hyperbolic spaces and fractals

Question

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 embedded in some hyperbolic space.

Mathematically I don't know how to define this idea, as it is somewhat vague and qualitative, but I'm believing that there is a relationship between the notion of hyperbolicity and fractals. Any help?

Answer 1

This is not a proper answer, is as vague as also in the question but hopefully impacts what you are attempting to say. At any rate I hope it survives down-votings.

Take a circular disk and let its boundary length increase indefinitely. The surface area in the neighborhood of boundary increases and spreads to the center by deformation compatibility definable by von Kármán compatibility relation ( i.e, it pulls in extra surface material towards its center to create a warped Pringles chip for example.

$$\frac{\partial ^2\epsilon_{\theta}} {dr^2}+..+..{}\approx -K $$

[ If border material is contracted on the other hand, the shape becomes like a bowl or dish.

]

Extra membrane energy /surface area created is constant for a soap film of minimal area as surface tension. So it can be the case of varying mean curvature $H \ne 0 $ that varies in a particular way to result in constant $ K<0. $

At first by addition of areas at circumference the disk has reduced or negative Gauss curvature $K$ somewhat like:

1. Pringles Chip with the saddle point at center and everywhere else and then

we have by exaggerating the same growth further on...

2. Sea Coral shapes like those made/ demonstrated by Prof. Daina Taimina by a process of introducing extra area at disk boundary with more loop areas continuously introduced due to extra area input the circumference.

which finally tend to, if allowed such an appellation, deeper warped "Koch surfaces":

3. In 3D Koch snowflake Fractal the area enclosed by the boundary is finite but boundary length approaches infinity. Fibonacci surface Fractals also can be found surfing the net. The first two

KochSurfaceGrowth

$$ Pringle Chip \rightarrow CrochetedCoral \rightarrow 3D\,KochSnowflake $$

These three shapes can be defined to have constant $K<0$. Imo it is a pity of hyperbolic geometry that none of the above has a mathematical description with a parameterization as of today. This aspect seems not been adequately researched or perhaps not published.

Answer 2

A regular pattern in the hyperbolic plane will appear self-similar and fractal in common models of the hyperbolic plane. You'd see copies of the same theme repeated at different scales. So there is at least some relation here.

To make things less vague, you'd first have to define the term “fractal”. On this topic, Wikipedia currently writes (text by Akarpe):

There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. […] The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions […].

While I confirmed self-similarity, I'm less sure regarding “iterated”. You could say that you start with one tile of the pattern, and then at each iteration you add the tiles directly adjacent to those you already have. That would give you an iterated construction, but contrary to what most fractals do it would leave the parts you already have unmodified, there is no refinement happening there. It would indeed be interesting to see whether you could compute a fractal dimension from this process. If I find the time, I may make some experiments regarding this, but don't count on it.

The characteristics section lists some characteristics, quoting Fractal Geometry: Mathematical Foundations and Applications by Falconer. Many of these are met by models of hyperbolic patterns.

Answer 3

One way more suitable is to think of it in this way:

$\mathbb{H}^n$ cannot be Isometrically (distance-preservingly) immersed $\mathbb{R}^n$ due to Hilbert's Immersion Theorem

And every Hyperbolic manifold of dimension $n$ is isometric to $\mathbb{H}^n$, in order for it to fit in $\mathbb{R}^n$, one requires a conformal model where angles remain the same, only distance gets shortened as we tend towards infinity (the boundary)

And it is the the boundary where we find the Fractal, as the Limiting set for such this projection

It is best imagined in terms of a projection of some struture drawn on a hyperboloid, onto the $\mathbb{R}^n$, and since the higher the object is from the plane over the hyperboloid, the furthest it'd be projected down the plane; therefore if the source of light (projection point) is just at the level set where the struture ends, then ideally the projection would tend to infinity. Therefore you abstractly create a new kind of projection, where further the projection are, smaller the projection would be. That's how you get the fractal in the boundary; since they've now become infinitesimally small.