SEx Education - Infinite Boundaries
Created | Updated Dec 7, 2006
Infinite Boundaries
Some words and phrases are guaranteed to get the SExperts' juices flowing. You just know they salivate at 'infinity', 'quantum' and 'Newtonian physics'. So when kea popped in with a question that featured all three in the thread title, there was no stopping them:
I heard someone a while ago trying to explain that if you measure the coastline of an island more and more accurately it gets longer and longer. And that as long as you have the technology to measure more and more accurately then there is no end to the measuring, so the coastline is in fact infinite rather than finite. I suppose that affects the measurement of area as well.
First up to the plate was Mu Beta, explaining why area wasn't an issue:
It's one of the big contradictions of mathematics that an object with infinite perimeter can have a finite area.
Traveller in Time, meanwhile, cast a little early doubt on whether a coastline could, in fact, really be infinite:
The infinite coastline is a model for fractal dimensions. This is a mathematical model of reality, one should not try to fit reality in this model. (Then you will end up with reality instead of a model.)
And indeed many scientific areas make use of the fractal dimensions to make models fit more realistic then using simple dimensions.
Looking at the question of area, Gnomon posted a series of explanations:
Even if a country has an infinite coastline, you can draw a circle which is bigger than the country, so that the whole of the country fits inside the circle. Since the area of the circle is fixed, the area of the country must be less than that, so it is finite, no matter how complex the coastline is.
Here's an example of the coastline increasing, but the area staying the same.
Consider a square. We're going to change one side of it. We could change all four sides, but I'm too lazy to do that much typing. Consider the top side of the square, which is a line.
Divide the line into four.
Leave the first quarter alone.
Construct an equilateral triangle on the second quarter, so that it is outside the square.
Construct an equilateral triangle on the third quarter, so that it is inside the square.
Leave the fourth quarter alone.
We now have a curious zigzag boundary consisting of six short lines, instead of the original straight line which was made of four short lines. The top boundary of the square is 1.5 times as long as it was. But the total area is the same, because we added and subtracted the same amount.
Now, repeat the same process for each of the short lines. The resultant spiky object has a coastline 1.5 * 1.5 times the original, but the area has not changed.
Repeat this process ad nauseam. The coastline keeps getting longer and longer without limit (which is the definition of infinite) but the area never changes.
While Woodpigeon did the maths:
It has something to do with limits. The following sequence is infinite:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....
However the sum never gets greater than the number 2. It gets very, very close to 2 but never actually reaches it. 2 is called the "limit" of the sequence. The area, if you like, is like a limit around which the perimeter can never exceed.
Kea was almost happy with that, but there was just one more anomaly to be ironed out.
Ok, I get that the area is finite, but not that it doesn't change if one measures more accurately. If you measure around an island and ignore the bays then you will get a different area than if you take the bays into account, surely? Isn't the reason that the area doesn't change in the square because you deduct the same amount as you add each time you make a change to the perimeter? In real coastlines that doesn't happen. Presumably when a coastline is measured they round off the measurements at some point. And depending on where that rounding off starts, it will affect the accuracy of the measurement of the area.
The succinct answer was, again, provided by Gnomon:
You're right. But the extra bits are so small that it doesn't matter.
... and Traveller in Time summed up with a graphic of his own:
Simple coastline ----------------------------------------
Fractured coastline \/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/
The dry area has not changed but the coastline has. Now we can make the jigsaw again on every straight line there is in the jigsaw line, and again. The area will not change as we extend a little but also take a little away.
Of course, this being SEx, that wasn't all there was to it. We haven't looked at the second part of kea's question yet, which was:
My next question then is where does one stop measuring using newtonian physics and start measuring using quantum physics (both in general and in terms of the above idea)? I often hear people say that you can only apply quantum physics to the subatomic part of nature not to the physical world as we know it. But when does it stop being one and become the other? Is it a gradual thing, or a sudden, crossed a line thing.
For the answer to that, and to find out why DaveBlackeye needed to post the wonderful quote below, you'll just have to go and read the thread!
It is theoretically possible to fire a Volkswagen Beetle through two slits at once and get an interference pattern. It is just very, very improbable.
QED
This article was based on a conversation at the SEx forum - where science is explained.
Why not pop over with your own questions? The pick of the bunch will feature in The Post's next issue.