Can fractals sum to a linear function?

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Discussion Overview

The discussion revolves around the possibility of summing fractals to yield a linear function or a differentiable field. Participants explore the nature of fractals, their dimensions, and the implications of superposition in this context, touching on theoretical and conceptual aspects.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions whether a set of fractals can sum to a differentiable field.
  • Another participant suggests that one can often combine "ugly" mathematical objects to produce a "nice" result through subtraction, although the application to fractals is not directly addressed.
  • A different participant inquires if a superposition of fractal fields could yield a linear field, expressing concern about their mathematical understanding.
  • One participant asserts that fractals are typically assumed to be discontinuous and nonlinear, questioning if their superposition could produce a continuous, linear pattern.
  • Another participant claims that it is possible to sum fractal curves of non-integer dimensions to achieve a "nice" result, referencing the embedding of fractals in the plane.
  • A participant asks for a simple proof that fractals are not differentiable, while another raises the question of whether fractals could be differentiable using fractional derivatives.

Areas of Agreement / Disagreement

Participants express differing views on the nature of fractals and their summation properties. There is no consensus on whether fractals can sum to a linear function or be differentiable, and the discussion remains unresolved.

Contextual Notes

Participants use terms like "ugly" and "nice" in a non-standard way, which may lead to confusion. The discussion also touches on the dimensions of fractals and their implications for differentiability, but these concepts are not fully clarified or agreed upon.

Loren Booda
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Does there exist a set of fractals whose sum defines a differentiable field?
 
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I'm not sure what you mean, but I think I can shed some light on it.


Generally, when you want to answer the question: "Can two ugly things sum to a nice thing?", you can usually answer in the affirmative by picking any ugly thing, then subtracting it from a nice thing. If this subtraction yields another ugly thing, then you have two ugly things that add to a nice thing.
 
Nice way to explain, really nice ... you should write a book Hurkyl
 
nice explanation Hurkyl...

marlon
 
Is it possible that a superposition of fractal fields yields a linear field? I hope this is sufficiently ugly, Hurkyl (please forgive my mathematical illiteracy).
 
The problem is, those terms just don't go together. I really have no idea what you are trying to say.
 
Hurkyl,

A fractal pattern is assumed discontinuous and nonlinear. I was wondering whether the "superposition" (sum of values for every correspondent point) over two or more such fractals could generate a continuous, linear pattern.

Consider a fractal of dimension 1.2 projected onto a fractal of dimension 1.8; might this mapping ever represent a differentiable geometry of dimension 3, or even of dimension 2?
 
Again, you're using words in an alien way. "represent a differential geometry"?


The simple answer is of course you can sum two fractal curves of non-integer dimension and get something nice:

embed your favourite fractal curve y=f(x), and 1-f(x) in the plane, then the sum will be 1just as hurkyl said.
 
Is there a simple proof that fractals are not differentiable?
 
  • #10
... or are fractals differentiable by fractional derivatives?
 

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