The coastline paradox states that due to their fractal-like properties, the length of a given coastline approaches infinity as the “ruler” used to measure it gets smaller and smaller. Suppose that I sketch a map of the coastline of Australia on a piece of paper, could the sketch be said to be infinite? If not, why not? Is it just the size and or level of “complication” of the coast? Neither of those explanations is very satisfactory to me when it comes to dealing with infinite issues.
I had not heard of this but the logic is pretty obvious.
Draw a sketch of Australia and the coast is clearly finite. Draw a small scale map and the coast becomes much longer with all the headlands and bays,
Draw a detailed, large-scale map and the coast becomes very long indeed, but if you tried to draw at an extremely large scale, say 1/1, the length would multiply many times - in fact, it would become impossible as the dimensions of a coastline are not totally fixed.
I am not sure about infinity though.
Apparently. Last night I was watching a Jeopardy! in whcich the final clue was:
“Excluding Russia, this nation has the longest coastline in continental Europe, at 15,900 miles.” (That sin’t the exact number but it was around there, close to 16000." The answer was Norway.
Now, Norway does have a lot of coastline, but 15,000 miles is more than a third of the way around the planet. It’s roughly equivalent to flying from New York to Mumbai and back again. Norway sure as shit doesn’t LOOK like it has that much coastline, so there’s clearly some fine slicing going on here and the use of every inlet.
According to Wikipedia, " At 30-meter (98 ft) linear intercepts, this length increases to 83,281 kilometers (51,748 mi)," which is far more than the diameter of Earth and about twenty percent of the way to the Moon.
At the risk of threadshitting - and with no intent of threadshitting…this “paradox” sounds like the “an arrow can never reach its target” paradox - the paradox that says that because an arrow must first fly half the length to the target, then half of the remaining half, then half of the remaining half, etc. that it can never reach its target.
In other words, it sounds clever, but has zero practicality since the arrow clearly strikes its target - and a coastline’s length is clearly not infinite.
Very wrong. It is a factual statement about fractal details and the scale of the ruler. Look at the coastline in this link–on a large scale map, it would be a straight line. On a smaller map, it would follow the contours of the “fingers” of land. On a very small map, it would follow the contours of individual rocks.
And to address the OP, if you look at your drawing under a microscope, you would see not infinitely smooth lines but irregularities in the shape of the graphite laid down (or ink absorbtion) that could be measured along, with more irregularities the more microscopic you got, down to tge bumps of individual atoms.
Not quite the same.
Zeno’s paradox is about the “impossibility of motion” which is pretty easily refuted by, well, moving.
The coastline paradox is about measurement, and how measurement is done. You could “refute” the coastline paradox by taking a trip along a coastline and measuring the distance you travel, but then the question becomes: how did you choose your path? Did you cut across some bay/inlet/fjord on the way? What if you traveled along a path that cut closer to the “coastline”?
The wikipedia article covers it well.
In terms of practicality: well, it’s not going to change, for example, the sailing distance between Boston and Miami, nor the driving distance between two points on the coast. Nobody follows the exact coastline, after all. But there is some applicability in, for example, chemical kinetics where the speed of a reaction is dependent on the surface area of a reagent.
You stopped too soon. What is it about “the contours of individual rocks” that will eventually make this coastline infinite? It must be that each side of every rock is also an infinite fractal.
But if that is correct, then I don’t see anything special about coastlines or rocks. It would apply just as well to any imperfect line. If I take a pencil and draw a 6 inch line on a piece of paper, and study that line under an electron microscope, would it be any less infinite than the coastline of Norway?
Ninjaed by Darren_Garrison!
The coastline question has a very special place in the history of fractals. So when discussing fractals in the real world it remains a common topic. But for the OP, when you start talking infinite quantities and the real universe you will always end up in trouble. For useful purposes there is effectively a smallest scale that is used to measure a coastline. But if you want to pursue the fractal nature as far as you can, you eventually reach the point where there is no contiguous boundary. It makes no sense to dive below an atomic scale as it just breaks up.
Infinity gets thrown around a lot when it is really good idea to stop a moment and consider if you really want to go there. Infinity is not just a really really really big number. It works completely differently. If you try to use infinity to relate to anything in the physical world, you probably have something wrong, and not just slightly, but in a very fundamental way.
Sure, but the coastline does not “approach infinity,” per the paradox stated in the OP.
Perhaps the coastline of Ireland is 6,226 kilometers. Perhaps it’s 6,231, if one uses incredibly tiny and accurate rulers to measure. But under no circumstances does it approach infinity; it couldn’t even approach 7,000.
Are you sure about that? Even if you take your ruler and start measuring around and between every quark that makes up the coastline?
I hope no one minds if I ask a related question. I understand that as you use progressively smaller units, the measured length of the coastline will increase. But how is this related to fractals? I thought that a fractal is when a subset of the unit is in the same shape as the larger piece (or however that is phrased in technical terms). But that’s not an inherent quality of coastlines, is it?
There are no perfect fractals in nature. But there are some things that behave an awful lot like fractals, over a very wide range of length scales. Coastlines are one of those.
You misunderstand. We’re not talking about differences in the third decimal place. We’re talking about measurements that differ by multiple orders of magnitude, when measured at different scales. You can’t even say what the approximate length of a coastline is, unless you specify a scale, and two different scales which each seem like perfectly reasonable scales to use can give very different answers.
By way of analogy, consider measuring the length of the edge of a saw. You could just lay a ruler down next to the saw blade, end to end, and say that the blade is 12 inches long. But if you look at a fine enough scale, you can measure up and down each tooth, and if the teeth are shaped like equilateral triangles, say, you’d find that the length was twice as long as you originally measured. But now imagine a saw that doesn’t just have teeth: The teeth themselves have teeth on them, and those teeth have yet smaller teeth, and so on. And every time you look at the next level of teeth, the length doubles.
What’s fundamental about fractals is that the measurements get larger as you look at smaller scales. Or more precisely, that they increase in a particular mathematical way. The most common way to get this behavior is if smaller scales look like copies of the larger scales, and this is in fact mostly true of coastlines: If I gave you a map of an unfamiliar piece of coastline, without a scale marked on it, you probably wouldn’t be able to figure out the scale, because small-scale coastline features look very similar to large-scale coastline features. But self-similarity isn’t the only way to get that behavior, and one could in fact construct fractals that, although they have structure at every length scale, it’s different structure at every length scale.
Self similarity.
The self similarity does not need to be a precise mathematical function, usually a recursive definition. The self similar nature of the Mandelbot Set is something of a surprise as well. There are processes that cross a range of scales which produce remarkably similar forms. OTOH, there has been a lot of total rubbish written about fractals in nature where no such relationship exists. In a previous life I worked in geophysics, and somewhere I have an entire book about the fracture nature of geology, that to a large extent is just bad science. There was a near religious belief that nature was fractal, and once you identified a pattern at one level that pattern could be found at other scales and that would let you predict the geology. Bollocks. There are self similar processes that span scales, mountain ranges are the well known example. But it doesn’t work in the vertical dimension. Processes that lay down geology over time don’t do fractals.
At sub-atomic scales measuring outlines is meaningless anyway. If you want to talk about particles, matter is almost entirely empty space, and in the vastness of this space sit insanely tiny particles. Even inside the nucleus, it is almost all just a vast empty space. There is no sensible semantic definition of a boundary to measure, certainly not one that is somehow extended in from macroscopic features.
There are several notions of fractal dimension, e.g. Hausdorff dimension, but the point here is that the coastline has Hausdorff dimension equal to e.g. 1.63 > 1 because it is jagged, not that the shape is precisely self-similar as in when you zoom it it will be exactly congruent to a segment of the original.
THAT was amazingly easy for me understand. Thank you! IGNORANCE FOUGHT!
(It wasn’t until just before clicking “reply” that I saw who I was responding to. Good to see you, Chronos! Thanks yet again!)
I came in to say something like @Francis_Vaughan, but he said it very well. An “actual” (i.e. mathematical) fractal does have infinite length, but no actual coastline is that. It is more that a fractal tends to look a lot like a coastline. BTW, to assign a length to something like Norway’s coastline is very misleading for all the reasons stated here. What is the coastline of Denmark? Remember to include Greenland.
There is a famous paper by a guy named “Mandelbrot”…
What I also find fascinating, is that there can be finite area inside an infinite perimeter !! (Like an island with finite area but the coastline length being dependent on the scale used to measure it )