That is what I’m saying. The process of getting bigger, of increasing area, will go on infinitely. Thus the area will be getting infinitely bigger (or infinitely getting bigger, if you prefer). What is incorrect about that?
The two phrases have different semantics.
getting infinitely bigger = becoming infinitely big
infinitely getting bigger = becoming bigger on every iteration forever
The second phrase allows for a bounded size, the first explicitly says the size is not bounded.
Infinite sums may be bounded. A fractal’s area is such a sum.
I think you understand correctly.
Assuming we are still talking about this snowflake:
you can see that it is not convex, i.e., the boundary does in fact constantly convolute inwards and outwards, just like a coastline. Even though no area is subtracted during the construction.
It is true that a convex curve cannot be “fractal” and each segment of it will have a well-defined length.
This discussion of “infinity” as applied to natural coastlines just seems to be confusing and obscures the original point that there is a genuine “coastline paradox”, namely, that the length of a real-world coastline is not defined unless you also specify at which resolution you measured it.
There is no need to (and I don’t think anyone ever did) say that “the coast of Norway is infinitely long” or to use the word “infinite” in that sense.
To quote a salient post from an earlier related thread:
The discussion here is strongly related to the not-quite-but-almost infinite threads we’ve had in the past asking if 0.999=1. Sorrowfully, the really major such thread seems to have gotten lost in the transition, as I can’t find it.
The whole question, at least at the level of theoretical mathematics, deals with the concepts of limits, and infinity, convergence and divergence, and sums of infinitely many terms. As almost always in these threads, nobody who understands these concepts really does much of a job explaining it to those others who aren’t really up on their advanced mathematics.
This particular threads seems to have gotten hopelessly bogged down in arguing over the theoretical possibility of measuring an irregular coastline with ever-smaller rulers, versus the practical, in-real-life impossibility of doing so.
The coastline of Norway is something that exists in the real-life universe. The snowflake curve, in its full infinite snowflakedness, exists only in the minds of those who contemplate it.
To explain how a finite area can be bounded by an infinitely long curve, and in particular the snowflake curve, requires some basic concepts from calculus (like limits).
And what are limits? And what can we possibly mean by “the sum of infinitely many terms (like 0.1 + 0.001 + 0.0001 + …)”.
As a matter of fact, yes we ARE all playing with definitions here. For starters, adding up an infinitely long list of terms isn’t just like adding a few numbers like you did in 2nd grade, because you can’t just add up infinitely many numbers. (Go ahead. Try it.) Instead, we must define a new meaning for addition that works well, specifically for adding up an infinite series. Only then can we say, with any meaning, what a sum like 0.1 + 0.001 + 0.0001 + … is, or 0.9 + 0.99 + 0.999 + … or the sum of all the sides of a snowflake curve, or the area of a snowflake curve.
Maybe I’ll try it again (but not in this post right now). I went through all that, in detail, in one of our lost 0.999=1 threads, the one I can’t find now.
The coastline of Norway is a story of what made people start thinking along these lines. It was not the end result of people thinking along these lines.
No, I am using the definitions as established and used by actual mathematicians who somehow or other convince the public pay them to sit around and think about these things.
I’m just trying to explain that definition.
Nothing in this thread can be explained by second grade math. It is well accepted that those participating in this thread would be willing to use or at least try to learn to use higher mathematical concepts.
Could you try to explain the Mandelbrot Set to someone who didn’t accept the existence of imaginary numbers? Would you be playing with definitions when you say that you can take the square root of a negative number?
One of the reasons that I brought up a calculator to use is because then one could try it. And it would show that certain algorithms converge towards a particular number, and other algorithms increase without bound. You don’t have to go on forever, just to the limits of what your calculator can do to see this in action.
The fun thing about fractal structures is that they have both algorithms. A convergent algorithm for the area, and a divergent algorithm for the perimeter.
As I said, the Mandelbrot Set has an infinite perimeter. The area is hard to calculate but upper bounds can be trivially determined.
No we don’t. Others have already done so.
We can attempt to explain their reasoning, their methods, and their results, but in no case are we, in this thread, actually creating anything new.
As regards to Norway or any other real physical object, they will display the same properties until you get to a level of resolution where shapes are no longer something that makes sense to define (ultimately at the molecular or atomic level). If you measure Norway’s coast with a 100 mile ruler, you will get a much smaller number than a 10, and that will be smaller than with a 1 mile ruler. It will not increase slightly, or converge on a particular number, but instead, every time you go down a scale, the perimeter will go up exponentially. If the water was perfectly still, so you don’t have to worry about waves or tides, you could keeps this up quite a while. If you got down to the point where you are measuring your way around sand grains, then the coastline is well into the millions of miles, probably orders of magnitude more. And don’t forget that sand grains have imperfections in them as well, if you then measure your way in and out of each of these tiny cracks and crevices, you are adding on even more orders of magnitude to the measured perimeter.
The only real difference between the coast of Norway and a fractal is that the fractal, not being made of physical material, is not subject to a minimum resolution size (until your computer rebels on you, anyway). You can download a fractal explorer that will work on an average modern computer, and by the time your computer gives up, the original shape will be larger than the observable universe.
The paradox is also sometimes stated as “how long is a piece of string?” * (e.g on the excellent BBC Horizons show Alan Davies did on fractals) , which gives an indication of the generic nature of the paradox.
* Which is not the exact same meaning as the idiom with that phrasing
That’s why it’s called a paradox.
Would something like pumice work?
I’d like to throw my own two cents in and try to help to Whack-a-Mole understand that it’s perfectly acceptable for Norway to have an infinite coastline:
Say we have two countries, one has a perimeter of 1,000 miles, the other a perimeter of 10,000 miles. Which country has more land area? Can you see that it’s impossible to tell? If not, let me help: Imagine the 1,000 mile country is a perfect circle, while the 10,000 mile country is a long skinny rectangle almost 5,000 miles long in one dimension but minuscule in the other (having virtually no area at all)
In fact, if two countries have perimeters x and y (both positive) it’s ALWAYS impossible to prove which is bigger in area, regardless of the relative sizes of x and y.
The reason I mention this is I think you’re confusing the size of an object (area of a country) with possible sizes for the boundary of that object. While those two measurements are not entirely independent (for example, given a fixed perimeter, there’s an upper bound on how much area it can contain), they can still be wildly different.
Of course, this doesn’t even begin to address the nooks and crannies of an actual fractal boundary, which is kind of what this thread is really about, but I wanted to address the difference between the size of an object versus the size of its boundary first; assuming that if one is small that the other must also be small is an easy mistake to make.
Speaking of Norway, unlike Britain, it is not an island, so you have additional issues like the precision and accuracy of border markers in cases where a border intersects the coast. After all, just move it by 1cm and you stand to gain or lose infinity km of coastline 
The boundary does convolute inward and outward, but all prior iterations entirely fit within subsequent iterations. That would not happen if you were measuring an actual coastline. Whereas the area of the snowflake is continually (infinitely) increasing, the area bounded by a coastline would increase and decrease as the scale of measurement changed.
As an aside, what if the snowflake iterations were such that every time an additional triangle was added tot the perimeter, it would randomly be added as either inward going or outward? That would seem to more closely model measuring a coastline. Would it still be fractal?
Thank you - that looks impressive. I have seen some catalysts like that.
That depends on what rule (not ruler) you use for measuring the coastline. If you use an approach where your ruler is on the water-land border on at least one point, but does not extend over water anywhere, the bounded area will strictly increase. If you go the other way, which would be my preference, it will strictly decrease.
It wouldn’t work for a Koch snowflake, since there’d be a high chance of the additional triangle for two adjacent line segments to go inward, and then their points would touch. But you still get a fractal (by at least some definitions of the word) if you randomly go in and out, and if you made the new triangles a fourth of the original, you’d avoid them touching (I think) and still get an infinite perimeter.
Possibly, depending on the exact procedure used to measure the area— but it will converge to a fixed finite value.
Not only would it still be fractal, this randomization will not alter the fractal dimension at all, because the resulting curve has the same self-similarity properties.
It works fine for a Koch snowflake:
You are right that the points can touch and therefore you get little islands and lakes.
I should have included my unstated premise “unless you’re fine with islands and lakes”. 
OK, let’s try to state the situation as accurately as possible, here:
1: The length of a real-world coastline is not well defined, because the smaller the scale you use to measure it on, the longer the total length you get.
2: If you use rulers of reasonable sizes (say, in the range from 1 meter to 100 km), you’ll find that, in any given region of the world, the total lengths you find will roughly follow a certain mathematical relationship. In fact, it’s the same sort of relationship that you find for objects of various dimensions, except that the number that would correspond to the dimensionality isn’t an integer.
3: This “fractional dimensionality” number of a coastline will vary somewhat from place to place on the globe (for instance, places with fjords will generally have a higher dimensionality than places with beaches), and even in any one location, the fit won’t be exactly perfect. And in any given location, the relationship will definitely break down eventually when you look at extremely small scales.
4: However, it is also possible to mathematically describe ideal shapes, for which these formulas are exact, and work at all scales.
5: In any of these mathematically ideal shapes, as you look at smaller and smaller scales, the total length increases without bound. That is to say, if you pick any finite number, there is some measurement scale small enough such that, if you measure at that scale or smaller, the total length will exceed your chosen length. In other words, the length is greater than any finite number.
6: In other words, these mathematically ideal shapes do, in fact, have infinite length. Real-world coastlines do not have truly infinite length, but they closely resemble these ideal shapes that do have infinite length, and certainly don’t have a well-defined finite length.
Anyone have any problems with any of these points?
Oh, and the usual construction of a Koch snowflake does always grow outwards as you increase the iteration. But one could also define the exact same curve in the opposite way, by starting with a hexagon and always cutting away material, so it shrinks inwards. One could even, if one were so inclined (though I don’t know why one would be) define it in a way that sometimes grows inwards and sometimes outwards as you iterate.
Activated charcoal is like this.
A little bit. And, I think this is where the confusion comes in:
I would throw more weasel words in there, like “there some theoretical measurement scale…” I think the whole confusion here comes from literalists saying “of course we can measure the coastline of Norway” because they are thinking of physically measuring the coastline with instruments that can be physically created vs. the mathematical fractal theory that says “using infinitely smaller rulers…” I suppose it would be possible to measure the coastline of Norway using electron microscopes; the mathematical theory assumes you can get 10X more accurate than that, and then you can get 10X more accurate than that, and then you can get 10X more accurate than that, and…
The coastline paradox is a great mathematical question that in its raw statement has no application in the real world. OK, a very small application - the coastline of Norway can be calculated at roughly X if you sail around it; Y if you include the fjords; Z if you include going around the rock formations within the fjords. Past that, measuring coastlines is pretty useless. The mathematical theory, however, is useful in, well, mathematics and limit theory; probably useful in physics and astrophysics. It’s called theoretical mathematics for a reason - great and necessary for mathematics; limited application in every-day life.
I mean, that is the application you would expect it to have. It seems a bit unfair to say a statement about the length of coastlines have “no applications, except with regards to the lengths of coastlines”.
And that’s why my point 5 was referring only to the mathematically ideal shapes, not to real coastlines. Nothing in the real world is ever actually a mathematically ideal shape, not even the simple shapes like circles and rectangles.