What on Earth's a Fractal?
Dripping with the yolk of the Universe, my legs dangling over the dark and alien abyss below I knew my answer had to be presented with caution.
I began "They are a late twenty-first century realization that the mechanical laws are not the only things that allowed the Universe to evolve.
However evidence for it comes from things well known to mankind. The Fibonacci number series with its relation to patterns in nature is one of these."
A voice dripping with skepticism called to the mob "Numbers alone can't do this. Numbers aren't able to create anything"
Another shouted "He's right. Numbers have no powers at all. These are just interesting patterns made by chance. "
I knew that my answer wouldn't be enough so I continued. "Fractals are a basis for nature's creative design. They are shapes set within any enclosure which nature tends to follow. And the shaping tool behind their existence is the pattern achieved when adding numbers."
I now felt an increasing interest in my words and the pictures that I took from my pockets. It wasn't just luck that I had them with me. I had prepared them the day before to sate my grandchildren's interest in what they saw on my laptop.
Einstein stood up alongside me, facing the mob. Respectful quietness settled as he spoke.
"These are pretty ideas and up until now they offer little new. Yet they have possibility. Let him speak and I will ensure his words don't go unchallenged if he goes astray."
The mob seemed happy that such a person had taken the lead and they settled down to hear my impossible fraud exposed.
Einstein turned a little while still facing the crowd and asked me: What are the properties of number on which you base your claims?"
Warily I started my reply being well aware that Einstein was the master at challenging underlying assumptions.
"I will take my lead from you and strip away the base that gives comfort to the unthinking mind. I begin with ideas that were true at the beginning of our Universe where terms like space and time were not applicable. I go to the ideas of a dynamic system experiencing immense change since such a scheme also applies to our Universe.
To that scheme I take up things that you yourself used when moving from the defined frames for your special theory involving steady speeds to the frame required to understand your general theory involving acceleration."
Einstein : "So which parts did you assume? "
Me: "Your ideas on a less rigid type of continuity as expressed in the mapping of your generalized coordinates.
I noted that continuity is a concept in itself that can apply to other systems. For example it can also relate to parts of a chaos where there is neither time nor space."
And once I accepted that idea could occur in chaos I had to include the concept of duration. I did so because every continuity has an extent between its end points.
And I took an implied concept you also assumed; that of adjacency of points. This seemed reasonable since without continuous neighbors any system is only a set of isolated singularities.
But I also took your generalized ideas of sequence where the gap between layers doesn't have to be uniform since the same principle works with more generalized systems.
From this set of concepts I knew labels could be applied to different paths connecting adjacent points.
And once I recognized the capacity for sequential labels I had to grant that such a system was open to the concept of numbers since any path can be given ordered labels as successive points are passed.
So with all this I knew I could give meaningful values to things in zones which existed before space-time. "
Einstein : "But so far you don't have anything that's real"
I pressed on knowing this wasn't a question he asked for himself but on behalf of the mob. I responded with " True. Yet it is enough when we note that all we need is the power of repetition to change the infinitely small into something real plus the natural power of addition."
Einstein who was obviously aware of the mobs askant looks towards him stated in an assertive way "There are of course issues with your definitions but they conform with ideas well known to me and others.
Even the idea of the infinitely small being used to define something real is at the heart of one of our most powerful tools, that of calculus. So proceed and show us how you think the power of repetitive addition makes it work."