**7**

votes

**2**answers

311 views

### Minimum number of contractions needed to obtain a particular invariant set

Consider the Koch curve $G \subseteq \mathbb{R}^2$. Clearly $G$ is the invariant set (IS) of the iterated function system (IFS) $\lbrace \phi_1, \phi_2, \phi_3, \phi_4 \rbrace$. Where (not wanting to ...

**13**

votes

**2**answers

1k views

### Does “Algebraic numbers coloured by degree” form a fractal?

This picture from Wikipedia's article on Algebraic numbers shows a visualization of Algebraic numbers coloured by degree.
I'm wondering if this is a fractal?

**29**

votes

**5**answers

2k views

### How to define a differential form on a fractal?

It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one ...

**-2**

votes

**2**answers

1k views

### What are the fractal parameters?

There are so many fractal which are not uniquely characterize by some fractal parameters like Fractal dimension, Succolarity, Lacunarity, Morphological entropy. Can you suggest some fractal parameters ...

**8**

votes

**2**answers

613 views

### Shortest Paths on fractals

How can one find shortest paths between 2 specified points on fractals, or (since I'm pretty sure this is quite complicated) make useful generalizations about them?
Since the above question is broad, ...

**2**

votes

**1**answer

315 views

### Is there an nontrivial function whose 'period paralellograms' are Gosper Islands?

The Gosper island tiles the plane, so I'm curious if a nontrivial elliptic? function exists which would have a 'period gosper-island' instead of a period parallelogram. In this case, I'm using ...

**12**

votes

**3**answers

861 views

### Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?

Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...

**3**

votes

**2**answers

378 views

### Systematization of deterministic and stochastic integrals

With this question I try to build up a systematization of different kinds of integrals. The following table differentiates between deterministic and stochastic integrals, the summation processes ...

**5**

votes

**4**answers

630 views

### Determining a lower bound on the Hausdorff dimension of a set

Does anyone know of a good method for finding a lower bound of the Hausdorff dimension of a set $G$?
The only method I could find is to find an $\alpha$-Hölder function $f \colon G \to H$ then ...

**5**

votes

**1**answer

1k views

### Self-similar matrices? [closed]

Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?