July 28, 2010 6:44 AM Subscribe

The British coast has infinite length. Doesn't everything?

I was watching a BBC documentary about the coast of Britain. They showed that the length of the rule you use to measure the coast changed the measured length of the coast.

For example, measuring with a 2 m rule would give a coast length which is a lot shorter than if you measured with a 50 cm rule, since the shorter rule could get into many of the nooks and crannies and follow the coast line much more accurately.

In summarising, they brought in fractal mathematics, and explained that they could always choose a shorter rule, and this essentially meant that the length of the British coast is infinite.

Which got me thinking. Presumably I am also infinitely tall, since you could always choose a more accurate way of measuring my height.

What am I missing here? What is the difference between the two cases?
posted by devnull to Education (40 answers total) 6 users marked this as a favorite

I was watching a BBC documentary about the coast of Britain. They showed that the length of the rule you use to measure the coast changed the measured length of the coast.

For example, measuring with a 2 m rule would give a coast length which is a lot shorter than if you measured with a 50 cm rule, since the shorter rule could get into many of the nooks and crannies and follow the coast line much more accurately.

In summarising, they brought in fractal mathematics, and explained that they could always choose a shorter rule, and this essentially meant that the length of the British coast is infinite.

Which got me thinking. Presumably I am also infinitely tall, since you could always choose a more accurate way of measuring my height.

What am I missing here? What is the difference between the two cases?

Your height is the distance between two defined points -- you have a top of your head and the bottom of your feet. This distance does not depend on the size of the ruler. The coastline, however, can theoretically be defined in a different way, such as you describe.

Also see Zeno's Achilles and the Tortise Paradox

posted by Comrade_robot at 6:49 AM on July 28, 2010 [2 favorites]

Also see Zeno's Achilles and the Tortise Paradox

posted by Comrade_robot at 6:49 AM on July 28, 2010 [2 favorites]

This is basically Zeno's Paradox (see the dichotomy paradox in the wiki link).

posted by backseatpilot at 6:49 AM on July 28, 2010

posted by backseatpilot at 6:49 AM on July 28, 2010

Fractal geometrydoesnt apply to things like one's height. You may be on to something with one's surface area, however....

posted by dfriedman at 6:50 AM on July 28, 2010

posted by dfriedman at 6:50 AM on July 28, 2010

What was stated was incorrect, because the coastline isn't truly fractal. While it is true that the length will increase when using a smarter rule, it will not increase consistently -- that is, it will continuously approach some upper limit.

Your height isn't a fair comparison, either, because you're only measuring in one dimension. There are no nooks and crannies involved in height.

posted by bfranklin at 6:50 AM on July 28, 2010

Your height isn't a fair comparison, either, because you're only measuring in one dimension. There are no nooks and crannies involved in height.

posted by bfranklin at 6:50 AM on July 28, 2010

Wow, smarter rule? Really? Perhaps smarter word choice is needed. I meant _smaller_ rule.

posted by bfranklin at 6:51 AM on July 28, 2010

posted by bfranklin at 6:51 AM on July 28, 2010

I doubt it. If you were to plot on a simple Cartesian axis the difference in total length of coastline (y) against the length of the rule (x), I'm prepared to bet a plate of beans that the curve would flatten out to the point where a further reduction in length of rule would bring zero increase in length of coastline.

posted by aqsakal at 6:51 AM on July 28, 2010

And actually, no matter what unit of measure you use, the length of the British coastline is not "infinite." I could say, accurately, that the British coastline is somewhere between 1 cm and 100 billion miles long. No matter what type of ruler we used to measure it, the outcome would fall somewhere in between those two measures. So, not infinitely long.

posted by decathecting at 6:58 AM on July 28, 2010 [3 favorites]

posted by decathecting at 6:58 AM on July 28, 2010 [3 favorites]

You're not missing anything... For any object with a complex shape (a coastline, humans, whatever), the length or resolution of your measuring tool always influences the measurement you're taking. This is a general result. Sometimes, when we have a standardized method for a certain measurement (ie human height is always the distance from the floor to the top of your head), this result doesn't matter.

In other applications, though, being aware of this result is important, and "best" scales of measurement depend on context and objectives... THink about how you might measure the coastline if you were 1) a real estate agent, 2) a geographer comparing the size of different islands, 3) an engineer measuring fine-scale sediment transport, or 4) a crab picking its way among stones on a beach...

posted by JumpW at 6:59 AM on July 28, 2010

In other applications, though, being aware of this result is important, and "best" scales of measurement depend on context and objectives... THink about how you might measure the coastline if you were 1) a real estate agent, 2) a geographer comparing the size of different islands, 3) an engineer measuring fine-scale sediment transport, or 4) a crab picking its way among stones on a beach...

posted by JumpW at 6:59 AM on July 28, 2010

You don't measure your height by stepping off the perimeter of your body (ignoring the problem of you being 3d for a minute) but by comparing it to a known standard length. You can increase the accuracy of that measurement by using a smaller length scale, but the measurements will converge on a single value (okay, there would be some fuzziness at very small length scales, but you could get down to a millimeter or so).

So with a 1m rule you could say you were between 1 and 2 metres. With a 10cm rule you could say it was between 1.9 and 2.0, with a 1cm rule between 1.95 and 1.96.

Similarly you can define a length of Britain (N-S say) that would be accurate down to a few metres based on how much beach you included.

The coastline has a fractal dimension, so as you choose smaller and smaller step lengths, the measured length doesn't converge, it just gets bigger and bigger. If we measured your perimeter something similar would happen once we got down to very small length scales where your skin stops looking smooth.

posted by crocomancer at 7:01 AM on July 28, 2010

So with a 1m rule you could say you were between 1 and 2 metres. With a 10cm rule you could say it was between 1.9 and 2.0, with a 1cm rule between 1.95 and 1.96.

Similarly you can define a length of Britain (N-S say) that would be accurate down to a few metres based on how much beach you included.

The coastline has a fractal dimension, so as you choose smaller and smaller step lengths, the measured length doesn't converge, it just gets bigger and bigger. If we measured your perimeter something similar would happen once we got down to very small length scales where your skin stops looking smooth.

posted by crocomancer at 7:01 AM on July 28, 2010

It is important to understand that mathematics does not precisely correspond to the physical universe that we live in. In mathematics, there is no limit to the sub-division of any measurement or of any object into ever smaller sections. If you look at a number line that runs from, let us say, 2 to 3, there are infinitely many numbers in that sequence, since you can make infinitely small increments or sections. In the real world this is not the case! This is the fundamental discovery that gave us the science of quantum mechanics. The word "quantum" simply means an amount or quantity. We now now that all quantities in the real world come in the form of packages or quantities which cannot be further reduced or cut in half. In other words, there is a minimum quantity (or a quantum) of space, of time, of matter, and of energy. You can't have any less than that minimum amount, it doesn't exist. Yet another way of looking at this is to call the universe, on a very small scale, digital rather than analog. If you look at anything closely enough, it is granular, or lumpy, rather than smooth.

Consequently, although the coastline of Britain could be extremely long if you measured it with sufficient precision, on a molecular level, it is certainly not infinite. You could have a mathematical model that looks like the coastline of Britain and has a fractal shape with infinite length, but in the real world there are no infinite fractals, only approximate fractals.

posted by grizzled at 7:02 AM on July 28, 2010 [9 favorites]

Consequently, although the coastline of Britain could be extremely long if you measured it with sufficient precision, on a molecular level, it is certainly not infinite. You could have a mathematical model that looks like the coastline of Britain and has a fractal shape with infinite length, but in the real world there are no infinite fractals, only approximate fractals.

posted by grizzled at 7:02 AM on July 28, 2010 [9 favorites]

So what was the purpose in claiming the coast line is infinitely long?

posted by devnull at 7:02 AM on July 28, 2010

posted by devnull at 7:02 AM on July 28, 2010

@aqsakal - it was precisely this type of experiment that led people to conclude it didn't converge.

posted by crocomancer at 7:03 AM on July 28, 2010

posted by crocomancer at 7:03 AM on July 28, 2010

That is to say, we now know. I left out a letter in my above reply.

posted by grizzled at 7:04 AM on July 28, 2010

posted by grizzled at 7:04 AM on July 28, 2010

So, the question is, given that the coast of Britain is actually not infinitely long (no matter how you measure it) why did the BBC documentary claim that it is? Answer: they are wrong. It happens. Even the BBC is fallible. I would say that someone at the BBC had learned about fractal theory but did not understand its full application in the real world.

posted by grizzled at 7:06 AM on July 28, 2010

posted by grizzled at 7:06 AM on July 28, 2010

As a person who specialized in catalysis and porosity, I know the analytical techniques and we could get to a finite number for an accurate measurement of a human's surface area. Yes that would potentially get up into the 1000's of m^2, but not infinite. Again, definite boundaries.

An accurate measurement of the earth's surface area, however, would involve the same paradox as the coastline.

posted by lizbunny at 7:14 AM on July 28, 2010

If you're really interested, you might investigate Mandlebrot's paper "How Long is the Coast of Britain":

Mandlebrot's paper

Wikipedia Summary

*Note that the paper does not claim that any coastline or geographic border actually is a fractal. It simply states that the measured length of a coastline or border may be empirically observed to behave like a fractal over a range of measurement scales.*

posted by Comrade_robot at 7:18 AM on July 28, 2010 [2 favorites]

Mandlebrot's paper

Wikipedia Summary

posted by Comrade_robot at 7:18 AM on July 28, 2010 [2 favorites]

this doesn't really have to do with zeno's paradox. speaking loosely, the coast was a familiar example of how a fractal-like curve can have a length much longer than it would appear if you looked at things coarsely. yes, it is probably not actually a fractal when you look finely.

check out this article for an example of the point that the analogy tries to illustrate. in particular, the "properties" section.

Koch Snowflake

the point is, the total length of the curve goes to infinity as you repeat the process of adding more line segments. this is a little surprising, since the curve always remains in a bounded area.

it's not the case that the fact you can measure your height with smaller units means you are infinitely tall. you can say you are 2 meters tall or 2000 milimeters tall, but since a millimeter is 1/1000 the length of a meter the measurements are equivalent and indicate you are of finite height.

posted by alk at 7:19 AM on July 28, 2010 [1 favorite]

check out this article for an example of the point that the analogy tries to illustrate. in particular, the "properties" section.

Koch Snowflake

the point is, the total length of the curve goes to infinity as you repeat the process of adding more line segments. this is a little surprising, since the curve always remains in a bounded area.

it's not the case that the fact you can measure your height with smaller units means you are infinitely tall. you can say you are 2 meters tall or 2000 milimeters tall, but since a millimeter is 1/1000 the length of a meter the measurements are equivalent and indicate you are of finite height.

posted by alk at 7:19 AM on July 28, 2010 [1 favorite]

I always understood that British coastline question to be about how it was impossible to measure definitively, not that it was infinite. In addition to using different tools to measure, there is also the issue of tides and surf constantly changing, making any true measurement impossible. The coastline isn't a static thing so it can't be measured as such.

That said, a person's height is subject to change as well, throughout the day, throughout a lifetime, etc. So some of the same measurement issues could apply, but again, it's about precision rather than something being infinite.

posted by headnsouth at 7:27 AM on July 28, 2010

That said, a person's height is subject to change as well, throughout the day, throughout a lifetime, etc. So some of the same measurement issues could apply, but again, it's about precision rather than something being infinite.

posted by headnsouth at 7:27 AM on July 28, 2010

also, why doesn't my argument in my last paragraph about meters and millimeters work for fractals? because with a fractal curve, when i measure with millimeters, i am able to measure the nooks and crannies (like the bumps in the koch snowflake) that i couldn't with a more blunt measurement. and every time i choose a finer measuring device, i find more nooks and crannies which end up adding to the total length i measure. for a true fractal curve, this keeps happening no matter how small i make my measuring device. so a koch snowflake has infinite length.

in the case of the coastline, eventually when your measurement device got small enough you probably wouldn't find any more nooks and crannies. but for a good range of length scales, you do, which is the point of the analogy.

posted by alk at 7:33 AM on July 28, 2010

in the case of the coastline, eventually when your measurement device got small enough you probably wouldn't find any more nooks and crannies. but for a good range of length scales, you do, which is the point of the analogy.

posted by alk at 7:33 AM on July 28, 2010

The coastline is made of the division line between sea and land. This division line is not static - as waves advance and recede, it would be up to you to pick the moment to measure. Therefore, it makes no sense to measure with anything less than about a meter or half a meter. Even if you had the water freeze all along, there's still the size of atoms that puts a limit to measurement. And then there's the planck scale.

There are a lot of things that are a whole lot more infinite than the coast of Britain.

posted by rainy at 7:36 AM on July 28, 2010

There are a lot of things that are a whole lot more infinite than the coast of Britain.

posted by rainy at 7:36 AM on July 28, 2010

I think I remember seeing that documentary. I think they were just trying to introduce fractals and the idea that they can have infinite perimeters with finite areas using something relatable, if not entirely accurate.

posted by lucidium at 7:48 AM on July 28, 2010

It's been a while since I studied, but I think Wittgenstein 'dissolved' these kind of questions. "Philosophy is a battle against the bewitchment of our intelligence by means of language"

The 'bewitchment' is that the thought experiment is taking language beyond the boundaries of what it is normally used to mean, whilst still appearing grammatically to make sense.

All things are measured to a degree of accuracy, so normally if I say 'It's 5 km to Exeter' you won't call me a liar if I measure by satellite say and it's 5k whereas if I was to actually measure it with a string it's 5.005.

If you said 'you don't know it's 5km, because you haven't measured accurately with string' you're invoking a different 'language game' where the rules are different. You're saying in 'string terms' I don't know, even though in 'satellite terms' I do know.

So this thought experiment is saying that because you can always invoke a greater degree of accuracy, the coastline is infinite because an infinitely small rule will give an infinite length.

If you look at what I just wrote, 'an infinitely small rule will give an infinite length' it makes grammatical sense. However, there is no infinitely small rule, so the 'language game' of measurement cannot apply. The 'language game' for talking about infinite things is mathematical language.

So in a specialised mathematical sense the coastline*is* infinite. The reason why this is jarring, is that it seems like it's somehow questioning normal reality where it isn't infinite. Whereas in fact they're not really saying any information about the world, they're just saying look at it in this mathematical way.

posted by Not Supplied at 7:57 AM on July 28, 2010 [3 favorites]

The 'bewitchment' is that the thought experiment is taking language beyond the boundaries of what it is normally used to mean, whilst still appearing grammatically to make sense.

All things are measured to a degree of accuracy, so normally if I say 'It's 5 km to Exeter' you won't call me a liar if I measure by satellite say and it's 5k whereas if I was to actually measure it with a string it's 5.005.

If you said 'you don't know it's 5km, because you haven't measured accurately with string' you're invoking a different 'language game' where the rules are different. You're saying in 'string terms' I don't know, even though in 'satellite terms' I do know.

So this thought experiment is saying that because you can always invoke a greater degree of accuracy, the coastline is infinite because an infinitely small rule will give an infinite length.

If you look at what I just wrote, 'an infinitely small rule will give an infinite length' it makes grammatical sense. However, there is no infinitely small rule, so the 'language game' of measurement cannot apply. The 'language game' for talking about infinite things is mathematical language.

So in a specialised mathematical sense the coastline

posted by Not Supplied at 7:57 AM on July 28, 2010 [3 favorites]

It highlights the fact that measurements of length depend on the size of the ruler. Questions of philosophy aside, this result has important operational consequences for many fields such as geography, ecology, and engineering, where "best" methods for measuring length depend on the context of the problem.

posted by JumpW at 8:25 AM on July 28, 2010

The Planck constant sets an upper bound for the perimeter of any real world object, including the island of Great Britain.

posted by atrazine at 8:29 AM on July 28, 2010 [1 favorite]

posted by atrazine at 8:29 AM on July 28, 2010 [1 favorite]

You are not infinitely tall. If we define height as "the straight line distance between the top of your head to the ground when you are standing upright", I can easily put an upper bound to your height by putting a flat board just above your head, marking a line on the wall and measuring the vertical distance to the ground. "Height" is not the same as "the distance around my body".

Similarly, whatever the length of the coastline of Britain, the (straight line) distance between Land's End and John o'Groats is finite.

posted by Electric Dragon at 8:45 AM on July 28, 2010

Similarly, whatever the length of the coastline of Britain, the (straight line) distance between Land's End and John o'Groats is finite.

posted by Electric Dragon at 8:45 AM on July 28, 2010

What alk said:

*when i measure with millimeters, i am able to measure the nooks and crannies (like the bumps in the koch snowflake) that i couldn't with a more blunt measurement. and every time i choose a finer measuring device, i find more nooks and crannies which end up adding to the total length i measure. for a true fractal curve, this keeps happening no matter how small i make my measuring device. so a koch snowflake has infinite length.*

in the case of the coastline, eventually when your measurement device got small enough you probably wouldn't find any more nooks and crannies. but for a good range of length scales, you do, which is the point of the analogy.

posted by 5Q7 at 8:53 AM on July 28, 2010

in the case of the coastline, eventually when your measurement device got small enough you probably wouldn't find any more nooks and crannies. but for a good range of length scales, you do, which is the point of the analogy.

posted by 5Q7 at 8:53 AM on July 28, 2010

As all the quantum comments are pointing out, there is an upper bound where one theoretically could count the atoms along the coast/measure with a planck-length ruler (since space is not, in fact, infinitely divisible). This is presumably a very large upper bound, since using their measurement technique your coast measurement will have to follow every crack in the rocks that the ruler can fit into (which is to say, 'every crack in the rocks').

posted by Lady Li at 9:02 AM on July 28, 2010

posted by Lady Li at 9:02 AM on July 28, 2010

Has this actually been established for space and time? I know that it's theorized by some, but is there any hard evidence for it?

And yes, I'm aware that you can you can manipulate the fundamental constants h, c, and G to get the Planck length and Planck time, which are indeed very very small, but given that the similarly-derived Planck mass is many many orders larger than the smallest known mass, the existence of the Planck length and Planck time as mathematical quantities, in and of themselves, does not seem to be sufficient evidence for the quantization of space and time.

posted by DevilsAdvocate at 9:27 AM on July 28, 2010

The difference between measuring the coast line and your height is the number of line segments. When measuring your height, it is always a single line segment. The only question is the accuracy of the measurement. The smaller the unit of measurement, the closer you converge to the true height. But the important thing is that the measurement is always a single segment and it has a limit.

When measuring the coast line, considering it as a fractal, the number of segments increases the smaller your unit of measurement. If a true fractal, the number of segments would be infinite and the length would be infinite. This is because the number of segments increases at a faster rate than the length of each segment decreases.

So the difference boils down to the fact that height is not a fractal. It is a single line segment. The coastline is a fractal with an infinite number of segments. (Although Mandbrot does not claim that the coastline is a true fractal -- only that it behaves as a fractal over a certain range of fractal dimensions).

posted by JackFlash at 9:40 AM on July 28, 2010

When measuring the coast line, considering it as a fractal, the number of segments increases the smaller your unit of measurement. If a true fractal, the number of segments would be infinite and the length would be infinite. This is because the number of segments increases at a faster rate than the length of each segment decreases.

So the difference boils down to the fact that height is not a fractal. It is a single line segment. The coastline is a fractal with an infinite number of segments. (Although Mandbrot does not claim that the coastline is a true fractal -- only that it behaves as a fractal over a certain range of fractal dimensions).

posted by JackFlash at 9:40 AM on July 28, 2010

It's only infinite if you assume infinite resolution. But if you subscribe to the laws of physics, you have to place a limit on where reality becomes unreality (aka the level of quantum foam). If we assume that is at the level of a Plank Length, then you can determine absolute length in plank units and scale up.

posted by blue_beetle at 10:19 AM on July 28, 2010

posted by blue_beetle at 10:19 AM on July 28, 2010

You are familiar with the famous equation e=mc^2, giving the relationship between matter and energy. Further study of relativity shows that there are interconversions between matter, energy, space, and time. They are all interconvertible. So necessarily, if there is a smallest possible amount of one of these, it applies to all of them. And exactly the same theoretical basis for the quantization of matter or energy applies to space and time. They are all quantized.

posted by grizzled at 10:43 AM on July 28, 2010

posted by grizzled at 10:43 AM on July 28, 2010

I'm aware that there are interconversions between matter and energy. I'm also aware that there are interconversions between space and time. I was unaware that there are interconversions between matter/energy and space/time—could you go into a bit more detail, or perhaps point me at some relevant reading?

And if space and time are quantized, why is it widely believed

posted by DevilsAdvocate at 11:34 AM on July 28, 2010

This is somewhat related: Banach_Tarski paradox, I think.

posted by Green With You at 4:42 PM on July 28, 2010

posted by Green With You at 4:42 PM on July 28, 2010

Your height can't be infinite, but of course, if you kept measuring it more and more accurately, the number you recorded for it could get infinitely long.

I am 1.83 meters tall.

Measuring more accurately, I might be 1.831

Measuring even more accurately, I might be 1.8314

Measuring even more accurately, I might be 1.831425

*[snip a few iterations]*

Measuring even more accurately, I might be 1.83142565489765798659875189846519827784054

and so on.

posted by AmbroseChapel at 11:04 PM on July 29, 2010

I am 1.83 meters tall.

Measuring more accurately, I might be 1.831

Measuring even more accurately, I might be 1.8314

Measuring even more accurately, I might be 1.831425

and so on.

posted by AmbroseChapel at 11:04 PM on July 29, 2010

It's not saying that though, it's saying if you have a fixed length ruler and keep making it smaller and smaller then the length will tend towards infinity.

For example if you had a ruler one molecule in length, it would have to navigate around all the molecules of the OP, making the measured length very long.

posted by Not Supplied at 3:33 AM on July 30, 2010

For example if you had a ruler one molecule in length, it would have to navigate around all the molecules of the OP, making the measured length very long.

posted by Not Supplied at 3:33 AM on July 30, 2010

No, for height the ruler wouldn't have to navigate around all the molecules of the OP. For girth or surface area, maybe. But height is the distance between two defined points.

posted by Comrade_robot at 3:50 AM on July 30, 2010

posted by Comrade_robot at 3:50 AM on July 30, 2010

I get what your saying, it might be slightly different from the coastline example, but I think maybe your mixing the mathematical model with the thought experiment in physical space.

If you took the distance between two point, it would have to navigate around some molecules one way or another air molecules or whatever. So maybe this makes the example not a simple visualisation like the coastline one, but the same principle applies.

posted by Not Supplied at 3:58 AM on July 30, 2010

If you took the distance between two point, it would have to navigate around some molecules one way or another air molecules or whatever. So maybe this makes the example not a simple visualisation like the coastline one, but the same principle applies.

posted by Not Supplied at 3:58 AM on July 30, 2010

If you look above, I've given a link to Mandlebrot's original paper on the length of the British coastline. From his introduction:

*The concept of ‘‘length’’ is usually meaningless for geographical curves. They can be considered superpositions of features of widely scattered characteristic sizes; as even finer features are taken into account, the total measured length increases, and there is usually no clear-cut gap or crossover, between the realm of geography and details with which geography need not be concerned.*

Right? Notice he doesn't say that the concept of length is meaningless, generally. He says that the concept of length if meaningless for**geographical curves**, because geographical curves have finer and finer features, and measuring those finer and finer features increases the length of the coastline.

Since your height has a standard definition that is not path dependent -- that is to say, your height is the distance from the top of your head to the bottom of your feet when you're standing up straight -- there are no finer and finer features which must be taken into account to measure your height. It doesn't matter how fat you are, or if there are air molecules or whatever.

posted by Comrade_robot at 6:27 AM on July 30, 2010

Right? Notice he doesn't say that the concept of length is meaningless, generally. He says that the concept of length if meaningless for

Since your height has a standard definition that is not path dependent -- that is to say, your height is the distance from the top of your head to the bottom of your feet when you're standing up straight -- there are no finer and finer features which must be taken into account to measure your height. It doesn't matter how fat you are, or if there are air molecules or whatever.

posted by Comrade_robot at 6:27 AM on July 30, 2010

In that case it seems originally I was answering a different question to what Mandlebrot is talking about.

I see the difference between the two cases in your way of looking at it. And presumably Mandlebrot would concede that the coastline is not fractal down to an infinite scale anyway, he was just illustrating the maths.

Therefore it's just point about mathematical models and neither physics or philosophy come into it, it seems to me.

posted by Not Supplied at 7:23 AM on July 30, 2010

I see the difference between the two cases in your way of looking at it. And presumably Mandlebrot would concede that the coastline is not fractal down to an infinite scale anyway, he was just illustrating the maths.

Therefore it's just point about mathematical models and neither physics or philosophy come into it, it seems to me.

posted by Not Supplied at 7:23 AM on July 30, 2010

This thread is closed to new comments.

posted by grumblebee at 6:49 AM on July 28, 2010 [1 favorite]