Understanding fractional and higher dimensions

AI Thread Summary
The discussion centers on the concepts of fractional dimensions and higher dimensions, particularly through the example of the Koch snowflake, which has a fractal dimension of 1.2. It highlights the difference between drawing such shapes on a 2D page and their actual dimensional properties, questioning the nature of these representations. The iterative process of creating the Koch snowflake is explained, emphasizing that it is not merely a transformation of an original shape but a unique construction that leads to an infinite perimeter with a finite area. The conversation also touches on the visualization of higher-dimensional objects, such as a 4D cube, as projections in lower dimensions. Overall, the thread explores the mathematical implications of dimensions beyond the traditional integer values.
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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