Thread: Koch snowflake

I understand how the perimeter of a Koch snowflake is infinite, but I'm having a difficult time understanding that its area is finite. I'm guessing that after a couple steps, the area of the newly added triangles is negligble, but is there an equation to show that the area doesn't keep getting larger and larger along with the perimeter...?

If A is contained in B, the area of A cannot be greater than the area of B. Can you bound the snowflake in a big rectangle?

Or you could just calculate the area of the koch snow flake. Let A_n be the area of the n'th equilateral triangle you add at each step. Well then its easy to see that A_n+1 = 1/9 A_n and so A_n = (1/9)ⁿ A where A is the area of the first equilateral triangle. Now at each step of the construction we add a triangle to each straight line segment so we need to count those too. Well lets draw a few pictures and we notice that the number of line segments increase as 3, 12, 48, ..., 3*4^n-1 and so we increase the area by a factor of 3*4^n-1/9ⁿ at each step.

So we need to sum A + 3/9 A + 3*4/9²A = A(1+1/3(1 + 4/9 + (4/9)²+...) = A(1 + 1/3(9/5)) = 8/5 A.

So the area of the Koch curve is finite.

The perimeter is given by the sum 3L + 3*4/3 L + 3*(4/3)² L + ... which obviously diverges. i.e. the Perimeter is infinite.

but is there an equation to show that the area doesn't keep getting larger and larger along with the perimeter...?

Who says the area doesn't increase with the perimeter? It does. However, the additional area with each increment approaches zero as the perimeter tends to infinity. Hence, the area enclosed by the curve approaches a finite value as the number of increments goes to infinity.

Depending on convergence, the sum of an infinite series can be a finite number, which a lot of people find counter-intuitive at first. Have you heard the story of Archimedes racing a turtle?

Originally Posted by M

but is there an equation to show that the area doesn't keep getting larger and larger along with the perimeter...?

Who says the area doesn't increase with the perimeter? It does. However, the additional area with each increment approaches zero as the perimeter tends to infinity. Hence, the area enclosed by the curve approaches a finite value as the number of increments goes to infinity.

Depending on convergence, the sum of an infinite series can be a finite number, which a lot of people find counter-intuitive at first. Have you heard the story of Archimedes racing a turtle?

i know the area increases with the perimeter, i just meant that the area obviously doesn't grow infinitely large, which the perimeter does. thank you for your answer, that's the kind of thing i was looking for. the area of each new section "approaches zero". yes. exactly what i wanted to hear. and no, i haven't heard any stories about racing turtles... though at the nature center i volunteer at, they have turtle races occasionally. it's fun.

Well, here it is as far as I remember (i'll make the rest up as I go):

It was actually Achilles, who was said to be a very fast runner, not Archimedes. Some joker once challenged Achilles to race his pet turtle and Achilles accepted the challenge. Because he didn't think the turtle had any chance, Achilles agreed to allow it a head start. So they let the turtle start ahead of Achilles by a specified distance (let's call it point A0). Both Achilles and turtle started running at the same time. Achilles sped off to follow the turtle, with full intentions to pass the animal on the track and be first to arrive at the finish line. However, by the time Achilles reached the initial position (A0) of the turtle, the turtle, although running slower, had obviously gotten a little closer to the finish line (point A1). And, again, when Achilles finally reached point A1, where the turtle had just been, the animal had gotten a little further (point A2). I think you'll get the point. The same story happens in each iteration: Achilles reached point A(n), but the turtle had already advanced, however slowly, to a further point A(n+1). This is true even as the number of iterations goes to infinity!

Did Achilles ever catch up with the turtle? Does this take an infinite amount of time, i.e. are they still running out there? It seems paradoxical at first, but it really isn't. The point is that the story line breaks up the event (Achilles catching up) into an infinite number of segments (of distance or time), however, the sum of all those segments (in both distance and time) is a finite number! This is because the series converges, i.e. iterations after iteration the segments become smaller and smaller, approaching zero.

I think you can see the analogy to the Koch curve. An infinite series can have a finite sum.

That is one of Xeno's paradoxes. The Ancient Greeks had trouble with them because their math wasn't up to dealing with convergent series. (Try to write out how far each has gone, and how long they've been running, at step i using modern notation, then try it again with Roman numerals! )

Originally Posted by M

Well, here it is as far as I remember (i'll make the rest up as I go):

It was actually Achilles, who was said to be a very fast runner, not Archimedes. Some joker once challenged Achilles to race his pet turtle and Achilles accepted the challenge. Because he didn't think the turtle had any chance, Achilles agreed to allow it a head start. So they let the turtle start ahead of Achilles by a specified distance (let's call it point A0). Both Achilles and turtle started running at the same time. Achilles sped off to follow the turtle, with full intentions to pass the animal on the track and be first to arrive at the finish line. However, by the time Achilles reached the initial position (A0) of the turtle, the turtle, although running slower, had obviously gotten a little closer to the finish line (point A1). And, again, when Achilles finally reached point A1, where the turtle had just been, the animal had gotten a little further (point A2). I think you'll get the point. The same story happens in each iteration: Achilles reached point A(n), but the turtle had already advanced, however slowly, to a further point A(n+1). This is true even as the number of iterations goes to infinity!

Did Achilles ever catch up with the turtle? Does this take an infinite amount of time, i.e. are they still running out there? It seems paradoxical at first, but it really isn't. The point is that the story line breaks up the event (Achilles catching up) into an infinite number of segments (of distance or time), however, the sum of all those segments (in both distance and time) is a finite number! This is because the series converges, i.e. iterations after iteration the segments become smaller and smaller, approaching zero.

I think you can see the analogy to the Koch curve. An infinite series can have a finite sum.

interesting. i enjoyed that.

Originally Posted by MagiMaster

That is one of Xeno's paradoxes. The Ancient Greeks had trouble with them because their math wasn't up to dealing with convergent series. (Try to write out how far each has gone, and how long they've been running, at step i using modern notation, then try it again with Roman numerals! )

That is actually a fallacy, the greeks were well equiped to handle convergent series and did so often - archimedes being the prime example, in fact he developed a rudamentary form of integral calculus to solve certain geometry problems (i could supply the details if anyone is interested, most of it relies on the fact that the reals are totally ordered).

The problem is more metaphyiscs in nature, is space and time discrete or continuous? Each one of Zeno's paradox's were designed to eliminate one of those four options - and that is the paradox.

Huh. Never knew that.

The problem is one of supertasks - is it possible to do an infinite number of things in finite time. The Stanford encyclopaedia of philosophy has a nice article on Zeno and supertasks.

Originally Posted by river_rat

Originally Posted by MagiMaster

That is one of Xeno's paradoxes. The Ancient Greeks had trouble with them because their math wasn't up to dealing with convergent series. (Try to write out how far each has gone, and how long they've been running, at step i using modern notation, then try it again with Roman numerals! )

That is actually a fallacy, the greeks were well equiped to handle convergent series and did so often - archimedes being the prime example, in fact he developed a rudamentary form of integral calculus to solve certain geometry problems (i could supply the details if anyone is interested, most of it relies on the fact that the reals are totally ordered).

The problem is more metaphyiscs in nature, is space and time discrete or continuous? Each one of Zeno's paradox's were designed to eliminate one of those four options - and that is the paradox.

Details must be provided or else i will not believe you. I was under the impression the greeks didn't understand how an infinite series could converge, it says it in a book i read.

As for the Koch snow flake. A simple minded folk like myslef just looks at it and realizes that even though the area is getting bigger if you drew a circle around the whole thing it would never break out of the circle so clearly its area converges.

Originally Posted by billiards

Details must be provided or else i will not believe you. I was under the impression the greeks didn't understand how an infinite series could converge, it says it in a book i read.

Your book would be wrong then - Archimedes played with infinite series way back in 225BC. Their treatment of infinite series was very different then ours though as their mathematics was based on geometry. Google the method of exhaustion if you want more info.

As for the Koch snow flake. A simple minded folk like myslef just looks at it and realizes that even though the area is getting bigger if you drew a circle around the whole thing it would never break out of the circle so clearly its area converges.

Ah but you have used that which you want to prove to prove the result (you can only bound it by a circle if it has finite area etc).

Originally Posted by river_rat

The perimeter is given by the sum 3L + 3*4/3 L + 3*(4/3)² L + ... which obviously diverges. i.e. the Perimeter is infinite.

I think of the perimeter of the Koch snowflake like this. Initially (stage 0) you have 3 edges of length L, so the perimieter is . At stage 1, you have a shape with edges of length L/3, so the perimieter is . At stage 2, it becomes a shape with edges of length so . And at stage 3, with edges of length , . Hence the perimeter at stage n is and so as .