#CoastlineMeasurement #FractalGeometry #AccuracyofMeasurement
Have you ever wondered why it’s so difficult to accurately measure the length of a coastline? 🏖️ It’s a question that has puzzled many people, but the answer lies in the mathematical concept of fractals and the limitations of measurement tools. Let’s dive into the fascinating world of coastline measurement and uncover the reasons behind this perplexing phenomenon.
The Complexity of Coastline Measurement
Coastlines are known for their irregular shapes, with countless nooks, crannies, and intricate details that make them incredibly challenging to measure accurately. In fact, the smaller the scale of measurement, the longer the coastline appears to be. This paradox has led many to question the reliability of coastline measurement and has sparked debates among mathematicians and geographers.
The Fractal Geometry Connection
The key to understanding the difficulty of coastline measurement lies in the concept of fractal geometry. Fractals are complex geometric shapes that exhibit self-similarity at different scales, meaning that the same patterns repeat at smaller and smaller levels of magnification. Coastlines exhibit fractal-like characteristics, with their intricate details repeating at different levels of observation.
Why Small Scale Measurement Matters
When it comes to measuring coastlines, the scale of measurement is crucial. Using a large-scale measurement tool will result in a simplified representation of the coastline, while using a smaller-scale tool will capture more of its intricate details. The smaller the scale of measurement, the longer the coastline appears to be, highlighting the challenges of accurately capturing its true length.
The Mathematical Challenge
The complexity of coastline measurement extends beyond the physical characteristics of coastlines and delves into the realm of mathematics. The coastline paradox, as it is known, arises from the self-similar nature of coastlines, making it impossible to arrive at a single definitive measurement of their length. This poses a significant challenge for cartographers, geographers, and researchers working to accurately represent coastlines on maps and charts.
Addressing Common Misconceptions
In the context of the videos and discussions claiming the inaccuracy of coastline measurements, it’s important to address common misconceptions. While it’s true that the small-scale measurement of coastlines can result in longer measurements, this is not a mathematical problem, but rather a demonstration of the intricacies of coastline geometry. While coastlines exhibit fractal-like characteristics, they are not strictly fractals, which further adds to the complexity of their measurement.
Applying Modern Techniques to Coastline Measurement
Despite the challenges posed by coastline measurement, modern techniques and technologies offer innovative solutions. Geographic Information Systems (GIS) and remote sensing technologies allow for more accurate and detailed mapping of coastlines, taking into account their complex geometric features. These advancements have revolutionized the way we study and represent coastlines, providing valuable insights for environmental conservation, urban planning, and coastal management.
Conclusion: Embracing the Complexity of Coastlines
In conclusion, the measurement of coastlines is a complex and multifaceted endeavor that reflects the intricate nature of these natural formations. The interplay of fractal geometry, scale of measurement, and mathematical challenges adds to the rich tapestry of coastline measurement, inspiring ongoing research and exploration in the field. While the accurate measurement of coastlines may pose challenges, it also opens doors to new perspectives and insights, shaping our understanding of these dynamic and ever-changing landscapes.
In conclusion, the measurement of coastlines is a complex and multifaceted endeavor that reflects the intricate nature of these natural formations. The interplay of fractal geometry, scale of measurement, and mathematical challenges adds to the rich tapestry of coastline measurement, inspiring ongoing research and exploration in the field. While the accurate measurement of coastlines may pose challenges, it also opens doors to new perspectives and insights, shaping our understanding of these dynamic and ever-changing landscapes.
This is on a problem/paradox when you introduce precision for no practical purpose. When we ask for the length of a coastline we are asking for practical purposes, e.g. if I were to motor a boat along the coastline, how far would this be, so for a given speed how long would this take? Likewise, if I were to walk the coastline, how far would this be, so for a given speed how long would this take? The answers are different in these two cases, and this seems fairly intuitive, in the same way ‘as the crow flies’ is an intuitive concept.
Sure, things get more jagged the closer you look, like looking at a sharp knife edge under a microscope, but there is little or no practical purpose for this in terms of determining a ‘useful’ distance.
I think the whole concept is more useful in terms of uncovering intuitions at play when discussing distances. By reducing it to absurdity we realise there is actually some human-related subjectivity at play.
The length of the coast changes every time the water moves back and forth with the tide and the waves and stuff.
> a coastline is not a fractal
True. A coastline is not a mathematically exact fractal, but it is a practical fractal, and has many of a fractal’s characteristics – characteristics such as self-similarity over extended scale ranges. It is this property that makes coastlines impossible to measure accurately.
A coastline *is* indeed fractal. You get different results when you measure it with different rulers. [That is fractal](https://en.m.wikipedia.org/wiki/Fractal).
It is not self-similar at smaller scales (patterns don’t repeat as infinitum), but that is not a requirement for a fractal. Just check out the Mandelbrot set, it doesn’t repeat itself in smaller scales.
In practice the problem is no in measuring with infinitely small rulers. But the results will be quite different if you measure with a 1 km, 100 m or 10 m ruler. And if the coastline is said to be 2500 km long, you’d want to know if that was measured with a 1 km or 10 m ruler.
There are similar challenges with the size of lakes and even the number or lakes and islands. How big must an island be to be counted? When you set that limit you must also include how you measure the area.
Interestingly there are similar problems in eg science such as chemistry. If you have a graph with peaks placed on a baseline (eg a [chromatogram](https://en.m.wikipedia.org/wiki/Resolution_(chromatography))), how do you measure the are under the peaks if the baseline is not perfectly flat? You must define a way to separate what is a “peak” from what is just a bump in the baseline.
In both cases scientists have found many ways to do the analysis and what is the best method usually depends on the problem. That’s why it is so important for scientist to know what method was used and exactly how it was performed.
Hold up your fingers in front of your face with them held together.
Trace along the from the bottom of your pinky to the bottom of your index finger.
That’s one interval of measure.
Now do the same, but trace along the tiny gaps between your fingers.
That’s a different, more precise interval of measure.
With the larger interval, the small gaps between your fingers are too large to be measurable, with the smaller interval, the gaps becomes measurable and therefore add the total distance between each of your fingers.
That’s the issue with measuring coastline’s, the more precision you try to use the smaller features you have to measure and the greater total “distance” you get.
A coastline has the same property that makes fractals problematic. The finer the details you measure, the longer the coastline will appear. Of course you won’t measure every pebble, but are you measuring in 1 meter intervals? 10 meter intervals? You’ll get very different answers.
Okay but where do you stop between “measure by miles” and “measure by feet” and “measure by inches” and “measure by Planck lengths” if the resulting measurements are several orders of magnitude different? If you’ve been hired to give an accurate length of the coastline, what do you do?
This is the Problem.
Why isn’t the coastline a fractal according to you?
I guess you can argue that if your unit of measurement is a Planck length, you may get an accurate result as it’s kind of a “real world limit”. But mathematically it doesn’t work out that way. It ultimately is a fractal problem so on paper you can always increase accuracy. You can curve your Planck length to the next one that is slightly rotated, then introduce the half Planck length on paper to measure the curve. Mathematically that’s not a problem.
I just feel like the whole paradox is just saying it’s a fractal problem, and that in practice the accuracy should always be mentioned with a coastline measurement so they can be compared accurately.
Don’t coast lines change with tides? Wouldnt when you measure play a role?
Imagine a beach with a straight shore line. You and some friends decide to dig a channel perpendicular from the shore inland to a hold you dug. The channel is about 50 feet long and it fills with water as it also fills your hole. Did you increase the shore line by 100 feet or so? Why or why not? What if it was naturally forming? What of a bay with a very narrow inlet?
The question is not only do you count it or not, but who gets to make that decision?
The coastline paradox isn’t necessarily stating that you “can’t accurately measure a coastline” because making that statement would depend on a definition of “accurately.” Even if your definition of “accurately” was on the sub-atomic scale, then measuring a coastline would be difficult but, in principle, not impossible. (Though this is true about measuring anything.)
Instead, the coastline paradox says that as your definition of “accurately” changes, the resulting measure of the length of the coastline will change in unexpected ways. It’s not a paradox to say that greater accuracy will change the measurement in some way. We might expect that there is some “true” answer that inaccurate measures will only approximate. Sometimes they will be too high, and sometimes they will be too low. What’s unexpected about coastlines is that increasing accuracy will almost always *increase* the measurement. This has to be taken into account in a couple of ways:
* When comparing coastline measurements, it’s important to ensure they were both taken with the same “ruler”
* Unlike a case with symmetric noise, it’s harder to use statistical tricks to glean the “true” measure from several noisy measures. If the noisy measurements lay both above and below the most accurate measurement, you could take a bunch of noisy measurements and average them to get a good idea of the true measurement. This doesn’t work with coastlines, so that’s the sense in which you “can’t accurately measure a coastline.”
It seems that once you get down to Planck units you ought to be able to come up with a reasonable answer, since any smaller unit is meaningless.
also due to coastal erosion and sedimentary accumulation its an ongoing dynamic process…so there isn’t something static that can be measured either.
Why wouldn’t you measure down to every pebble? I thought you wanted to know how long the coastline was. The coastline is fractal right up to the point where you go to the quantum realm and then it’s just uncertainty.
that doesn’t mean we can’t come up with a number that’s useful, so long as everyone agrees on the ruler. But even then the number would change over time.
The real point here is that there is no such thing as a line, or a curve or an exact measurement. Those are abstract concepts that have no real world direct correlation.
1. Coastlines are not static. Anything you do measure, will be instantly invalid to a certain degree
2. The length of something depends on how you measure it. The longer your measuring stick, the harder it is to approximate curves. You cant measure the perimeter of a circle with a straight line.
1. If you use your ruler to measure a diamond shape from the circle, you will get one length. Reduce your ruler, and now measure the octagon, you will get a new longer length despite the circle not changing.
Imaging a completely flat coastline of 100km. Simple.
Now take that same coastline except added with a large square notch of 1km on three sides. That coastline is now 100km + 1km + 1km = 102km. (Two of the three sides represent new length while one side is part of the original 100km but pushed “inward”). Still fairly simple. And a 1km “notch” is fairly significant, you could build a lot of new ocean front homes, harbors etc on the extra 2km of coast.
Now imagine the same original flat 100km coastline but I cut a very narrow creek 100km long but negligibly wide, say 10cm but very deep so ocean water always fills this creek. Is that coastline 100km + 100km + 100km = 300km? Kind of. But is this coastline really meaningfully 3x the original? Obviously not!
If I’m building ocean front property along this creek, it wouldn’t work as such for people especially those 100km away from the “main” coast.
A lot of people would completely discount this creek as meaningful additional coast.
The question then is what is meaningful to consider. That is not easy to answer. 1km “wide” notch seems meaningful but not 10cm. So where is the dividing line?
Most coastlines are full of these “notches” that technically add length but how meaningful are these notches?