Koch snowflake and planck length

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Discussion Overview

The discussion revolves around the properties of the Koch snowflake and its implications regarding the concept of infinity in mathematics versus physical reality. Participants explore whether the infinite perimeter of the Koch snowflake can exist in the physical universe, particularly in relation to the Planck length and the nature of fractals.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that as smaller sides are added to the Koch snowflake, they could theoretically become smaller than the Planck length, raising questions about the feasibility of an infinite perimeter in a physical context.
  • Others assert that the Koch curve, being a mathematical construct, cannot represent a physical object since all physical objects are composed of atoms, thus limiting their length.
  • There is a suggestion that the Koch snowflake could theoretically be made of space or energy, but this would challenge its classification as a fractal.
  • Some participants question the relevance of fractals if they do not have infinite perimeter in reality, noting that real-world examples of fractals, like coastlines, have finite lengths.
  • One participant introduces the distinction between scale-invariant fractals, which lack physical representation, and scale-dependent fractals, which can apply to real-world structures and have fractal dimensions that depend on scale.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the Koch snowflake and its implications for physical reality, with no consensus reached on whether it can exist as a physical object or the significance of fractals in representing real-world phenomena.

Contextual Notes

Participants highlight the limitations of mathematical models in representing physical situations, particularly when distances approach the Koch length, suggesting that continuous segment geometries may not be appropriate in such cases.

fromage123
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surely if you keep adding smaller and smaller sides to the snowflake they will become even smaller than the Planck length and so how can the perimeter be infinite- or is the infinite perimeter only theorteically possible with maths, but not actaully acheivable?
 
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fromage123 said:
surely if you keep adding smaller and smaller sides to the snowflake they will become even smaller than the Planck length and so how can the perimeter be infinite- or is the infinite perimeter only theorteically possible with maths, but not actaully acheivable?

The Koch curve cannot be a physical object because in our universe every physical object is composed of atoms.

The Koch curve is a mathematical object that actually has infinite length.
 
but why can't the snowflake be made of space or energy itself directly (theoretically?)
and if it it is as you say- that it cannot exist as a phyical object what is the point in fractals if in reality they do not have infinite perimeter? (since all the 'real' examples of fractals such as coastlines will in actual fact have a finite length)
 
fromage123 said:
but why can't the snowflake be made of space or energy itself directly (theoretically?)
It can, but then it would no longer be a "fractal".

and if it it is as you say- that it cannot exist as a phyical object what is the point in fractals if in reality they do not have infinite perimeter? (since all the 'real' examples of fractals such as coastlines will in actual fact have a finite length)
The same point as any mathematical model of a physical situation- by using the right mathematical model we can get arbitrarily good approximations to what is "physically" happening. If a physical situation involves distances near the Koch length, then "fractal geometry" or any geometry using continuous segments is no long the "right" mathematical model.
 
fromage123 said:
... what is the point in fractals if in reality they do not have infinite perimeter?

One may add that you can distinguish between scale-invariant fractals, that do not have a physical representation, and then scale-dependent fractals where the fractal dimension depends on the scale. The later is very much applicable to real world structures. See for instance [1] for some more details.

[1] http://en.wikipedia.org/wiki/Fractal_dimension
 

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