# Is the coastline paradox only applicable to physical coastlines? **URL:** \ **Category:** Factual Questions\ **Created:** [June 22, 2021, 4:27pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844 "2021-06-22T16:27:12Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![k9bfriender](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/k9bfriender/32/3283_2.png) [@k9bfriender](https://boards.straightdope.com/u/k9bfriender)\ **Post date:** [June 22, 2021, 9:52pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/21 "2021-06-22T21:52:04Z")
As mentioned, in the real world, you are never going to get infinites out of this. If nothing else, it will break down once you get to the molecular or atomic level. However, purely mathematical objects don’t have this problem. The perimeter of the Mandelbrot set is infinite yet the area is very finite. ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [June 22, 2021, 10:00pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/22 "2021-06-22T22:00:38Z")
Veritasium covered this one. He’s one of the good guys when it comes to breaking science (and math) down into bite size pieces. [![](https://img.youtube.com/vi/I_rw-AJqpCM/maxresdefault.jpg "What Is The Coastline Paradox?") ](https://www.youtube.com/watch?v=I_rw-AJqpCM) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 22, 2021, 10:34pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/23 "2021-06-22T22:34:37Z")
> [@Hari\_Seldon](#): > > BTW, to assign a length to something like Norway’s coastline is very misleading for all the reasons stated here. If we start from an image of the coastline in question and plot the length evaluated at several length scales (obviously not microscopic) and estimate its Hausdorff dimension as a best fit, ISTM that is some meaningful and not misleading data about the geometry of said coastline, and does tell you something about how to approach defining its “length”, like Mandelbrot’s approach where there is a well-defined function telling you how long it is when measuring at scale ε. And if we get 1.52 for Norway but 1.25 for Britain, that is a significant difference. > [@Little\_Nemo](#): > > If you consider the line as a physical object then it will have width. That means at some point you will be able to draw a line that overlaps the coastline (because the variations in the coastline will now be smaller than the width of the line) and at that point, the line is complete and you will be stopping before you reach infinity. That is exactly what Mandelbrot did, as I understand it. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [June 22, 2021, 10:40pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/24 "2021-06-22T22:40:16Z")
> [@Darren\_Garrison](#): > > 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. I think in order for the paradox to work, you have to treat the drawn line as if it is a one-dimensional line that’s curving around in a two-dimensional plane. If you consider the line as a physical object then it will have width. That means at some point you will be able to draw a line that overlaps the coastline (because the variations in the coastline will now be smaller than the width of the line) and at that point, the line is complete and you will be stopping before you reach infinity. And I feel if you’re treating the line as an ideal in order to produce the paradox, you can’t then turn around and treat the same line as an actual physical object. ---
**Author:** ![Darren\_Garrison](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/darren_garrison/32/92_2.png) [@Darren\_Garrison](https://boards.straightdope.com/u/Darren_Garrison)\ **Post date:** [June 22, 2021, 10:56pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/25 "2021-06-22T22:56:01Z")
> [@Little\_Nemo](#): > > And I feel if you’re treating the line as an ideal in order to produce the paradox, you can’t then turn around and treat the same line as an actual physical object. I’m treating the line as an actual physical object because the OP was specifically asking about an actual physical object. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [June 22, 2021, 11:21pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/26 "2021-06-22T23:21:25Z")
> [@Darren\_Garrison](#): > > I’m treating the line as an actual physical object because the OP was specifically asking about an actual physical object. Yes, but that’s the distinction I’m talking about. If you drew a line on a paper then the line itself, as a physical object, would not be infinite in length. But if you then measured that physical line with an ideal line, the ideal line that measured the physical line would be infinite in length because it is measuring the variations in the physical line. ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [June 22, 2021, 11:32pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/27 "2021-06-22T23:32:52Z")
People keep saying “infinite” in this thread. Would it truly be infinite? It seems to me once you are measuring Norway’s coast using a Planck Length scale you are done getting smaller and, while you might get an almighty number out of it, it will not be infinite. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [June 23, 2021, 1:15am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/28 "2021-06-23T01:15:40Z")
> [@Whack-a-Mole](#): > > Would it truly be infinite? It seems to me once you are measuring Norway’s coast using a Planck Length scale you are done getting smaller and, while you might get an almighty number out of it, it will not be infinite. If you’re one of those proverbial primitives who can only count to three, then it will be as infinite as it can get. To demonstrate (more simply than the arcane [Mandelbrot Biscuit](https://toriavey.com/images/2011/01/TOA67_5.jpg)) how region of finite area can have an infinite perimeter, consider the simple snowflake curve (which seems to appear in the video that @Joey_P linked above). > **[Koch snowflake](https://en.wikipedia.org/wiki/Koch_snowflake)** > > The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch. > The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is forme... Successive iterations of this curve have steadily increasing area yet never exceed a finite limit, a little larger than the initial triangle. Yet successive iterations have increasing perimeters that increase without limit. (ETA: This is indeed simpler than anything like the Mandelbrot Set. I wrote a recursive program to draw these snowflake curves, and it proved to be a trivially simple exercise.) ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [June 23, 2021, 1:17am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/29 "2021-06-23T01:17:36Z")
If you do that without end then sure. But in our universe you will bump up against the Planck limit and will not be able to go any smaller. Eventually, you simply cannot make a smaller triangle in the real world. So, you have a finite set which seems to me you end up with a finite result. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [June 23, 2021, 1:18am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/30 "2021-06-23T01:18:41Z")
> [@Whack-a-Mole](#): > > But in our universe . . . Wrong universe. 🙂 ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [June 23, 2021, 1:25am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/31 "2021-06-23T01:25:01Z")
You mean the one where in Norway a Møøse once bit my sister? No realli! She was Karving her initials on the møøse with the sharpened end of an interspace tøøthbrush given her by Svenge - her brother-in-law - an Oslo dentist and star of many Norwegian møvies: “The Høt Hands of an Oslo Dentist”, “Fillings of Passion”, “The Huge Mølars of Horst Nordfink” 😉 ---
**Author:** ![dtilque](https://avatars.discourse-cdn.com/v4/letter/d/d6d6ee/32.png) [@dtilque](https://boards.straightdope.com/u/dtilque)\ **Post date:** [June 23, 2021, 4:40am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/32 "2021-06-23T04:40:41Z")
> [@Whack-a-Mole](#): > > You mean the one where in Norway a Møøse once bit my sister? That Møøse was just pinin’ for the fjords. ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [June 23, 2021, 4:52am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/33 "2021-06-23T04:52:25Z")
👍 (some extra text for Discord) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 23, 2021, 9:20am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/34 "2021-06-23T09:20:38Z")
The OP asked whether the coastline paradox only applies to physical coastlines. Again, the right answer is that it applies to both physical coastlines and mathematical fractals, and in the first case any Planck-level stuff is not relevant. The formula is that the (1-dimensional) length of the coast of Norway, as measured by rulers of length δ, is proportional to δ−0.52, at least for meaningful values of δ like 0.5km–100km. It is probably not relevant to worry about whether the coastline (or space-time) is smooth on the Planck scale or not, if our focus is on what the coastline looks like sketched out on a piece of paper. (The “infinite issues” come when taking the limit of an expression like δ−0.52 as δ tends to zero. In that case, the number gets bigger and bigger without limit. One way to get a finite measure for the size of such a fractal is to use fractional-dimensional Hausdorff measure, but then we are not dealing with normal “length” any longer but rather with something intermediate between 1-dimensional length and 2-dimensional area.) ---
**Author:** ![bob\_2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bob_2/32/3341_2.png) [@bob\_2](https://boards.straightdope.com/u/bob_2)\ **Post date:** [June 23, 2021, 10:45am UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/35 "2021-06-23T10:45:21Z")
I blame the mice. And Slartibartfast of course. ---
**Author:** ![am77494](https://avatars.discourse-cdn.com/v4/letter/a/d2c977/32.png) [@am77494](https://boards.straightdope.com/u/am77494)\ **Post date:** [June 23, 2021, 12:02pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/36 "2021-06-23T12:02:20Z")
Are there **non-living** natural fractals with finite volume, but surface area approaching infinity ? I know about the blood vessels in our lungs / many other organs are like fractals; also know about trees / branches… ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 23, 2021, 12:14pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/37 "2021-06-23T12:14:50Z")
IIRC, interstellar dust grains tend to have (approximately) fractal shapes, resulting from how they’re formed, with a few atoms here and there colliding with the seed of the grain and sticking. ---
**Author:** ![k9bfriender](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/k9bfriender/32/3283_2.png) [@k9bfriender](https://boards.straightdope.com/u/k9bfriender)\ **Post date:** [June 23, 2021, 12:17pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/38 "2021-06-23T12:17:09Z")
Seems as though watersheds and river deltas would be fairly fractal in nature. ---
**Author:** ![am77494](https://avatars.discourse-cdn.com/v4/letter/a/d2c977/32.png) [@am77494](https://boards.straightdope.com/u/am77494)\ **Post date:** [June 23, 2021, 12:21pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/39 "2021-06-23T12:21:46Z")
Thanks @k9bfriender - should have thought of them. Was hoping for something more impressive like the Menger Sponge. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [June 23, 2021, 12:24pm UTC](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844/40 "2021-06-23T12:24:29Z")
They are. From memory they span about 3 levels of self similarity before the pattern breaks down. Possibly one of the nicest natural examples are craters on moons and planets. The range over which self similarity extends is huge. [Previous page](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844.md?page=1) [Next page](https://boards.straightdope.com/t/is-the-coastline-paradox-only-applicable-to-physical-coastlines/944844.md?page=3)