A Fractal Question

[hiflier]

As a Koch snowflake perimeter increases iterally towards an infinite length, I am told that the area within the snowflake is always exactly 1.6 times that of the original triangle. But how can this be if at each iteration you are adding a discreet number of triangles onto each straight edge of the snowflake, each of which has an area which must surely add to the original area. Yet this suggests an area that tends to infinity too, which is absurd. Where have I gone wrong?

[Zacs-Master]

1.6 times that of the previous iteration, surely?
http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htm

[Old_Geezer]

If the added areas tend towards the infinitesimal then the summation of an infinite number of them to get the total area will tend towards a finite value. One can add an infinite series of values and still end up with a finite value.

[Old_Geezer]

To show this for yourself try using your day to add the following (or until your reach an infinite number, whichever comes first).

1/1 + 1/2 +1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...

[Peter Pedant]

You have gone wrong where everyone went wrong over the last 2000y in thinking without thinking through the concept of an infinitely long line can enclose a finite space

Same thing for the serenyi sponge - by drilling out bits you can increase the surface area - as the volume goes down. This one I would say is rather obvious (!) as we have been trying for years to increase the surface area for catalysis ( o level chem 1966 ) with a decrease in the amount needed ( esp Pt )

[Jonathan-Joe]

"the area within the snowflake is always exactly 1.6 times that of the original triangle" No, with each iteration, the area increases, tending towards the limit of 1.6 (8/5) times the original triangle.https://en.wikipedia.org/wiki/Koch_snowflake#Area_of_the_Koch_snowflake

[maggiebee]

You lost me after the first five words lol. Science definitely not my forte.

[ToraToraTora]

Calculus dear boy!

[jim360]

Basically, limits are weird. Each one has to be examined on its own, and without any bias beforehand, in order to see how it behaves as things tend to infinity.

Possibly the most simple example to intuit would be comparing the two series

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

(sum of reciprocal numbers), which, despite each term getting successively smaller, will never reach a limit ever, and

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + ...

(sum of reciprocal squares), which this time does reach a limit. But that this limit is "pi squared over six" is an answer that hardly leaps out at you. Why the heck would summing square numbers give you something to do with circles? But there you go. Limits are weird.

And they can get weirder still, as limits can be defined in multiple, and subtle, ways. Quite a famous example is that the sum of all whole numbers, 1+2+3+4+5..., etc, is:

1 + 2 + 3 + 4 + ... = -1/12

as long as you wave your hands over what you mean by "=" here, at least.