a Koch curve. I can find the third point of a equilateral triangle (and i need it for build a koch curve) but ...