# Tagged Questions

Questions on fractals, irregular-looking mathematical objects that display the property of self-similarity.

86 views

### Examples of smooth fractals

A classic example of a fractal curve is the Koch Snowflake. This is a topological manifold (as opposed to many other fractals which are not), but it also clearly not smooth. Question: Are there ...
21 views

### 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 ...
265 views

### Fractal dimension of the function $f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}$

Consider the function $$f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}\, .$$ This is a bizarre and fascinating function. A few properties of this function that SEEM to be true: ...
79 views

### Perfect circles in the Mandelbrot set?

It is known that the boundary of the period 2 hyperbolic component of the Mandelbrot set is a perfect circle of radius $\frac{1}{4}$ centered at $-1$. Moreover it is known that the boundaries of the ...
42 views

### Radius of inner circles given radius of outer circle and number of inner circles in circular fractal

I am trying to create a circular fractal in which each circle is composed by a given number $n$ of smaller circles. It would look something like this for $n = 8$: However, I don't know how to ...
41 views

### Sierpinski triangle formula: How to take into account for 0th power?

The formula to count Sierpinski triangle is 3^k-1 .It is good if you don't take the event when k=0.But how can you write a more ...
47 views

### Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
17 views

### Hausdorff dimension of a Cantor Set: attaining a lower bound

I'm considering the problem of calculating the Hausdorff dimension of a Cantor set, according to the following lemmas: Lemma 1 Let $C: [0, 1] \rightarrow [0, 1]$ be a Cantor staircase function. Then ...
34 views

### a conundrum regarding integrated Brownian motion and fractals

Let $X(t)$ be a Brownian motion. I know that the integral $$Y(t) = \int_0^t d\tau ~ X(\tau)$$ is well-defined, since Brownian motion $X(\tau)$ is a.s. continuous. Thinking ...
75 views

### Fractal identification

I was trying different algorithms out, and after a while, I found this fractal: The generation has similarities to Koch's curve, but instead of putting triangles on triangles, I put circles on top ...
40 views

110 views

15 views

### What are the Geometric Properties of Non Integer Vector Spaces?

I found a paper from Princeton called "Axiomatic Basis for Spaces with Non Integer Dimension" that presents five axioms and then starts to create a framework similar to what I'd think the subject ...
41 views

### Hausdorff dimension via ergodic theory

This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of ...
203 views

### Pythagoras tree bounding size

The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed ...
53 views

27 views

### Exact value of Hausdorff measure of middle-third Cantor set

Is there any result about the exact value of $\log_3 2$-dimensional Hausdorff measure of the middle-third Cantor set? And is there any fractal (in $\mathbb R^n$) which is not contained in a $p$-...
37 views

### What´s the dimension of a Sierpinski fractal?

I know the dimension of a Koch snowflake (log4/log3), but what numbers do I have to put in to obtain the dimension of a Sierpinski fractal?
49 views

### What is the condensation set of a fractal?

Is there a definition of the condensation set of a fractal that is both clear and rigorous? I've been searching around to get a sense of what exactly the condensation set of a fractal is - I've ...
73 views

### Can the fractal dimension of a surface be less than 2?

I have two surfaces represented as raster images with heights as grayscale values. One is natural landscape elevations; the other is just distance from a line. I have computed Minkowsky D = 2 - H ...
29 views

### How was one derivied from the other?

In the geological paper entitled The power–law relationship between landslide occurrence and rainfall level by C. Li et al, a power-law cumulative probability distribution is derived. However, I don't ...
32 views

### How to determine constant $C$ in $p(x) = Cx^{-D}$?

Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L_{cf}(> X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given ...
56 views

### Scaling factor closest to 1 in an infinite sequential rectangle packing

The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest ...
61 views

### hyperbolic spaces and fractals

Is there a relation between hyperbolic spaces and fractals? In group theory, if we take the Cayley graph of a free group on two generators, we get a fractal quaternary tree, which I'd like to think as ...
80 views

### Is this a valid definition of “self-similar fractal”?

I have always been fascinated by self-similarity, particularly in fractals. I was always wanted to find a simple definition of a self-similar fractal. Of course, saying "is self-similar, and is a ...
40 views

### Why c>1/4 is not in Mandelbrot set

As title: $f_c(x)=x^2+c$ I got to the step: $f_c(x)>x$ (for all x) But what's next? How to show that after k iterations, $f^k_c \to \infty$ as $k \to \infty$ Thanks,
67 views

### Fractal fundamentals

I am a programmer by trade, and am very interested in fractals. To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and ...
86 views

### Is every basin of attraction completely invariant?

I can't seem to find a definitive answer in the literature. I believe the answer is yes, but my focus has been on the rational maps on the Riemann sphere. At the very least I'm confident that if the ...
61 views

### Is there a general metod to construct a fractal?

I would like to construct a fractal (traditional, self-affine, and fat fractal) with a given embedding and fractal dimension, but I don't know how to do it programmatically. The shape of the fractal ...
41 views

### How to generate/validate unique fractal?

There are many known fractals that exist such as Mandelbrot, Cantor set, or the Koch curve, Sierpinski Triangle. What I am curious about, is how one could go about creating their own, unique fractal ...
90 views

### How is this fractal produced?

It is stated here: Iterating the above optimized map $$f(z)=\frac{1}{4}(1 + 4z - (1 + 2z)\cos(\pi z))$$in the complex plane produces the Collatz fractal. The point of view of iteration on ...
36 views

### Is the generalized mandelbrot set a fractal in the $d$ dimension?

The $d$-mandelbrot set is defined as the set of $c$ such that the iterations of $$z \mapsto z^d + c$$ starting with $z=0$ is bounded in absolute value. Here is a picture of the mandelbrot sets from \$...
32 views

### On a formulation in Hilberts original paper about the space-filling Hilbert curve

I have a question on the famous paper Über die stetige Abbildung einer Linie auf ein Flächenstück (which translates roughly as On the continuous mapping of a line onto a square) by D. Hilbert. Let the ...
58 views

### Is there a plane filling function calculator online?

I recently read about the "Hilbert Curve" and found it very interesting. Does anyone know of a place online where I could extrapolate different shapes and explore this field of mathematics?
I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some ...