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No One Knows How Long the US Coastline Is (discovermagazine.com)

Posted by msmash on from the closer-look dept.

How long is the U.S. coastline? It's a straightforward question, and one that's important for scientists and government agencies alike. From a report: The U.S. Geological Survey could give you an answer, too, but I'm going to tell you right now that it's wrong. In fact, no one could give you the right answer, and if you look around, you'll find a number of estimations that differ by seemingly improbable amounts. One government report lists the number as 12,383 miles. The same report admits that a different government agency says the figure is actually 88,612 miles. That's an almost eight-fold disparity for a fact that seems simple to obtain. We all know how to use a ruler, right?

Well, we all know how to measure a straight line, but what about a curve? And what if that curve has curves? The crux of the problem comes down to geometry, and the fundamentally uneven nature of coastlines. Though the border between land and sea may look fairly straight when seen from far away, they're anything but. Coastlines jut and dip, curve and cut, and each deviation from a straight line adds distance. Some of these features are massive, like bays, while others are miniscule.

26 of 189 comments (clear)

  1. It's infinite. by jd · · Score: 5, Insightful

    Coastlines are fractal and have 1.4 dimensions. This does mean Cthulhu could break out at any moment.

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    1. Re:It's infinite. by Obfuscant · · Score: 5, Insightful

      Coastlines are fractal and have 1.4 dimensions.

      And being fractal makes the initial question of "how long" meaningless. To what resolution? If you look at the molecular level, it is nearly infinite. If you look at the sand grain level, less so. If you look at one foot intervals, even less so.

      That of course leaves the question of how you define the coastline. It's not just bays and similar features that create issues, but AT WHAT TIDE LEVEL? Do you define the "coast" as being at the mean high high water (MHHW), mean sea level (MSL), high water line, or where? Do you count both sides of the Outer Banks in North Carolina, both sides and the inner edge of the sounds behind them, or just the outer edge?

      The question is also meaningless because it changes nothing. Nothing changes if you say that the US coast is 1,000,000km or 200,000km or 1km. If you're estimating how much it will cost to install coastal protection you will measure how long the protection measure is, not how long the coast is behind it.

      So, must be a slow news day at /.

    2. Re:It's infinite. by Anonymous Coward · · Score: 2, Informative

      It is certainly wrong to say it is 1km. There is a lower bound to the length of a coast line. It can not be shorter than the longest straight line distance between any two points on the coast line.

    3. Re:It's infinite. by sexconker · · Score: 3, Interesting

      It can not be shorter than the longest straight line distance between any two points on the coast line.

      Of course it can.

      A: Not all land between points of coastline is coastline. We don't even have a solid definition of coastline to start measuring.

      B: In strict 3 space, measuring coastline makes little sense. Lines are 2D, the coast is 3D, and you first need strict rules for tracing the path before you can measure it. How do you handle a cliff on a beach? What if it juts out over the water? When considering 2D projections, it makes a bit more sense. But "the longest straight line distance between any two points on the coast line" in such a projection is infinite. Consider "the longest straight line distance between" your dick and your ass based on a 2D projection (flat map) of Earth. That line would simply wrap around the Earth forever in an infinitely tight spiral. Or consider the simpler scenario of wrapping around Earth once. That's "the longest straight line distance", yet we can show the actual distance is shorter, even if we don't know the true position of either point. Start by drawing a bounding box around each point with whatever accuracy/precision we want / can achieve, then measuring the distance between the outer edges of the bounding boxes. This gives us an upper bound for the actual distance. (You can simply draw a line segment between the points, then draw lines perpendicular to that segment at each point, and find where those lines intersect the bounding boxes to determine the "outer edge" of each bounding box.)

    4. Re:It's infinite. by ath1901 · · Score: 2

      Don't forget waves or kids playing on the beach.

      I'd argue the length of the coastline is exactly what the surveyors office says it is. It is a matter of definition, not measurement. There is no coastline on the beach, only on maps.

      While I'm at it, if a tree falls in the forest, there is a sound. The standard definition of sound does not require a human listener. The sound of one hand clapping? I can clap with one hand. It sounds almost like clapping with two hands.

      Fractals are cool. Cooler than coastlines.

    5. Re:It's infinite. by Dragonslicer · · Score: 2

      It can not be shorter than the longest straight line distance between any two points on the coast line.

      Of course it can.

      A: Not all land between points of coastline is coastline. We don't even have a solid definition of coastline to start measuring.

      Instead of a straight line through three-dimensional space, the shortest possible length would be the length of the shortest line between two points (for the surface of a sphere, I think it's part of a Great Circle). Any way that the coastline deviates from that line must increase the length, since by definition that line is the shortest possible line connecting those two points. You also have to add the requirement (which the GP probably just left unstated) that you measure each contiguous coast separately; obviously Atlantic and Pacific coasts must be separate, maybe also the Gulf of Mexico. For example, the shortest possible length of the Atlantic coast is the "straight" distance from the northern-most point on the coast to the southern-most point.

      But "the longest straight line distance between any two points on the coast line" in such a projection is infinite. Consider "the longest straight line distance between" your dick and your ass based on a 2D projection (flat map) of Earth. That line would simply wrap around the Earth forever in an infinitely tight spiral. Or consider the simpler scenario of wrapping around Earth once. That's "the longest straight line distance", yet we can show the actual distance is shorter, even if we don't know the true position of either point. Start by drawing a bounding box around each point with whatever accuracy/precision we want / can achieve, then measuring the distance between the outer edges of the bounding boxes. This gives us an upper bound for the actual distance. (You can simply draw a line segment between the points, then draw lines perpendicular to that segment at each point, and find where those lines intersect the bounding boxes to determine the "outer edge" of each bounding box.)

      You're misinterpreting the GP. By "the longest straight line distance between any two points on the coast line", I believe the GP meant the longest of the shortest-length lines between every pair of points. In other words, the shortest line between the two farthest points.

    6. Re:It's infinite. by vux984 · · Score: 2

      "The sound of one hand clapping? I can clap with one hand."

      The action you are likely performing is not generally recognized as 'clapping'. So the sound it makes is irrelevant.

      "The standard definition of sound does not require a human listener."

      What so-called "standard definition of sound" are you using? I just checked two separate dictionaries and both of them defined sound in terms of sensing vibrations on the auditory structures of an ear as their first definition of sound (noun).

      "I'd argue the length of the coastline is exactly what the surveyors office says it is. It is a matter of definition, not measurement. "

      Clearly there are wrong answers though. 1 mile is clearly wrong. Equally clearly there is no limit to the number of correct answers, but it depends on methodology -- so first you set a methodology, then you make your measurements according to that methodology. The stricter and more well defined the method, the stricter and more restricted the range of values that will satisfy it.

      Furthest extent of land, at low tide, on the vernal equinox, 2020, to a resolution of 5 meters. With suitable definitions to decide when and where to leap over water to reach islands, cross stream and river mouths, whether to cross over the opening to a bay or inlet, or to follow it around based on the size of the opening, salinity of the enclosed body, depth at the opening, etc.

    7. Re:It's infinite. by Immerman · · Score: 2

      Ah, but atoms are composed of quantum particles, which strictly speaking have neither a definite size nor position, but rather a wave-function that extends to the farthest reaches of the universe with non-zero probability. The circumference of even a single electron is thus exactly equal to that of the entire universe, which may or may not be infinite.

      Unless you impose some completely arbitrary cut off point (e.g. there's a 99.99999999% chance that the atom will resolve as being within this area when its wavefunction momentarily collapses), *all* truly accurate measurements of size will be infinite.

      --
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    8. Re:It's infinite. by dryeo · · Score: 2

      And where is the NW tip of Washington State? Point Roberts? Then you have to consider the coast enters Canada for a while before rejoining Washington (Point Roberts is only accessible by land from Canada). One of the San Juan Islands? Then you have a length of no coast line.

      --
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    9. Re:It's infinite. by jrumney · · Score: 2

      The longest point on the west coast would start in Alaska, and the GP's point then becomes very relevant.

  2. Where do you draw the line? by jellomizer · · Score: 2

    We have rivers that fan out into the ocean. How far up the river do we go until we are no longer at the coastline. I want to sell my home to be on the coast even though it is hundreds of miles away from the ocean and thousands of feet above sea level, however there is a little stream next to my home, which leads to a creak, that leads to a river, which leads to a larger river then goes into the ocean.

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  3. Someone just discovered fractals! by Anonymous Coward · · Score: 5, Informative

    The coastline problem is literally a textbook example of fractals. The "size" of the object (perimeter, area, etc.) depends on the scale at which it is measured (the size of the "ruler" one uses). A coastline has finer and finer features as one zooms in, so the overall length/perimeter one computes is larger and larger as one uses a finer resolution. For a perfect/mathematical fractal, the coastline could actually be infinite in length.

    This is absolutely interesting. But also not new. Everyone with a passing familiarity with measuring coastlines knows about this issue.

    1. Re: Someone just discovered fractals! by bill_mcgonigle · · Score: 4, Funny

      Fifth-grade news for nerds.

      --
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    2. Re:Someone just discovered fractals! by Dast · · Score: 2

      And if you go "around" an atom, and at what state of excitement do you measure it in?

      --

      This sig is false.

    3. Re:Someone just discovered fractals! by Immerman · · Score: 2

      Except quarks, like all quarticles, don't have a well-defined circumference. Instead they have a quantum wave function that extends to infinity with non-zero probability. To make any non-infinite measurement you must first impose an arbitrary cut-off point on that probability.

      --
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  4. Mandelbrot beat them by dabadab · · Score: 5, Informative

    Mandelbrot's* well-known "How Long Is the Coast of Britain?" article (published in 1967) starts with this question - and it goes on to discuss self-similar curves that are a type of fractals.

    *: yes, he is the guy who came up with the Mandelbrot fractal

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  5. It seems straightforward, but it's not. by arcade · · Score: 2

    This comes down to the resolution used. Think fractals. What's your minimum measurement unit? 10km? 1km? 100m? 10m? 1m? 10cm? 1cm? 1mm? Smaller?

    The smaller the unit of measurement, the larger the coastline, as you can cover smaller and smaller details.

    Then it's the question of where to place the coastline. High tide? Low tide? Middle? What about the "type of coastline"? It seems obvious if it's rock .. but what if it's sand? Where do you put the line?

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  6. Coastline Paradox by Issildur03 · · Score: 2

    There's a famous paper by Mandelbrot on the question of coastlines: "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension".

    In short, coastlines are fractal and effectively have infinite length.

    https://en.wikipedia.org/wiki/...

  7. Non-Scientific Language Question by holophrastic · · Score: 4, Insightful

    As usual, the question isn't specific enough to have an answer. Therefore, people think it's difficult to answer.

    The answer is very simple, ask for what you want.

    Maybe it's how far a ship would need to travel to get to any point -- ships take gentle curves. Maybe it's how long would it take to see it all on-foot -- humans take 1-yard-long straight lines.

    Another stupid question: which came first, the chicken or the egg? Again, a language question. Define the word "egg" and it's easy.

    If "egg" is any egg, then dinosaurs had eggs long before chickens.
    If "egg" is "chicken egg", then define "chicken egg".
    If "chicken egg" is an egg laid by a chicken, then the chicken came first, by your own definition.
    If "chicken egg" is an egg from which a chicken hatches, then the egg came first, by your own definition.

    Stupid questions are questions that only exist because of the manner in which you formed the question itself.

    As my associate likes to say: "the answer is 6. there. now, what are you going to do with it?"

    Decide what you actually want to know -- that means how you're going to use the information. Ask that question. There won't be anything crazy about it.

    How long is a shoreline? Tide in or out? At what depth -- does a puddle ruin everything? Footprints? What about waves lapping on the beach?

    Hey, what about the birth from a river?
    Do we include the entire river?
    Where does the river become the ocean?
    Right, that's easy, it's where the fresh water changes to salt water.
    How do you want me to measure that, given that the river's "fresh" water has dirt in it.
    At what level of salinity does your version of "fresh" water become your version of "salt" water?

    Arbitrary trivia is arbitrary from the start. Decide what you're doing. Are you designing battleships, or water-ways for salmon? I promise, the navy doesn't agree with the fish, and they don't need to.

  8. Easy, just make a new unit! by WilliamGeorge · · Score: 5, Funny

    The length of the US coastline is equal to... 1 US Coastline. Good luck with conversions to metric, though - we hate that here in the States ;)

    --
    William George
    1. Re:Easy, just make a new unit! by Aighearach · · Score: 2

      How many laps around the library of congress building is that?

      Exactly 1, the Library of Congress is already inside the land border!

  9. Re:+/- 12,383 miles by Shikaku · · Score: 2

    https://en.wikipedia.org/wiki/...

    The best answer I can give is >0.

  10. Re:+/- 12,383 miles by Joce640k · · Score: 4, Informative

    Isn't "coastline" the very first example they give in Fractals 101?

    It certainly was back in the 1980s when I was at school but here it is as front page news on Slashdot. FFS.

    Now get off my lawn.

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  11. Re:+/- 12,383 miles by Actually,+I+do+RTFA · · Score: 3, Funny

    No, coastlines are the second example. Snowflakes are the first.

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  12. A fractal measurement challenge for you all. by az-saguaro · · Score: 3, Interesting

    Read the article. It turns out that this is not a scientific journal or communique, not a technical report or abstract, not detailed information written by experts for experts. It is a general interest blog discussing items of scientific interest. It makes no claim to be novel or current. Furthermore, the article is not about the coastline per se. The second half of this very brief article discusses fractals and the relevant concepts about measuring length with respect to scale. While many people on Slashdot know this subject and its implications, many other people out there might not. So, as an informative article for laymen, it is perfectly reasonable for the forum it was published in. Even by those standards, it was still brief and naive, but if you have never encountered the concept before, it was a reasonable enough introduction to the idea. It does make one wonder though why it was posted on Slashdot, being as basic as it is.

    However, the post has elicited many comments, and now, a challenge. For those who say the coast length is moot, well no, not really so. True, we can quibble the details, and the coastline is dynamic rather than static, and it all depends on length of your ruler. But that does not invalidate that the measure is important based on context. Examples:

    - A boat is tasked to follow the coastline, maintaining a tangent or parallel course at all times, 200 meters of the shoreline. The boat has an aft screw, a certain length (e.g. 60 meters), and a certain rudder turning radius. Assume that the boat is laying cable and furthermore that it must to perform to perfect efficiency so that it can maximize the amount of cable it carries rather than excess fuel. How many kilometers will it ply, how many kilometers of cable are needed, how much fuel in its tanks?

    - A coastal highway is being built 100 meters back from the high tide waterline. The road will be 10 meters wide. It will go from town A to town B, 20 kilometers from each other as the crow flies. Concrete and asphalt must be specified. How much of each are needed to complete the project?

    - Recent seismic or volcanic activity has altered a coastline, creating a new large rocky mass along the coastline near an urban area. The altered contour creates new wave or current or tidal patterns that threaten erosion to coastline. How much rock, timber, concrete, or whatever will be needed to create a new seawall or jetty to protect human structures? Or, based on the metrics of those waves and tides, what will be the erosion rate along nearby beaches?

    In each example, the length of the coastline has a tangible meaning. A rowboat that wants to follow the coast 10 meters away will have a different measure than an oil tanker following 2 kilometers away, but for the problems presented, their relative lengths matter. Based on the physical scales of each problem, the shorter rulers with longer coastlines, and the longer rulers measuring shorter coastlines must all be filtered out to yield the Goldilocks answer. As Obfuscant stated in a response above, "If you're estimating how much it will cost to install coastal protection you will measure how long the protection measure is, not how long the coast is behind it."

    So, here is the challenge or invitation. Please respond below with realistic scenarios of a scientific, mathematical, engineering, or commercial nature where the length of the coast does matter for the problem or project at hand. They could be hypothetical or imagined, or they could be real world examples of prior endeavors or ordinary practices.

    Post here . . . . . .

  13. Re:+/- 12,383 miles by GrpA · · Score: 2

    Australia defined a standard about how big a feature had to be to be regarded as "coastline" - same with rivers mouths and the like. Once this was defined, then anything outside of this definition no longer counted, and the coastline was measured quite accurately, in a way that if measured in the future, could identify whether the coastline was shrinking or expanding.

    eg: https://ebrary.net/10179/manag...

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