Like this, then?
As others have mentioned, this is the specific comment that I question.
I think it’s an open question in our universe, hinging primarily on whether space is quantized or not. At any rate, I think it’s quite obvious that your “finite” vs “infinite” vs “fill the universe” is quite off the mark:
Do you actually think those criteria are sufficient to prove that an infinite coastline is impossible? Do you think it’s impossible even in the abstraction? I still think you’re confusing dimensions, erroneously thinking that a finite 2 dimensional area must necessarily have a finite 1 dimensional boundary.
Consider a two dimensional finite universe where the space is modeled by a unit square (I^2, where I is the closed and bounded interval from 0 to 1, for example). Put a Koch snowflake inside, and Voila! You have an infinite boundary surrounding a finite area which does NOT fill up the finite universe.
You may have other objections to why this is physically impossible (if so, I would like to hear them) but “a place of finite size on a planet of finite size has an infinite coast which, by definition, would fill the universe” is simply not sufficient to demonstrate such an impossibility here. It simply isn’t so.
Stepping back a moment, there are, I think, some interesting questions that don’t require instantly worrying about a coastline or fractals.
So, how might we define a line in physical space? “breadthless length” is probably not going to cut it. In a physical realm I would hope we could come up with physical definitions of things like congruence, and moreover lack of congruence.
A classical ideal of physics once had the physical universe running as a perfect deterministic automaton. Perfect knowledge allowed perfect prediction. This isn’t the physical universe we exist in. The universe as we know it does not support perfect knowledge and in particular does not contain infinite information. It certainly does not allow for infinite information to exist within a finite volume. It even places an upper bound on the information that can be placed within any finite volume before the volume collapses.
So, I remain unconvinced that it is possible to provide definitions of objects that can adhere to the usual definitions of geometric objects and also provide the needed properties to allow reasoning in the same manner as we accept for abstract geometry. For instance congruence, or betweenness.
We can obtain some form of algebra on physical universe geometry, but it won’t have the same properties as the Euclidian geometry we place our usual fractals in, and I don’t think it will be possible to define fractals in the manner we want.
In the limit, if we want to define say an infinite curve via a physical object, that curve must be defined by the information content of the physical object. So, the intersection of the sea with the land is defined by the physical properties of the sea and land, each of which is defined by a finite amount of information. Either the intersection has finite information, in which case it is not a fractal and has finite length, or the intersection obtained its infinite structure from infinite information contained in the structure of the land and sea. At which point we have something of a problem.
Except that a fractal, even a mathematically-ideal one with structure at all scales, can be defined with a finite and in fact fairly small amount of information, if you use that information correctly. We do know (or at least, think we know) that there’s an upper limit to the possible information density of the Universe, but we don’t know how the Universe encodes that information.
Sure, but that isn’t the point. The claim is that the physical universe presents the fractal as a physical thing. That is not the same thing as a simple deterministic algorithm defining an abstract curve in space.
We don’t claim, say, that the information within a black hole can be re-encoded by correctly using a limited amount of information less than that that fell in. There is a strict amount of information in the volume. The claim that there is a definable fractal coastline does not include any algorithm for creating that fractal from limited information. It claims that as an intrinsic part of the operation of the laws of nature that fractal exists and can be recovered from the coastline. There are an infinite number of ways of defining a fractal algorithm, but we assume only one actual coastline.
Civil and legal definitions get complicated: besides the “instantaneous” water level there is an apparent shoreline, a mean water level, mean high water line, mean higher high water level, etc., and of course you have to agree on the methods and data for determining all of those…
Except it exists as a physical thing. The interface between the ocean, land and air, all 3 of them physical things, is itself a physical thing. What it isn’t, is well-defined. That doesn’t make it any less real than, say, an electron that exists in a probability cloud.
Now, obviously, there’s a lot of room there for ambiguity, such as the way the ocean can be considered just a really water-saturated part of the atmosphere. But we have ways to account for ambiguity other than just saying “it’s not real”
Oh, and this definition:
is self-referential (how’s a “shore” defined without referencing the water?) and therefore invalid.
We can perhaps define some limitations on what is expected, and maybe limit things a little.
The coastline, or for these discussions, any outline fractal outline, lies on a plane. (The geoid is not a sphere, but for simplicity we can just say the fractal must lie in a plane and the island is small enough. It doesn’t matter.)
The coastline cannot intersect or otherwise touch itself. (If it does, it creates an island, which is not part of the coastline.)
The coastline is within a finite sized continuous region of the plane.
The coastline must be defined by the physical properties of the matter comprising the island.
Worrying about the waterline or the like is really not needed. We don’t need to be considering a coastline. The same argument that Mandelbrot made about the coastline of Britain could be made about the surface are of the moon. The moon is cratered, and craters are a fabulous fractal, much better than a coastline. So in the same vein we could define the surface of the moon, and require that it lie within a finite volume of space, never intersect or otherwise touch itself, and be defined by only the physical nature of the moon’s matter. Same argument.
Bingo! Dat’s the one! A mere 2175 posts. The thread deals extensively with the meaning and interpretation of infinite decimals, and the use of limits to define them. As such, it is relevant to this thread.
Now, in THIS thread, the main contention seems to be a dispute between the theoretical idea of an infinite fractal shoreline, vs the physical possibility of such a thing. I’ll stay out of that aspect of this thread.
Several posters here appear to be unclear on the theoretical notion of infinite sums, which may in total be finite (e.g., the area enclosed by the snowflake curve) or infinite (e.g., the perimiter of the snowflake curve).
The thread linked here attempts to give a basic understanding of all that, especially starting at Post #37 there (by me). I highly recommend that all participants in this thread review this old thread, at least the first 100 or so posts. (It gets pretty much off into the weeds eventually.)
The mathematical difference between curves that have finite length, and curves of infinite length, is that the former are differentiable (a well-defined tangent line, i.e., no sharp corners) almost everywhere. However, when talking about such distinctions one is clearly thinking of platonic objects and not stringing together atoms or force fields.
You can also have a curve of infinite length (contained in a finite-area region) that’s differentiable everywhere, though. Like, a spiral that asymptotically approaches some maximum radius.
It wouldn’t be too hard to define something very much like a Koch curve that was smoothed out with tiny (relative to the length of the line segments they were connecting) pieces of circles at the corners.
You are absolutely right; we would have to say something about compactness if we wanted to rule out such cases.
Those Koch curves are all corners.
Yeah, you might say that you’re making your fillet with a radius of, say, 1% of the length of your segments, and thus have it nicely out of the way where it won’t cause any problems, but five or six iterations later, you’ll find that you need to put a small triangle right on top of where that fillet was.
There are some points that are not on corners (in fact, the vast majority of them). But every point is close to corners: For any distance you choose, there is some iteration number at which there’s a corner within that distance (and hence, at the infinite iteration for the full curve, an infinite number of corners within that distance). It’s like the relationship between rational and irrational numbers: Most numbers are irrational, but there are always rationals close by (in fact, I’m pretty sure that you can make a direct correspondence between corners of the Koch snowflake and the rational numbers).
Although, now that you mention it… I think you might be able to make something similar to a Koch snowflake, with all corners filleted, just by using constant ratio of fillet to segment length, and once you get down to where you would be putting triangles on a fillet, just skip those triangles. At every spot on the curve, then, there would be some maximum iteration beyond which you get no further detail… but there would always be some other spot on the curve that does still have detail.
Could you instead specify that you’re shrinking the fillet for every new and existing corner with every iteration?
You are correct. I didn’t think it through.
I think this would work.
Start with a straight line segment.
For every iteration replace every straight line segment with four line segments and three arcs of circles as follows:
Choose a basis so the endpoints of the segment are (0, 0) and (1, 0).
Segment 1: (0, 0) to (1/4, 0).
Segment 2: (3/8, sqrt(3)/24) to (11/24, sqrt(3)/8).
Segment 3: (13/24, sqrt(3)/8) to (5/8, sqrt(3)/24).
Segment 4: (3/4, 0) to (1, 0).
Arc 1: The part of the circle with center (1/4, sqrt(3)/12) and radius sqrt(3)/12 that smoothly connects segments 1 and 2.
Arc 2: The part of the circle with center (1/2, 5 * sqrt(3)/36) and radius 2 * sqrt(3)/72 that smoothly connects segments 2 and 3.
Arc 3: The part of the circle with center (3/4, sqrt(3)/12) and radius sqrt(3)/12 that smoothly connects segments 3 and 4.
The limit of iterating infinitely will be both infinitely long and differentiable.
And, unlike the asymptotic spiral, a fractal.
ISTM this merely defines a curve that is smooth everywhere except for a set of measure zero. Can you explain why it should be infinitely long?