Understanding fractional and higher dimensions

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Discussion Overview

The discussion revolves around the concepts of fractional dimensions and higher dimensions, particularly in relation to the Koch snowflake and its representation in two dimensions. Participants explore the implications of these dimensions in geometry, questioning how such constructs can be visualized and understood in reality.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion about how the Koch snowflake, which has a fractional dimension of 1.2, can be represented on a 2D page, questioning the relationship between its drawing and its actual dimensionality.
  • Others mention that fractional dimensions arise from measuring complex shapes, such as coastlines, where the length can vary based on the measurement scale.
  • A participant suggests that the process of creating the Koch snowflake is iterative, involving specific geometric transformations rather than simple scaling or translation.
  • There is a discussion about the nature of space-filling curves, with some participants challenging whether the Koch snowflake and similar shapes can be classified as such due to their dimensions being less than 2.
  • Some participants propose that the notion of continuity in mathematics may not be accurately represented by the idea of drawing curves without lifting a pencil, suggesting a more nuanced understanding of continuous functions.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the interpretation of fractional dimensions and their implications for visualization. There is no consensus on how to reconcile the mathematical definitions with physical representations or the nature of space-filling curves.

Contextual Notes

Limitations in understanding arise from the abstract nature of fractional dimensions and the iterative processes involved in constructing fractals like the Koch snowflake. The discussion highlights the dependence on definitions and the challenges of visualizing higher-dimensional objects in lower dimensions.

PcumP_Ravenclaw
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Halo,
I was reading about geometry from Tim Gowers book titled "A very brief introduction to mathematics". I came across fractional dimensions and the 4th dimension. The koch snowflake has dimension 1.2 yet he could comfortably drawn it on a 2d page (or is it complete?). Has not he just transformed the original "snowflake", scaled it, translated it, etc.. what does it have to do with dimensions? maybe, its like drawing a 3d structure on a 2d page? so this snowflake is not how it actually appears in the 1.2 dimension as drawn on 2d page? This just a extrapolation of the formulas in 1d, 2d & 3d? How do we prove such domensions exist in reality, if at all? So a 4d cube as being a cube inside a bigger cube is also for sake of visualization, like drawing a 3d cube on a 2d page?

Please confirm!

Danke.
 
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Fractional dimensions come out of the notion of measuring a coast line the length depends on how close you are.

https://en.wikipedia.org/wiki/Coastline_paradox

and this on fractal dimensions:

https://en.wikipedia.org/wiki/Fractal_dimension

You could compare it to how integers were extended into rational number and then rational into real with the notion of irrational...

And so Mandelbrot extended the notion of dimension into a fractal (aka fractional) dimension.
 
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PcumP_Ravenclaw said:
Halo,
I was reading about geometry from Tim Gowers book titled "A very brief introduction to mathematics". I came across fractional dimensions and the 4th dimension. The koch snowflake has dimension 1.2 yet he could comfortably drawn it on a 2d page (or is it complete?). Has not he just transformed the original "snowflake", scaled it, translated it, etc.. what does it have to do with dimensions? maybe, its like drawing a 3d structure on a 2d page? so this snowflake is not how it actually appears in the 1.2 dimension as drawn on 2d page? This just a extrapolation of the formulas in 1d, 2d & 3d? How do we prove such domensions exist in reality, if at all? So a 4d cube as being a cube inside a bigger cube is also for sake of visualization, like drawing a 3d cube on a 2d page?
A line segment has dimension one. A rectangle has dimension two, but its perimeter is one-dimensional. The Koch snowflake, in contrast, is so intricate that its perimeter is unbounded, even though it has a finite area. For that reason, it is considered to be one of several kinds of space-filling curves. Another space-filling curve is the Sierpinski triangle.
 
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Mark44 said:
A line segment has dimension one. A rectangle has dimension two, but its perimeter is one-dimensional. The Koch snowflake, in contrast, is so intricate that its perimeter is unbounded, even though it has a finite area. For that reason, it is considered to be one of several kinds of space-filling curves. Another space-filling curve is the Sierpinski triangle.

These are certainly not space-filling curves in the classical sense.
 
micromass said:
These are certainly not space-filling curves in the classical sense.
You're right - I misspoke. If they were truly space-filling, their dimension wouldn't be less than 2.
 
wha
Mark44 said:
You're right - I misspoke. If they were truly space-filling, their dimension wouldn't be less than 2.
what is a space-filling curve?
 
jedishrfu said:
Notice you can draw the curve without picking up your pencil and basically cover an entire area ie every point in the area is a part of the curve

Since the curve has infinite length, I doubt that.
 
This is interesting though. Continuous functions are often introduced as "those curves that can be drawn without picking up the pencil." But I think this is not the right motivation for continuity. The curves that can be drawn without picking up the pencil are (according to me) rather the continuous functions of bounded variation. Those seem like a much better contender.
 
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Micro, we have all the time in the world... :-)
 
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ii
PcumP_Ravenclaw said:
Halo,
I was reading about geometry from Tim Gowers book titled "A very brief introduction to mathematics". I came across fractional dimensions and the 4th dimension. The koch snowflake has dimension 1.2 yet he could comfortably drawn it on a 2d page (or is it complete?). Has not he just transformed the original "snowflake", scaled it, translated it, etc.. what does it have to do with dimensions? maybe, its like drawing a 3d structure on a 2d page? so this snowflake is not how it actually appears in the 1.2 dimension as drawn on 2d page? This just a extrapolation of the formulas in 1d, 2d & 3d? How do we prove such domensions exist in reality, if at all?
What do you mean by "reality"? This is, after all, mathematics, not physics.

So a 4d cube as being a cube inside a bigger cube is also for sake of visualization, like drawing a 3d cube on a 2d page?
Basically, yes. They are representations of a higher dimensional object in two dimensions.

Please confirm!

Danke.
 
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PcumP_Ravenclaw said:
Has not he just transformed the original "snowflake", scaled it, translated it, etc.. what does it have to do with dimensions?
 
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PcumP_Ravenclaw said:
Has not he just transformed the original "snowflake", scaled it, translated it, etc.. what does it have to do with dimensions?
No. There is no "original snowflake." There is no scaling or translation going on. Creating the Koch snowflake is an iterative process, that starts with an equilateral triangle. Let's say the triangle has sides of length 1. In the first step, each of the three sides is divided into thirds. The middle segment is deleted, and two segments whose length is equal to the deleted segment are attached to the gap formed by deleting the middle third.

Graphically, we're going from ___ to _/\_ for each side. This same process is applied to the other two sides of the original equilateral triangle. In this first step, the perimeter increased from 3 to 4. (We removed a segment of length 1/3 from each of the three sides, but added two segments of length 1/3 to each side, so each side increased from 1 to 4/3 units.) Also, in this step, the triangle went from three sides to a polygon made up of 12 segments.

In the next step, each of the 12 segments has its middle third removed, and replaced by two more segments, in the same way as described above. This is much easier to show graphically than to explain in words.
 
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