A mathematical curiosity: A fractal tree object with branches in every dimension, up to infinity

I stumbled upon a curious mathematical beast: a fractal with infinite dimensions. I will be producing 3 dimensional renderings to get an idea what this ‘looks’  like, but let’s start with the maths.

Imagine a trunk of a 1-dimensional tree, i.e. a fat line. After a while it splits into two branches in a 2-dimensional plane. Now imagine that each branch splits again towards the front and the back, in a 3-dimensional space.

What if each new set of branches split off into a next dimension, i.e. from the 3rd into the 4th dimension, from the 4th into the 5th etc. Forever branch into a next dimension.

Since the branches curve one way and the other, the overall hyper dimensional tree will be in a limited hyper volume. Just imagine that, an infinitely dimensional object that fits in the palm of your hand.

You could only see 3 dimensions at any one time so if we rotate the infinitely dimensional object in some way, we would be able to rotate the invisible dimensions into view and this should give us a glimpse of what infinite dimensions looks like. Even then, it would only be a glimpse because we can never visualize such an object in the same way that it would be hard to get an idea of the Taj Mahal if we were only looking at 1 square millimetre at a time.

The model  of this fractal is an extension of the 2-dimensional generic fractal I have developed and blogged about before. Here is the equation:

\check{p}=\int \dot{r}e^{i\check{u}\int \dot{\varphi }ds}ds

As is explained in this paper, the \check{u} function is a multi valued fractal function that is 0 at a branch point and 1 and -1 along the branch. Since this is a periodic function, the branches will split at every point when \check{u} = 0 .

This equation could be rewitten with a subtle change by including the imaginary unit “i” in the fractal unit function:

\check{p}=\int \dot{r}e^{\check{u}\int \dot{\varphi }ds}ds

and \check{u} would be defined as 0 at the branch points and i or -i along the branches.

Having redefined the fractal unit function, it now includes the dimensional vector “i” and we can extend this further. If we define the fractal unit function, renamed “w” for clarity, to evolve from branch to branch as:

1st branch: \check{w} = 0 or \{i,-i\}

2nd branch: \check{w} = 0 or \{i,-i,j,-j\}

2nd branch: \check{w} = 0 or \{i,-i,j,-j, k, -k\}


where i,j and k are imaginary and  perpendicular unit vectors, then if we replace these unit vectors by a generic notation i_{n} for n=0..\infty then we can define the infinite dimensional tree as:

\check{p}=\int \dot{r}e^{\check{w}\int \dot{\varphi }ds}ds


\check{w} = \left \{ 0 \ or \ s_{n}\mid n=0..\infty \right \}

The mathematical language needs to be tidied up but as innocent as this looks, these two equations represent an object that is very far beyond our imagination. Now I need to generate a dozen or so iterations to generate a 12 dimensional object and look at it through a 3 dimensional viewer.


So what do you think?

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