Fractal Manifolds

Welcome to my page on Fractal Manifolds (FMs).

Over the past several years I have been working on FMs and developing them as a purely mathematical object from which smooth (or differentiable) manifolds form a subset.

Recently, several definitions of FMs have begun to emerge which, in my opinion, are either way too complicated (assuming too much structure) or developed with just a particular physical problem in mind, making them to specific to be useful.

My definition of a Fractal Manifold is that it is simply a self-similar manifold such that every iterated function system there exists at least one positive maximal Lyapunov exponent. That’s it! I will show that there is quite a lot you can do with this definition – for example, you get all of fractal geometry for free.

Later in the series I will show that Fractal Manifolds are no-where smooth and that Fractal Manifolds are generalisations of other important manifolds.

Finally, I will present the correspondence between simply connected, compact 4-manifolds and Fractal Manifolds and show that by Freedman’s Theorem (1982) the E8 manifold is a Fractal Manifold.

Using this correspondence we will investigate the naturally fuzzy behaviour of E8 at small scales and that if Lisi is correct then the physics of E8 will inherently have a fractal nature and maybe find a link between E8 and the operators of quantum mechanics.

The Fractal Manifold Series

In the following series of articles which can all be accessed from this page we will cover the fundamentals of a fractal manifold and see what we get from a mathematical point of view by dropping the differentiability structure. We will then extend in to four dimensions to prepare ourselves for the application to spacetimes.

Fractal Manifolds will then be shown to be a special case of the more general simply-connected topological 4-manifolds.

Finally, with some logical constraints we show that the Fractal Manifolds arise naturally from the E8 manifold (which is also non-smooth)

Table of Contents

CHAPTER I

1. Introduction
2. Manifolds
3. Self-Similarity
4. Iterated Function Systems
5. Fractal Manifolds
6. Fractal Curves
7. The Fractal Metric
8. Dimension and Entropy
9. Fractal Spacetimes
10. Curvature on Fractal Spacetimes
11. Goodbye Calculus Hello Fractals

CHAPTER II

1. Introduction to Quantum Mechanics
2. Operators & C*-algebras
3. Fourier & the Uncertainty Principle

CHAPTER III

1. Introducing the E8 Lie Group
2. Compact and Simply Connected
3. The E8 Manifold
4. Is Nature Described by the E8 Manifold?
5. From E8 to 4-Space
6. Why E8 Must be Fractal