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  • I've been driving along Australia's famous Great Ocean Road.

  • And I'm stopped here near the Twelve Apostles, which are these big sandstone bluffs.

  • Actually, there's only eight of them left because the others have eroded over time.

  • And erosion is really what's given us this coastline the way it looks now.

  • So that brings to mind a question for me.

  • Which is "How long is the Australian coast line?"

  • Well, if you were to measure it out in lengths of 500 kilometers,

  • You would find that it's about 12 and a half thousand kilometers long.

  • But the CIA World Factbook puts the figure at more than double that. Over 25,700 kilometers.

  • But how can it be that we have two different estimates for the length of the same coastline?

  • Well this is called "The Coastline Paradox."

  • The answer is, it depends on the length of measuring stick that you use.

  • So, if you connect up the dots from cliff to cliff to cliff

  • You get a shorter length of coastline than if you measure with a smaller measuring stick

  • And measure into every inlet.

  • So what length of measuring stick should we use?

  • Well in theory, you can go all the way down to the size of a water molecule

  • And if you do that, then the length of Australia's coast is virtually infinite.

  • Do you believe me that you could have finite area object like Australia bounded by an infinite perimeter?

  • It doesn't seem to make sense.

  • But I can give you another example of this, it's called the Koch snowflake

  • so what you do is you take a triangle with sides of length 1

  • and then on each side

  • add another triangle with sides of length a third

  • Continute doing that again and again forever

  • What you end up with is a shape which is a finite area but an infinite perimeter

  • shapes like these are called fractals and many coastlines have the same

  • fractal structure which means they have some sort of self-similarity on many

  • different scales so you can zoom in and zoom in

  • and the coastline looks roughly the same

  • so if you want to know the length of a coastline you need to first specify the

  • length of your measuring stick

  • because that's what the answer depends on.

I've been driving along Australia's famous Great Ocean Road.

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海岸線のパラドックスとは? (What Is The Coastline Paradox?)

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    林宜悉 に公開 2021 年 01 月 14 日
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