I respectfully submit that Mandelbrot probably did not really expect infinity. That might be the answer for a mathematical fractal, but for a physical thing such as an island, the fractal self-similarity breaks down when you get down the scale of individual atoms (or before that). A completely separate issue is that the length of a coastline cannot even *have* a definite value, even in principle. Say you define the coast as the outer-most point with an altitude of the mean sea level at that point. Every time a grain of sand shifts by a fraction of an angstrom, which is constantly, the length of the "coastline" changes. One cannot measure something with a greater precision than the thing possesses. Of course the 'coast' used to define territorial waters and economic zones is merely defined by official charts with special rules for islands and inlets and such.
That's a fair and important distinction. The "infinite" answer works as a mathematical idealisation, but you're right: physical coastlines have a natural lower bound where self-similarity breaks down. I was perhaps too quick to present it as the answer rather than as a useful way of grasping the paradox. Well taken.
This is my physicist's take on it. It is a classic measurement problem.
The problem is one of definition - in defining a perimeter you need to include time, physical constraints and precision. Time is the most tricky: sea level varies continuously so the precision can never be better that the precision of the sea level measurement. There is constant erosion and deposition, but that is probably handleable at a precision of 1 month. The physical constraint is that what is land or sea is not clear cut - you need to make a decision. These impose a limit on the precision that could ever be obtained. Forget about atom by atom, just consider that a you need a way to decide whether each grain of sand is in or out. What is certain is that physical and temporal constraints make the length measurable to a precision determined by the way in which it is measured. 1m precision would probably mean measuring all points along the sea edge simultaneously within a 10 minute interval.
And back to the post, which is not about reality, rather about human perception. . The question is, what is the length as measured on a map. The map is a real object and the border length can be measured to the precision at which the map is printed. But obviously there is not just one map. Choose a different map and you will get a different answer. Each individual map however does deliver a precise finite answer, demonstrating that they are artifacts which we have created
The physicist's framing is really useful here. The point about maps as artifacts (each of them providing a precise but constructed answer) is a clean way of putting something I didn't explicitly mention.
It goes much further into the mathematics than I did, and probably much further than I could (even if I tried). The Slartibartfast opening is difficult to beat.
The only real problem is that there is no degree upon definition of what the „coast line“ actually is.
There is no legitimate reason to choose to model it as a fractal curve over a continuous and differentiable one and in the later case there is a well defined answer and the tape measuring process will converge.
Just another case of a bad model leading to nonsense answers.
I respectfully submit that Mandelbrot probably did not really expect infinity. That might be the answer for a mathematical fractal, but for a physical thing such as an island, the fractal self-similarity breaks down when you get down the scale of individual atoms (or before that). A completely separate issue is that the length of a coastline cannot even *have* a definite value, even in principle. Say you define the coast as the outer-most point with an altitude of the mean sea level at that point. Every time a grain of sand shifts by a fraction of an angstrom, which is constantly, the length of the "coastline" changes. One cannot measure something with a greater precision than the thing possesses. Of course the 'coast' used to define territorial waters and economic zones is merely defined by official charts with special rules for islands and inlets and such.
That's a fair and important distinction. The "infinite" answer works as a mathematical idealisation, but you're right: physical coastlines have a natural lower bound where self-similarity breaks down. I was perhaps too quick to present it as the answer rather than as a useful way of grasping the paradox. Well taken.
This is my physicist's take on it. It is a classic measurement problem.
The problem is one of definition - in defining a perimeter you need to include time, physical constraints and precision. Time is the most tricky: sea level varies continuously so the precision can never be better that the precision of the sea level measurement. There is constant erosion and deposition, but that is probably handleable at a precision of 1 month. The physical constraint is that what is land or sea is not clear cut - you need to make a decision. These impose a limit on the precision that could ever be obtained. Forget about atom by atom, just consider that a you need a way to decide whether each grain of sand is in or out. What is certain is that physical and temporal constraints make the length measurable to a precision determined by the way in which it is measured. 1m precision would probably mean measuring all points along the sea edge simultaneously within a 10 minute interval.
And back to the post, which is not about reality, rather about human perception. . The question is, what is the length as measured on a map. The map is a real object and the border length can be measured to the precision at which the map is printed. But obviously there is not just one map. Choose a different map and you will get a different answer. Each individual map however does deliver a precise finite answer, demonstrating that they are artifacts which we have created
The physicist's framing is really useful here. The point about maps as artifacts (each of them providing a precise but constructed answer) is a clean way of putting something I didn't explicitly mention.
You wrote that Richardson wrote in 1961… but he died in 1953 (see https://en.wikipedia.org/wiki/Lewis_Fry_Richardson)
Thanks for the catch! You're absolutely right: Richardson died in 1953, and the article was published posthumously in 1961.
I've already corrected the text to make that clear.
Sorry. Wasn’t trying to be adversarial.
Fry Richardson is a hero of mine
See https://craigavad.org/2026/01/09/where-do-weather-forecasts-come-from-and-why-are-they-sometimes-so-wrong/
No! The comment was greatly appreciated! I rather be corrected in time than just keep going with the mistake. :)
Here’s a discussion of this topic that I wrote a few weeks ago
https://craigavad.org/2026/05/09/how-long-is-a-coastline-well-it-all-depends/
Thanks for sharing this, it is a great piece!
It goes much further into the mathematics than I did, and probably much further than I could (even if I tried). The Slartibartfast opening is difficult to beat.
The only real problem is that there is no degree upon definition of what the „coast line“ actually is.
There is no legitimate reason to choose to model it as a fractal curve over a continuous and differentiable one and in the later case there is a well defined answer and the tape measuring process will converge.
Just another case of a bad model leading to nonsense answers.