Koch snowflake and planck length

In summary, the conversation discusses the concept of fractals and whether they can have infinite perimeter in reality. While the Koch curve is a mathematical object with infinite length, it cannot exist as a physical object because all physical objects are composed of atoms. The purpose of fractals is to provide mathematical models for physical situations, but they may not always be the most accurate at very small distances. There are also scale-dependent fractals that can be applied to real-world structures.
  • #1
fromage123
2
0
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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  • #2
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.
 
  • #3
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)
 
  • #4
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.
 
  • #5
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
 

1. What is the Koch snowflake?

The Koch snowflake is a mathematical fractal, named after Swedish mathematician Helge von Koch. It is created by repeatedly replacing the middle third of each line segment with an equilateral triangle, resulting in a self-similar shape with infinitely many sides.

2. How is the Koch snowflake related to the concept of self-similarity?

The Koch snowflake exhibits self-similarity, meaning that it looks the same at any scale. This means that if you zoom in on any part of the snowflake, it will look identical to the whole snowflake. This is a common characteristic of fractals.

3. What is the significance of the Koch snowflake in mathematics?

The Koch snowflake is a significant mathematical concept because it is an example of a geometric object with a finite area, but an infinite perimeter. This challenges our traditional understanding of geometric shapes and has led to further exploration and understanding of fractals and infinity in mathematics.

4. What is the Planck length and why is it important?

The Planck length is the smallest possible length in the universe, at approximately 1.616 x 10^-35 meters. It is significant because it is believed to be the scale at which quantum gravitational effects become significant and our traditional understanding of space and time breaks down. It is also the smallest length that can be measured with any degree of certainty.

5. Can the Koch snowflake be used to explain the concept of the Planck length?

While the Koch snowflake and the Planck length are both fascinating mathematical concepts, they are not directly related. The Koch snowflake is a geometric shape, while the Planck length is a fundamental unit of measurement in physics. However, both concepts challenge our understanding of traditional geometry and scale, making them interesting to explore together.

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