Can we learn anything about international boundaries if we model them as fractals?

Question

Many country borders are determined by rivers, which are fractal in nature. Other country borders are determined by other natural phenomena such as mountain ranges and coastlines.

(edited) Main question

Because they are often defined by natural phenomena, Can we learn anything about international boundaries if we model them as fractals? (Thank you to @XanderHenderson for both this re-framing of the question and his thorough answer.)

Rivers

For country borders determined by rivers, See this list of international Border Rivers. The start of the European list:

• Ardila River: Spain and Portugal
• Bug River: Belarus and Poland
• Bug River: Ukraine and Poland
• Danube: Austria and Slovakia
• Danube: Croatia and Serbia

See also The fractal nature of river networks.

(edit) In Calculation on fractal dimension of river morphology, the authors have calculated the fractal dimensions of various rivers:

The fractal dimensions of stream length in Haihe River baisn vary from 1.01 to 1.14, and its mean value is 1.10. The fractal dimensions of Lanhe River and Daqing River are larger because their stream channels are zigzag.

Coastlines

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines, i.e., the fact that a coastline typically has a fractal dimension (which in fact makes the notion of length inapplicable). The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded upon by Benoit Mandelbrot. Coastline Paradox

Exceptions

Some country borders were drawn through treaty, such as the 38th parallel in Korea and the 49th parallel between the US and Canada.

Related questions

What sorts of problems can fractals solve? OP states "I already know people use fractals to map coastlines and borders"

What exactly are fractals Accepted answer says, "[A] fractal is one thing, and certain methods for constructing particular fractals are another."

From English Language and Usage / Irregular shape with projections and recesses that interlock with other shapes This OP's answer was "The borders between countries may be fractal in nature."

Motivation

This question is intended for the mathematics community, which has an understanding of fractal that is distinct from the English community.

Presuming that the answer to this question will be a "yes" or a "yes with caveats," I'd also like to solicit a researched, cogent "no" response.

(edit) This OP was surprised that the answers were populated with cogent "no" responses to the original question: "Are country borders fractal?" The main objections are that fractal is not well defined XanderHenderson and that the fractal is mathematical concept and "There are no instances of fractals in the physical world." YvesDaoust

Answer 1

It is a gross and naive approximation to state that a river describes a fractal curve.

In the first place because the exact geometry of the banks is not at all defined, as this would require an infinite (uncountable) amount of information. Merely stating "the delimiting line between water and land" is completely insufficient because water diffuses in a continuous way in the ground and the separation is very fuzzy. And continuously changes over time. In the short term (say daily), the uncertainty of the bank position is expressed in centimeters (more with tide effects). In the longer term, it can be inaccurate to meters or more. Were the banks made of very stable Platinium cooled to absolute zero, the fractal model would hold down to the scale of atoms. On smaller scales, we don't know what the world is made of, and thermal motion renders any description hopeless. And by quantum mechanics, atoms don't even have a shape.

In my opinion (I can't be authoritative in this respect), the precise border of countries is just left undefined where accuracy doesn't matter (say to the width of a river), or relies on a line drawn on a reference map (maybe on several independent maps to make things worse). This line will correspond to a large stripe on the ground.

So in practice borders are not fractal lines (which probably do not exist in our physical world) and there was certainly never any intent to make them so. Due to the finite thickness of the border lines, the exact border lengths are comprised in some uncertainty interval, the bounds of which are finite are finite.

Answer 2

Short answer:

Fractals are a mathematical concept. There are no instances of fractals in the physical world.

The "length of the coasts of Brittany" is just a piece of folklore, introduced for pedagogical purposes. Just like the Romanesco cabbage.

Answer 3

The question is ill-posed and cannot be answered. If an answer is demanded, the most reasonable answer is probably "No, international boundaries—even those which follow natural features, such as ridges and rivers—are not fractal." That being said, I don't think that you are even asking the right question.

Why is the question ill-posed?

The term "fractal" is not well-defined. There is no commonly agreed upon definition of fractal in mathematics. In order to even answer the question, you would have to first give a solid mathematical definition of what a fractal is.

For example, Mandelbrot suggested that a fractal is a space with "fractal dimension" (Hausdorff-Besicovich dimension) strictly exceeding the topological dimension. Of course, Mandelbrot quickly backed off of that definition when it was noted that it had problems (e.g. it excluded certain sets which are generally considered fractal, and included other sets which are generally not considered fractal). Falconer suggests that the term continue to be left undefined, and uses a kind of "quacks like a duck" test:

In his original essay, Mandelbrot defined a fractal to be a set with Hausdorff dimension strictly greater than its topological dimension... This definition proved to be unsatisfactory in that it excluded a number of sets that clearly ought to be regarded as fractals. Various other definitions have been proposed, by they all seem to have this same drawback

My personal feeling is that the definition of 'fractal' should be regarded in the same way as the biologist regards the definition of 'life'. There is no hard and fast definition, but just a list of properties characteristic of a living thing...

[1, pp. xx]

As I've addressed this issue on Math StackExchange before, I won't go into more detail on this issue here.

In any event, if you want an answer to your question, the first task is to give a mathematically rigorous description of what a "fractal" is. Without that, there is no reasonable answer.

Why should the answer be "No"?

While there is no universally agreed upon definition of what a fractal is, the kinds of objects which most people who work in the area would like to call "fractal" have many properties in common. One of those properties is a kind of "complexity" at all scales. That is, fractals are objects which continue to show detail as you "zoom in" on them.

For example, consider the von Koch snowflake. In the animation below (taken from Wikipedia), as we zoom in on a corner of the curve, we see more and more detail. Indeed, the snowflake is self-similar in a fairly strong sense. Contrast this with a circle. Zoom in enough on a circle, and it starts to look like a straight line. Indeed, this is the fundamental insight of differential calculus.

In the natural world, there appears to be a natural "smallest scale", i.e. the Planck scale. Nothing smaller than this scale can be measured, nor said to exist. This automatically implies that nothing in nature is actually fractal (whatever that means). Eventually, after zooming in enough, everything "smooths out" and there are no new details to observe. New experiments may overturn this understanding, but, given the current state-of-the-art, it seems that physical realizations of ideal fractals simply don't exist.

Why isn't this the right question?

If you pick any arbitrary mathematical object and ask "Does this object exist in nature?" the answer is almost certainly "No". For example, perfect circles don't exist in nature, nor does the number one. We use mathematics to abstractify and simplify the real world. We make unrealistic assumptions which help us build mathematical models which, hopefully, give us insight into the way in which the universe works. The question shouldn't be "Does this mathematical object exist?" but rather "Does this mathematical model give any useful insight?"

Therefore, I think that the correct question is "Can we learn anything about international boundaries if we model them as fractals?"

But that is another question entirely.

[1] Falconer, Kenneth, Fractal geometry. Mathematical foundations and applications, Hoboken, NJ: John Wiley & Sons (ISBN 978-1-119-94239-9/hbk). xxx, 368 p. (2014). ZBL1285.28011.