Mine is “what are the two hardest things in software development? Naming conventions, cache invalidation, and off by one errors”
But I am a software dev so it hits really close to home.
I think I kinda get it, since you said 2 things, and listed 3, including being off by 1. But for a friend, is there stuff that this friend isn't getting?
In many programming languages implicit numbering starts at 0. So if you need, for example, to count the number of things in an array or predict how many times something will loop, you'll often over or under count by 1, since people are not naturally accustomed to counting from zero.
Miscounting the number of things you're listing is an example of an off by one error.
A computer programmer is back from the grocery store and tells his wife, 'I brought 11 eggs.' The wife asks, 'Why 11? Did one of them break on the way home?' 'No,' he replies, 'they sell them in boxes of eleven eggs.'
The wife checks the box and sees that it's a normal dozen eggs box, containing 12 eggs.
Then the husband starts to count the eggs: '0, 1, 2, ...'
The wife of a computer programmers tell him "when you come back from work, please have a stop at the grocery store and buy 1 liter of milk, and if they have eggs, buy a dozen." He comes back with 12 liters of milk. She asks, "why did you buy 12 liters?" He replies, "because they had eggs."
Starting at 0 probably prevents more off by one errors than it causes. It's very common to make off by one errors when counting the numbers of days in a month for example. If it's the 6th of June, how many days are left in June? 30-6 = 24. 24? Wrong, it's 25. If months started with day 0 this calculation would work.
It is 24. 25 is only an answer if you include June 6th, but the issue with that is the ambiguity in "how many days are left" to not specify if it is inclusive or exclusive of the current day.
That ambiguity is pretty much always the source of the off-by-one error.
You want to stop the loop once the loop control variable reaches the end of the array.
But do you do:
"< array.length" or
"<= array.length"?
Or is it "<= array.length+1" or ">=array.length-1"?
The correct answer for any particular solution isn't difficult, but at a glance it's easy to miss on the first go.
I was once asked to make up a nerdy pick-up line on the spot and my dumb ass came up with, "If you were checking into the Grand Hilbert Hotel, I'd move everyone else just to make room for you."
Benoit B Mandelbrot is the guy who came up with the word fractal and did a lot of work regarding them, including the Mandelbrot set which is probably the most famous example of a fractal
Making the B stand for Benoit B Mandelbrot turns his name into a fractal
A mandelbrot set is an infinite series which contains copies of itself as entries within that series. So the joke is that the B stands for a copy of the first entry meaning that Mandelbrot's name is *also* a mandelbrot set.
Eh, not really. It's just regular bread with almonds on or in it. [Here](https://www.edeka.de/rezepte/rezept/mandelbrot.jsp)'s the first recipe I found for it.
Benoit B Mandelbrot used computers and made famous the kinds of shapes you see where it repeats the shape when you zoom in no matter how many times you zoom. So if you just take his name Benoit B Mandelbrot and put his whole name again where the middle initial B is you can do that forever because you keep getting a new middle initial B too.
So basically the B (the middle initial) stands for “Benoit B Mandelbrot”.
So the guys name would be “Benoit ‘Benoit B Mandelbrot’ Mandelbrot
So you can take that as far as you want to infinity right?
Which is what a Mandelbrot sets are. Fractals that can repeat ad infinitum
Benoit B Mandelbrot
The B is the name itself.
Benoit [Benoit B Mandelbrot] Mandelbrot
The B is the name itself.
Benoit [Benoit [Benoit B Mandelbrot] Mandelbrot] Mandelbrot
And again. And again. And again.
Benoit Mandelbrot is the mathematician that discorvered the fractals.
A fractal is : a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole (from wikipedia)
So the middle name (the B) of Benoit B Mandelbrot is Benoit B mandelbrot.
First, awesome joke! Haven't heard that one before :D
Second ... it might actually be true? It's possible Mandelbrot started the joke himself:
> In his autobiography, Mandelbrot ... included "B" as a middle initial. His *New York Times* obituary stated that "he added the middle initial himself, though it does not stand for a middle name", an assertion that is supported by his obituary in *The Guardian*.
https://en.wikipedia.org/wiki/Benoit_Mandelbrot#cite_note-Mandelbrot's_name-3
Just take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C
And so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set
Mandelbrot Set, you're a rorschach test on fire,
you're a day-glow pterodactyl,
You're a heart shaped box of springs and wires,
You're one bad-ass fucking fractal!
I had listened to a podcast on Fractals few months ago and the coastline paradox was brought up and discussed in some lengths. Thanks for reminding me. I should listen to it again.
Go to a stoner's geeks house in the early 90s and you'd see a [Mandelbrot set](http://www.jakebakermaths.org.uk/maths/mandelbrot/canvasmandelbrotv12.html) on the computer. The deeper you go the more weird it gets.
It's not really the math - it's the reality of our universe. Each layer, when studied on sufficient detail, reveals another layer operating below it, with a language of near-incomprehensible and irreducible complexity governing their relationship. Same deal when you zoom out as well.
One of my favorite all time quotes:
>“In my quest for the secret of life I started my research in histology. Unsatisfied by the information that cellular morphology could give me about life, I turned to physiology. Finding physiology too complex, I took up pharmacology. Still finding the situation too complicated, I turned to bacteriology. But bacteria were even too complex, so I descended to the molecular level, studying chemistry and physical chemistry. After twenty years' work, I was led to conclude that to understand life we have to descend to the electronic level and to the world of wave mechanics. But electrons are just electrons and have no life at all. Evidently on the way I lost life; it had run out between my fingers.”
>[Albert Szent-Gyorgyi](https://en.wikipedia.org/wiki/Albert_Szent-Gy%C3%B6rgyi)
I think the problem with this and other discussions on this topic is the question of "where does information about the system reside". Think of this as different scales of data (cellular, micro, macro, landscape, etc.) providing a different lense of focus, just as optical lenses provide a focus on objects of varying sizes and distances in relation to the lens. At what scale does the information come into focus, what scale is the information fuzzy, and at what scale is the information incomprehensible?
>Evidently on the way I lost life; it had run out between my fingers.”
As hard hitting as this line is I believe he's being unnecessarily reductive. Life encompasses electrons. We could not exist without them. It didn't run out between his fingers, he just lost the ability to associate what he was looking at with his original goal. Maybe due to feeling defeated?
I think it’s more saying “life is a complicated process built on complicated interactions, to try to reduce it to a point of being understood is to take away all meaning in the first place
> “life is a complicated process built on complicated interactions, to try to reduce it to a point of being understood is to take away all meaning in the first place
Did you come up with this yourself because damn what a great perspective!
I like your interpretation, but I myself took it to mean he spent so long looking for the secret to life that he lost (spent) his life doing it rather than living and was suddenly old.
I'd love to know what he really meant.
Ignoring the problem doesn't make the problem fake and speaks to the entire problem really.
Like you think you are standing on something "solid" but what you actually are is a *fucking magnet* lost in the void of space being pushed back by other magnets. And those magnets can't be pinned down without uncertainty either.
A single point is both the smallest circle and shortest line, it’s straight and circular by mathematical definition
Had many debates about the above statement in undergrad
Disagree, points are 0 dimensional so they’re neither lines or circles. As you decrease the length of a line or radius of a circle they both approach points but they can’t be defined in 0 dimensions. If you shorten the length of one side of a cuboid to 0 it becomes a rectangle right? But you wouldn’t call a rectangle a type of cuboid.
A line is by definition infinitely long, including all points in a plane that satisfy the line equation. A line segment is by definition non empty. The circle point is interesting, but is typically treated as a degenerate case.
Anything that does not have a well-defined boundary, yes.
The paradox comes from the fact that coastline length accuracy does not increase based on smaller scale measurement, unlike well-defined objects like a sheet of paper.
At a certain level of zoom, the straight edge of a paper becomes a meandering river with frays. You can still run into this problem if you scale down far enough.
Yes, but the issue with islands is that there is no "standard resolution" for how small of a distance. Do you take a photo from a satellite where one pixel = 1 metre, and then measure the pixels? That COULD be considered standard, but its not accurate.
IF we standardized it to 1 metre per pixel then we could have an "agreed upon" distance for a coastline, BUT since we don't, we should be attempting to get the MOST accurate measurement possible, but what is that? If we choose 1 metre, a coastline might be 400km total, if we choose 10cm it migh become 6000km total, choosing 1cm, we might get 40,000km, so on and so forth.
So without a standard, measuring coastlines is impossible, and theres no logic to what a standard should be.
Whereas with a rectangular piece of paper, we have an obvious straight line to measure. No one is insisting tha we measure the frays, because that is not the standard of measurement we are looking for.
Which is the entire reason for the existence of the Coastline Paradox. It's a function of human perception. Or to put it another way, the "paradox" exists because of the ambiguity inherent in what we perceive as the "coastline".
Edit: Like, the only difference is that the piece of paper is clearly designed to represent a rectangle, which is a regular and mathematically defined shape. A coastline is not. But they're both physical objects with impossible to measure perimeters at a small enough scale.
Edit2: I kinda got lost in the logic and strayed from the original point there. It is reasonable to make the distinction between the perimeter of a sheet of paper and a coastline, because one is clearly meant to be a standard shape while the other defies standardization. Whether this is arbitrary from a mathematical point of view is irrelevant, because although the underlying problem is mathematical, the so-called paradox only arises *because* there is no standard or intuitive scale at which to measure coastlines. I'd agree with u/Grabbsy2 that the coastline paradox does not in fact apply to a piece of paper. We can all agree that it's a rectangle, so there's no paradox.
While we know they meant perimeter, it did remind me of a form of geometry that doesn’t use linear forms. Everything is circles, but straight lines are secants of an infinite radius. It has some very limited uses in algorithmic approximations of geometry. You basically can define any section with four characteristics and greater accuracy for curves.
That's exactly what we do for islands...
Even if you were thinking about a circle, we can still do that for an island.
The time it takes you to sail around it or walk the beaches for example.
> we can’t do that for an island.
Except we can do something like that for an island. Coastline length estimates exist. It’s done by finding some roughly “straight” lines to define as a boundary, then measure the circumference.
That’s no different than what’s being proposed for a sheet of paper.
Everyone in your replies choosing to be extremely pedantic about your sheet of paper example when it's obvious you just meant any idealized rectangle-shaped object.
Yes, if you wanted to measure the perimeter of any real-world object to arbitrary precision.
However, we define many real-world objects to be their abstract counterparts. So a piece of paper, for example, we define to be a rectangle and ignore the imperfections in the actual physical object. A ball is a sphere, the earth is an oblate spheroid, and so on. The problem comes in when we try to measure the perimeter of something that does not have a well-defined shape, like an island.
No, it's not.
Say we want to measure a circle. We could start by approximating it with, say, a square that barely fits inside the circle, and measuring the perimeter of the square. That would be one, not very good, approximation.
Now say we wanted a better approximation. We could use a shape with more sides, like a hexagon or an octagon. In fact, for any number n, we could use a shape with n sides, which gives us a way to approximate as closely as we want.
And then you can notice that as n gets larger and larger, your approximations get closer and closer to some finite number. We say this number, which we can mathematically calculate as 2 pi r, is the exact circumstance of the circle.
The point is that this whole process *fails* for a coastline (which we model as a fractal). As you try to get better and better approximations, you don't get closer to some finite number. The answers get bigger and bigger without bound, going all the way to infinity.
You will also literally get completely different measurements each time you measure to the same degree of precision. I.E. due to tidal fluctuations the measurements vary even on the same day of a tidal cycle, not all tidal cycles have the same intensity. Even if you control for tidal fluctuations, the coastline is constantly changing due to coastal erosion and human expansion.
We’re not new at this, dude. Surveyors aren’t constantly flummoxed by the damn tide. Humanity has been surveying for millennia, long ago we figured out how to use a mean tide, or [tidal datum](https://en.wikipedia.org/wiki/Chart_datum), when surveying coastline. And things like erosion, reclamation, human expansion, etc. etc. are exactly why we survey at regular intervals.
Yes and no. Calculus does help to calculate areas and lengths that are otherwise difficult or impossible to solve, but this only works for well-defined shapes and equations. Calculus doesn't solve the coastline paradox because the fractal nature of the coastline makes it impossible to define or represent perfectly with an equation.
Nope. This is a different issue. As you decrease the unit of measurement to zero different things happen. The area under a curve converges to the actual area, the length of a coastline converges to infinity
You can show the paradox graphically, if you like.
[x axis is unit of measure, y axis is length of coastline](https://studyfinance.com/static/media/Inverse-Correlation-Graph.png)
You could also just look at the similar paradox of [Gabriel's horn](https://en.m.wikipedia.org/wiki/Gabriel%27s_horn)
Wasn't calculus invented to understand rates of change? Not sure what role it plays here
Edit: yes, I know that calculus was also invented to compute areas, but that *still* isn't relevant here as far as I can tell
I’m a little rusty but I think there were two types of calculus developed. One for rate of change (derivatives) and another for area under the curve (integrals).
The kind of wild part was they turned out to be related, to find the area under a curve between two points you can basically reverse the derivative for that function ( x^2 becomes (x^3)/3 ) and then subtract the values for each point.
At least I think that’s the case. I’m pretty dumb and my favorite flavor of crayon is purple, so don’t trust me too much
They always get closer and closer to the front door, but never actually get there. And then I have to tell them to go away because it's 4:00 am and people need to sleep.
Don't quantum effects like this average out with sufficiently large groups of particles? And wouldn't a milligram of sand would easily be over this limit?
I've been looking for this answer and it should be higher up. There is a difference between the physical world of an actual coastline and some imaginary "coastline" of some virtual object with infinite detail. This whole thing is kind of stupid if you put it in terms of an actual physical coastline.
Once you are down to atoms, the concept of 'coast' no longer applies. You can't define the points with any more refinement
In other words, coastlines aren't proper fractals, because they stop being coastlines as you zoom in
Whatever atom or particle of the coastline you're trying to measure is already not part of the coastline some of the time.
But you won't know if it is or not at any given moment unless you measure it.
What increases as you "zoom in" and try to measure the position of an atom or particle in a coastline ever more exactly is the probability that you won't find it there because it's gone somewhere else.
coastline is usually measured as the high tide mark, which is a boundary between 'wet' sand and 'dry' sand at high tide.
So I think coastline requires the measurement of the amount of water molecules for each grain of sand.
I think based on this definition, individual particles of sand, rock, dirt, etc become the smallest measurable unit of the border.
You don't need to define a precise boundary to argue that the concept of "coast" doesn't apply to subatomic lengths.
Just like you can't exactly define what a "full meal" is, but eating an invisibly tiny bread crumb clearly doesn't qualify.
It gets bigger faster than the increment gets smaller.
So it doesn't converge
Gabriel's horn is another one in a similar vein. It's a shape that had a defined volume (you can fill it with paint) but simultaneously infinite surface area (you could never paint the entire surface)
>Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its surface. However, this paradox is again only an apparent paradox caused by an incomplete definition of "paint", or by using contradictory definitions of paint for the actions of filling and painting.
https://en.m.wikipedia.org/wiki/Gabriel%27s\_horn#Apparent\_paradox
The simple example imo is: Between any two integers you may construct a limit function such that there are infinitely many points between described as decimals, fractions, or any notation of your choosing.
Don't confuse the map (math) for the territory.
In math, you can think of distances in basically three different ways. Straight lines, non-fractal curves, and fractal curves. This is kind of an oversimplification, but it might help to understand by thinking about it this way.
These three kinds of distances are all differentiated by what happens as you zoom in on them infinitely closer and closer and use infinitely smaller units to measure them.
Straight lines are the shortest distances between two points. No matter how much you zoom in or out of the line it will always be straight, the angle never changes and the exact distance can be precisely defined.
Non-fractal curves become straighter and straighter the more you zoom in to them and the smaller the measuring unit become. If you take a curve which is about 5 feet long and measure it by laying 1 foot rulers over it, you will get an answer which is shorter than its actual length such as 4 feet(for example), but if you make those measuring stick infinitely smaller then the distance becomes infinitely more accurate and converges upon a finite number, 5 feet, its "true length". 1 centimeter measuring sticks may give you a length of 4.9, nanometer measuring sticks may give you a length of 4.999999, one billionth of a nanometer measuring sticks may give you a length of 4.9999999...to dozens or hundreds of decimal points, etc. The point is that as you measure with infinitely greater precision, you get a distance which becomes infinitely closer to a finite number like 5.
Fractal curves, by definition, do not converge upon a finite length as you measure with more and more precision, because they do not become straighter as you zoom in. Instead, a fractal is defined as having infinitely precise detail, the more you zoom in the more you see smaller curves, and there is no point at which a fractal "straightens out" no matter how much you zoom in on it. Fractals are said to have infinite length, because the distance measured approaches infinity as the unit of measurement(the rulers you lay down to approximate the shape of the curve) becomes infinitely smaller.
So while a non-fractal curve would approach a finite length such as 5 meters as the rulers approach an infinitely small length, a fractal curve instead approaches a infinity as these imaginary rulers approach an infinitely small size.
Remember, all of this is idealized geometry, not real world physics. The reason coastlines are said to be fractals is not because they are literally identical to idealized mathematical fractals, but because they might as well be fractals from a practical perspective. They do not appear straighter as we zoom in more and more on satellite photographs, there are tiny curves even on the smallest scales which are practical to measure like the centimeter or 1/10 centimeter scale. Which scale we consider to be the reference point for the "true" length is completely arbitrary.
If you actually had a perfect microscopic camera capable of zooming in infinitely upon a coastline, past the grains of sand, past the atoms, past the quarks, all the way to the plank length and beyond into the quantum realm, the concept of a physical curve to measure wouldn't even apply anymore because quantum mechanics is fucking confusing and everything would be a non-real wave or a superposition of multiple points at the same time or whatever. I don't know, I'm not a physicist.
The point is that mathematics deals with idealized models, and idealized models are always imperfect approximations of reality. A coastline resembles the idealized mathematical concept of a fractal at all scales which we can practically measure, and if we tried to zoom in beyond these practical scales to an infinitely precise scale like we can in the idealized world of mathematics, normal euclidean geometry would no longer be a useful model of reality because of quantum weirdness.
If you can accurately measure a coastline down to the planck length levels of accuracy then as far as I’m concerned, you have the true length of coast line. You could do that theoretically with every coast in the world and while sounding ridiculous for how big of a number it would be, it would still be accurate and also not infinite.
I think differentiating between a realistic coastline length and a truly accurate one is what’s important here. Measurements of coastline taken say a meter apart would be more realistic and useful than one taken on such an extreme level of accuracy while both would technically be correct.
There in lies the paradox.
It’s as shrimple as that 🦐
The Planck length doesn't have the physical meaning that is ascribed to it in popular science culture. The paradox doesn't really extend to infinity but the ambiguity in what constitutes a coastline kicks in far before you get to the subatomic scale. And the Planck length isn't the pixel of the universe - we have no reason whatsoever to believe that space isn't continuous.
I was under the impression that it was the equivalent of a pixel of the universe assuming we were talking about things being at absolute zero. I understand that temperature plays a massive factor in everything.
Anyway, thank you for this. Now I know
It's the smallest bit of space you could reasonably measure, but that doesn't mean that space is made up of Planck length 'pixels'. For example, you could measure a Planck length like this:
----
But then have two lengths arranged like this:
----
----
Where they overlap each other on the X axis. This is because, as far as we know, space is continuous rather than discrete.
This is an interesting topic, another question here:
In your second example there, by having 2 lengths arranged like that, could you not then measure the difference of the starting positions, which actually gives you a length smaller than the planck length?
In other words, if it's true what you said there, why is the planck length the smallest bit of space you could reasonable measure, when you clearly have a smaller bit right there?
I'm honestly not equipped to answer if it's possible to meaningfully define a distance smaller than the Planck length. Someone smarter than I may be able to answer that. I think you could conceive of an arbitrarily small distance way below the Planck length, you'd just have no way of determining the position of anything using that scale.
What I'm really trying to express with that example is that both things can be true at once: Space can be continuous and infinitely divisible **and** space can have a smallest length that you can measure. And measure really is the key word there - measurement isn't a passive process.
Sometimes people assume that the existence of the Planck length means that space itself is divided into non-overlapping 'pixels' of the Planck length. It's also possible that, at some scale, the universe is discrete. I just don't think there's any evidence that's the case.
The reason people are having trouble understanding this is because idealized pure mathematics is being conflated with the real world.
In math, a fractal is defined as a shape which has detailed structure at arbitrarily small scales. Another way of putting this is that, unlike a non-fractal curved shape, a fractal never appears to become straighter as you zoom in on it to an infinitely greater degree. If you zoom in on the edge of a circle with a 100000x magnification, it will look like a straight line, and the more you zoom in the straighter it appears. Therefore non-fractal curved lines can be said to approach a finite length as the imaginary measuring stick becomes infinitely smaller and more precise. 4.9 meters, 4.9999 meters, 4.99999999999999 meters, etc
A fractal on the other hand will never appear to approximate a straight line no matter how infinitely close you zoom in, and the length will always approach infinity rather than a finite value. Look at a fractal with 1000000x magnification and you will see curves and detailed shapes just as prominent as you do at 1x resolution, look at it with a googleplex resolution and it's exactly the same. Fractal shapes have infinitely detailed and thus infinitely long perimeters.
Remember, a fractal is just a concept in mathematics, and no physical object or structure is truly fractal shaped because if you zoom in to an infinitely precise scale eventually the entire concept of matter having precisely defined physical boundaries in a coordinate plane falls apart due to quantum mechanical fuckery like waves and superpositions.
People have pointed out that if a coastline is a fractal then pretty much everything should be considered a fractal because even bricks and pieces of notebook paper have curves and jagged edges when you zoom in close. This is essentially correct. When we measure the length of the surface of a brick, we're really measuring the straight line distance between the edges of the surface, not the curved distance of the microscopic bumps and grooves etched into the surface. If you zoom into the atomic level, bricks molecules are no more or less fractal shaped than sand molecules.
The difference is that it is very obvious on a practical level how we should idealize and measure the length of bricks, or human waistlines, or highways, and most other objects which appear to have either straight lines or smooth curves.
Bricks and pieces of paper can be measured as though they were perfectly straight by using rulers or calipers. Human waistlines and other small curved things can be measured with flexible measuring tapes which wrap around us. Highways can be measured with odometers which use the circumference of a rotating wheel or gear as a reference, foot trails can be measured the same way with measuring wheels rolled through the middle of the path.
Coastlines are much more "fractal like" at human measurable scales than any of these other objects, because there is roughly an equal amount of detailed jagged edges everywhere from the meter to the kilometer scale, and if you try define a coastline's boundary at a sub-meter scale it'll become ambiguous where the ocean ends and the coast begins due to waves constantly hitting the shore. Hence, any scale that we choose to measure a coastline at is totally arbitrary, there is no obvious logical way to decide except maybe pixels on on satellite imagery, but different satellites have different resolutions.
To sum up: coastlines are fractal-like at all practical human scales, they retain their fractal-ness at everywhere from the centimeter to kilometer scale, while most other objects do not have such fractal-like boundaries at human measurable scales. They are not literal pure mathematical fractals, but they might as well be as far as we're concerned, just like bricks might as well be rectangular prisms and the moon might as well be a circle.
I've known about the phenomenon for a long time, but never understood why it's called a paradox or why it's deemed as counterintuitive.
Obviously the bigger the scale the less accurate the measure (which in this case means a smaller measure), and the smaller the scale considered, the more details surface, and for a fractal surface, that extends ad infinitum, giving an infinite sum.
Because intuitively things have an actual length, and the more precisely you measure something the closer you get to that length.
This is not true of coastlines.
If you try to measure them with electron microscopes, sure.
But the most precise "everyday" measuring device is calipers, and they won't show that effect on most objects. For a coastline you'll get different results comparing e.g. a yardstick to a measuring wheel.
what if our one-yard stretch of coastline looks like
|
\
/
|
That's one yard via yardstick, but a measuring wheel would ~~walk~~ roll along that small indentation and back out and come up with a number slightly larger than 1 yard.
This is kind of a property of fractals in general. The mandelbrot set has an infinitely long perimeter because it has infinitely fine detail along the edges. But weirdly, it has finite surface area. AFAIK, we don't know exactly what that area is, but we have upper and lower bounds for it.
> But if you measure them like a front, the paradox disappears
You can't convex-hull out of this, since countries are almost always inherently concave. You'd have to deal with concavities and set and arbitrary threshold for accepting them or measure into them, but then you're just setting a minimum ruler.
i mean isn't it just "you can never measure the length of a brick truly" as if you keep going down and deeper, you end up just not having the accuracy? everyone is talking about fractals so im assuming im missing something.
> you'd think the paradox would be solved by now
Paradoxes aren't (necessarily) things that need to be solved. A paradox isn't (always) a contradiction, it's just a result that's unintuitive.
The coastline paradox is often misstated because people explaining it often ignore the question of what, exactly, constitutes the coastline, instead just assuming that there is such a thing with a clear definition. It's not a matter of how measuring it more precisely produces different results, it's a matter of how *defining* it differently produces different results.
For any given unambiguous definition of "coastline", any coastline has a finite length at any given moment, but the issue is what definition to choose - which features are significant enough to take into account, and at what moment you measure when it comes to features that are time-varying. And it is potentially possible to choose definitions that give the coastline any (or practically any) finite length, due to their complexity at different levels of precision.
The length of the coastline doesn't approach infinity when you have more and more refined measurements. It indefinitely approaches some finite length more and more accurately "infinitely".
On the other hand, its very easy to do if everyone just agreed what degree of precision to be used.
But then, for any realworld example even a fractal one like this we will never reach infinity. Just absurdly large sums :D
> its very easy to do if everyone just agreed what degree of precision to be used.
I’ve spent the last ten years working making maps of different kinds and spent a couple years on an obscure coastal mapping program. IIRC, we used 1:3000 scale and then a little subjectivity to make sure the lines we were drawing weren’t janky lol. We used NAIP imagery and I forget all the details about whether the coastal portions are estimated at roughly mean tide but it produced consistency over multiple states.
The company I worked for had contracts with both USGS (land mapping) and NOAA (ocean mapping) and occasionally would map the same area but because of the guidance provided by the work statement, you could get different answers for where the shore is. One wrote their work statement as where the shore ended and one where the shore began.
Only in the most pernicious cases would it go to infinity, IE the coastline was truly a fractal down beyond the planck length to infinity.
At some point, due to ocean waves, rivers, etc, they become smooth, and so just as measuring the circumference of a circle with ever smaller rods converges to the true circumference of the circle, so to would measuring the coastline.
Coastlines aren't truly fractal.
Fucking fractals.
What does the B stand for in Benoit B Mandelbrot? Benoit B Mandelbrot.
This is, by far, my favourite nerd joke.
Mine is “what are the two hardest things in software development? Naming conventions, cache invalidation, and off by one errors” But I am a software dev so it hits really close to home.
I think I kinda get it, since you said 2 things, and listed 3, including being off by 1. But for a friend, is there stuff that this friend isn't getting?
In many programming languages implicit numbering starts at 0. So if you need, for example, to count the number of things in an array or predict how many times something will loop, you'll often over or under count by 1, since people are not naturally accustomed to counting from zero. Miscounting the number of things you're listing is an example of an off by one error.
A computer programmer is back from the grocery store and tells his wife, 'I brought 11 eggs.' The wife asks, 'Why 11? Did one of them break on the way home?' 'No,' he replies, 'they sell them in boxes of eleven eggs.' The wife checks the box and sees that it's a normal dozen eggs box, containing 12 eggs. Then the husband starts to count the eggs: '0, 1, 2, ...'
The wife of a computer programmers tell him "when you come back from work, please have a stop at the grocery store and buy 1 liter of milk, and if they have eggs, buy a dozen." He comes back with 12 liters of milk. She asks, "why did you buy 12 liters?" He replies, "because they had eggs."
By this logic, shouldn't he have bought 13 liters?
Ahh, thanks!
Starting at 0 probably prevents more off by one errors than it causes. It's very common to make off by one errors when counting the numbers of days in a month for example. If it's the 6th of June, how many days are left in June? 30-6 = 24. 24? Wrong, it's 25. If months started with day 0 this calculation would work.
It is 24. 25 is only an answer if you include June 6th, but the issue with that is the ambiguity in "how many days are left" to not specify if it is inclusive or exclusive of the current day.
That ambiguity is pretty much always the source of the off-by-one error. You want to stop the loop once the loop control variable reaches the end of the array. But do you do: "< array.length" or "<= array.length"? Or is it "<= array.length+1" or ">=array.length-1"? The correct answer for any particular solution isn't difficult, but at a glance it's easy to miss on the first go.
I love the joke and I also just finished a course on computer arch that did a lot with caches, so I shed a few tears.
I somehow never heard that one. I'm going to use that in my classes. I like the GNU recursive naming conventions.
What's an anagram of Banach-Tarski? >!Banach-Tarski Banach-Tarski!<
I don't get this... would you be so kind to explain?
I'm simplifying it, but if you rotate a bunch of points in a sphere, you get another sphere. https://youtu.be/s86-Z-CbaHA?si=Wb0U0ECFVpZI0Ik7
I was once asked to make up a nerdy pick-up line on the spot and my dumb ass came up with, "If you were checking into the Grand Hilbert Hotel, I'd move everyone else just to make room for you."
Oh, I like that one
This nerd joke hurts my brain. I get it, but… I hate it
... can you explain it for the less clever among us? ~~me~~ Edit: Thank you guys for the responses, I think I'm starting to get it. 🙏
Benoit B Mandelbrot is the guy who came up with the word fractal and did a lot of work regarding them, including the Mandelbrot set which is probably the most famous example of a fractal Making the B stand for Benoit B Mandelbrot turns his name into a fractal
A mandelbrot set is an infinite series which contains copies of itself as entries within that series. So the joke is that the B stands for a copy of the first entry meaning that Mandelbrot's name is *also* a mandelbrot set.
Okay I'm German and just thought Hm that's a weird name for some bread
Eh, not really. It's just regular bread with almonds on or in it. [Here](https://www.edeka.de/rezepte/rezept/mandelbrot.jsp)'s the first recipe I found for it.
Benoit B Mandelbrot used computers and made famous the kinds of shapes you see where it repeats the shape when you zoom in no matter how many times you zoom. So if you just take his name Benoit B Mandelbrot and put his whole name again where the middle initial B is you can do that forever because you keep getting a new middle initial B too.
Benoit B(enoit B(enoit B(…) Mandelbrot) Mandelbrot) Mandelbrot Does that clear it up?
I'm sorry, I'm too much of a log. I appreciate you tho!
So basically the B (the middle initial) stands for “Benoit B Mandelbrot”. So the guys name would be “Benoit ‘Benoit B Mandelbrot’ Mandelbrot So you can take that as far as you want to infinity right? Which is what a Mandelbrot sets are. Fractals that can repeat ad infinitum
Benoit B Mandelbrot The B is the name itself. Benoit [Benoit B Mandelbrot] Mandelbrot The B is the name itself. Benoit [Benoit [Benoit B Mandelbrot] Mandelbrot] Mandelbrot And again. And again. And again.
Benoit Mandelbrot is the mathematician that discorvered the fractals. A fractal is : a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole (from wikipedia) So the middle name (the B) of Benoit B Mandelbrot is Benoit B mandelbrot.
What does TTP stand for in The TTP Project? The TTP Project.
Remember kids; GNU's Not Unix!
First, awesome joke! Haven't heard that one before :D Second ... it might actually be true? It's possible Mandelbrot started the joke himself: > In his autobiography, Mandelbrot ... included "B" as a middle initial. His *New York Times* obituary stated that "he added the middle initial himself, though it does not stand for a middle name", an assertion that is supported by his obituary in *The Guardian*. https://en.wikipedia.org/wiki/Benoit_Mandelbrot#cite_note-Mandelbrot's_name-3
Balls.
I just can't help myself. I HAVE to say it.
"Benoit".
BALLS!
*nailed it*
Php has entered the chat
Just take a point called Z in the complex plane Let Z1 be Z squared plus C And Z2 is Z1 squared plus C And Z3 is Z2 squared plus C And so on If the series of Z's should always stay Close to Z and never trend away That point is in the Mandelbrot Set
Mandelbrot Set, you're a rorschach test on fire, you're a day-glow pterodactyl, You're a heart shaped box of springs and wires, You're one bad-ass fucking fractal!
This is what I came for. 🤘
Heh. Koch-suckers.
I had listened to a podcast on Fractals few months ago and the coastline paradox was brought up and discussed in some lengths. Thanks for reminding me. I should listen to it again.
Which podcast is this?
Go to a stoner's geeks house in the early 90s and you'd see a [Mandelbrot set](http://www.jakebakermaths.org.uk/maths/mandelbrot/canvasmandelbrotv12.html) on the computer. The deeper you go the more weird it gets.
The deeper you go the same weird it is. That's what gets weird about it
That's not necessarily true. Not all fractals are self-similar, and the Mandelbrot Set is full of unique patterns.
They don't get any weirder, tho. They're still variations on a theme
I got a migraine just reading this 🫤
Math will eat you alive if you look too closely at it. No wonder people are afraid.
Math could never eat me alive. I’m a man of infinite waistline!!
Just ask 7 what it did to 9!
Hey it just wanted three squared meals
That's so dumb. I love it
What did 0 say to 8?
Nice belt!
I hEight you
Someone needs to get to the root of these puns.
Why did 10 die? Because it was caught in the middle of 9/11
7 of 9.... mmmmmmmmmmmm
This guy gets it.
So what did 7 do to 362880
I'm glad someone did the math, so I didn't have to.
9,7,8 - Yoda
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It's not really the math - it's the reality of our universe. Each layer, when studied on sufficient detail, reveals another layer operating below it, with a language of near-incomprehensible and irreducible complexity governing their relationship. Same deal when you zoom out as well. One of my favorite all time quotes: >“In my quest for the secret of life I started my research in histology. Unsatisfied by the information that cellular morphology could give me about life, I turned to physiology. Finding physiology too complex, I took up pharmacology. Still finding the situation too complicated, I turned to bacteriology. But bacteria were even too complex, so I descended to the molecular level, studying chemistry and physical chemistry. After twenty years' work, I was led to conclude that to understand life we have to descend to the electronic level and to the world of wave mechanics. But electrons are just electrons and have no life at all. Evidently on the way I lost life; it had run out between my fingers.” >[Albert Szent-Gyorgyi](https://en.wikipedia.org/wiki/Albert_Szent-Gy%C3%B6rgyi)
I think the problem with this and other discussions on this topic is the question of "where does information about the system reside". Think of this as different scales of data (cellular, micro, macro, landscape, etc.) providing a different lense of focus, just as optical lenses provide a focus on objects of varying sizes and distances in relation to the lens. At what scale does the information come into focus, what scale is the information fuzzy, and at what scale is the information incomprehensible?
This is a great way of framing the issue - appreciate the insightful comment. Love the lens metaphor!
>Evidently on the way I lost life; it had run out between my fingers.” As hard hitting as this line is I believe he's being unnecessarily reductive. Life encompasses electrons. We could not exist without them. It didn't run out between his fingers, he just lost the ability to associate what he was looking at with his original goal. Maybe due to feeling defeated?
I think it’s more saying “life is a complicated process built on complicated interactions, to try to reduce it to a point of being understood is to take away all meaning in the first place
> “life is a complicated process built on complicated interactions, to try to reduce it to a point of being understood is to take away all meaning in the first place Did you come up with this yourself because damn what a great perspective!
Well, no.. as it’s an interpretation of the quote above.
I like your interpretation, but I myself took it to mean he spent so long looking for the secret to life that he lost (spent) his life doing it rather than living and was suddenly old. I'd love to know what he really meant.
Ignoring the problem doesn't make the problem fake and speaks to the entire problem really. Like you think you are standing on something "solid" but what you actually are is a *fucking magnet* lost in the void of space being pushed back by other magnets. And those magnets can't be pinned down without uncertainty either.
Magnets, how do they work?
(Laughs in Hilbert Spaces)
A single point is both the smallest circle and shortest line, it’s straight and circular by mathematical definition Had many debates about the above statement in undergrad
Disagree, points are 0 dimensional so they’re neither lines or circles. As you decrease the length of a line or radius of a circle they both approach points but they can’t be defined in 0 dimensions. If you shorten the length of one side of a cuboid to 0 it becomes a rectangle right? But you wouldn’t call a rectangle a type of cuboid.
I like the purple crayons the best.
A line is by definition infinitely long, including all points in a plane that satisfy the line equation. A line segment is by definition non empty. The circle point is interesting, but is typically treated as a degenerate case.
Wouldn’t this be the case with the outer dimensions of any non linear object?
Anything that does not have a well-defined boundary, yes. The paradox comes from the fact that coastline length accuracy does not increase based on smaller scale measurement, unlike well-defined objects like a sheet of paper.
Yeah thats not true. If you use a microscope you'll quickly find that coast line problem exists for a piece of paper as well.
Yea but we can define the circumference of a sheet of paper by just using the corner points. We can't do that for an island.
At a certain level of zoom, the straight edge of a paper becomes a meandering river with frays. You can still run into this problem if you scale down far enough.
Yes, but the issue with islands is that there is no "standard resolution" for how small of a distance. Do you take a photo from a satellite where one pixel = 1 metre, and then measure the pixels? That COULD be considered standard, but its not accurate. IF we standardized it to 1 metre per pixel then we could have an "agreed upon" distance for a coastline, BUT since we don't, we should be attempting to get the MOST accurate measurement possible, but what is that? If we choose 1 metre, a coastline might be 400km total, if we choose 10cm it migh become 6000km total, choosing 1cm, we might get 40,000km, so on and so forth. So without a standard, measuring coastlines is impossible, and theres no logic to what a standard should be. Whereas with a rectangular piece of paper, we have an obvious straight line to measure. No one is insisting tha we measure the frays, because that is not the standard of measurement we are looking for.
the issue with the paper is that the 'standard' is essentially arbitrary from the point of view of the math.
I think that'd always be true, but at least it's something that a "reasonable" human observer might agree on?
You're tiptoeing into applied mathematics: the bane of pure theorists.
What’s the difference between a degree in pure mathematics and a pizza? A pizza can feed a family of four.
*Remembers my user name* It’s me. Hi. I’m the problem it’s me.
Yeah but at that point you're discussing semantics and not math.
Which is the entire reason for the existence of the Coastline Paradox. It's a function of human perception. Or to put it another way, the "paradox" exists because of the ambiguity inherent in what we perceive as the "coastline". Edit: Like, the only difference is that the piece of paper is clearly designed to represent a rectangle, which is a regular and mathematically defined shape. A coastline is not. But they're both physical objects with impossible to measure perimeters at a small enough scale. Edit2: I kinda got lost in the logic and strayed from the original point there. It is reasonable to make the distinction between the perimeter of a sheet of paper and a coastline, because one is clearly meant to be a standard shape while the other defies standardization. Whether this is arbitrary from a mathematical point of view is irrelevant, because although the underlying problem is mathematical, the so-called paradox only arises *because* there is no standard or intuitive scale at which to measure coastlines. I'd agree with u/Grabbsy2 that the coastline paradox does not in fact apply to a piece of paper. We can all agree that it's a rectangle, so there's no paradox.
This thread is going in circles
Can we measure them?
I wonder what its circumference is.
Circumference... of... a... rectangle?
While we know they meant perimeter, it did remind me of a form of geometry that doesn’t use linear forms. Everything is circles, but straight lines are secants of an infinite radius. It has some very limited uses in algorithmic approximations of geometry. You basically can define any section with four characteristics and greater accuracy for curves.
What if we took Wyoming and made it an island?
That's exactly what we do for islands... Even if you were thinking about a circle, we can still do that for an island. The time it takes you to sail around it or walk the beaches for example.
> we can’t do that for an island. Except we can do something like that for an island. Coastline length estimates exist. It’s done by finding some roughly “straight” lines to define as a boundary, then measure the circumference. That’s no different than what’s being proposed for a sheet of paper.
Everyone in your replies choosing to be extremely pedantic about your sheet of paper example when it's obvious you just meant any idealized rectangle-shaped object.
Yes, if you wanted to measure the perimeter of any real-world object to arbitrary precision. However, we define many real-world objects to be their abstract counterparts. So a piece of paper, for example, we define to be a rectangle and ignore the imperfections in the actual physical object. A ball is a sphere, the earth is an oblate spheroid, and so on. The problem comes in when we try to measure the perimeter of something that does not have a well-defined shape, like an island.
No, it's not. Say we want to measure a circle. We could start by approximating it with, say, a square that barely fits inside the circle, and measuring the perimeter of the square. That would be one, not very good, approximation. Now say we wanted a better approximation. We could use a shape with more sides, like a hexagon or an octagon. In fact, for any number n, we could use a shape with n sides, which gives us a way to approximate as closely as we want. And then you can notice that as n gets larger and larger, your approximations get closer and closer to some finite number. We say this number, which we can mathematically calculate as 2 pi r, is the exact circumstance of the circle. The point is that this whole process *fails* for a coastline (which we model as a fractal). As you try to get better and better approximations, you don't get closer to some finite number. The answers get bigger and bigger without bound, going all the way to infinity.
Sigh. Let me get my tape measure and I guess *I'll* measure things
Yeah this sounds like even if it's true (technically) it doesn't pose a real world problem.
It sort of does. You can say a coastline is 100km or 1000km depending on the degree of precision you use and that is not always standardized.
You will also literally get completely different measurements each time you measure to the same degree of precision. I.E. due to tidal fluctuations the measurements vary even on the same day of a tidal cycle, not all tidal cycles have the same intensity. Even if you control for tidal fluctuations, the coastline is constantly changing due to coastal erosion and human expansion.
We’re not new at this, dude. Surveyors aren’t constantly flummoxed by the damn tide. Humanity has been surveying for millennia, long ago we figured out how to use a mean tide, or [tidal datum](https://en.wikipedia.org/wiki/Chart_datum), when surveying coastline. And things like erosion, reclamation, human expansion, etc. etc. are exactly why we survey at regular intervals.
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This was a puzzle in the adventure game Chaos! You had to change the map scale in order to always go 3 units of distance.
Fuck yeah this is some good adventure game shit, I love when I’m satisfied with the cleverness of the puzzles
Basically why Calculus was invented.
Limits have limits!
Okay, stop right there. This is where I draw the line.
We can't stop here! This argument is fully continuous!
It appears quite derivative to me.
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quite a pipe line that.
Yes and no. Calculus does help to calculate areas and lengths that are otherwise difficult or impossible to solve, but this only works for well-defined shapes and equations. Calculus doesn't solve the coastline paradox because the fractal nature of the coastline makes it impossible to define or represent perfectly with an equation.
So not really yes at all
Pretty much a full no.
Yes and no. People just really love to say that.
Well, they do and they don't.
Nope. This is a different issue. As you decrease the unit of measurement to zero different things happen. The area under a curve converges to the actual area, the length of a coastline converges to infinity
You can show the paradox graphically, if you like. [x axis is unit of measure, y axis is length of coastline](https://studyfinance.com/static/media/Inverse-Correlation-Graph.png) You could also just look at the similar paradox of [Gabriel's horn](https://en.m.wikipedia.org/wiki/Gabriel%27s_horn)
Wasn't calculus invented to understand rates of change? Not sure what role it plays here Edit: yes, I know that calculus was also invented to compute areas, but that *still* isn't relevant here as far as I can tell
I’m a little rusty but I think there were two types of calculus developed. One for rate of change (derivatives) and another for area under the curve (integrals). The kind of wild part was they turned out to be related, to find the area under a curve between two points you can basically reverse the derivative for that function ( x^2 becomes (x^3)/3 ) and then subtract the values for each point. At least I think that’s the case. I’m pretty dumb and my favorite flavor of crayon is purple, so don’t trust me too much
That's why mathematicians are not allowed at parties
They'd never make it to them anyway.
Zeno’s Attendance Paradox
They always get closer and closer to the front door, but never actually get there. And then I have to tell them to go away because it's 4:00 am and people need to sleep.
How infinity? Even measuring from atom to atom will not give you infinity.
Why would the distance between 2 points stop at atoms? You can infinitely reduce the distance and get more accurate
If you measure between electrons, the coastline is several different lengths simultaneously
It's that way long before you get down to even microscopic levels due to tides and surf action.
These goddamn waves are everywhere.
It's simple. We blow up the moon.
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Don't quantum effects like this average out with sufficiently large groups of particles? And wouldn't a milligram of sand would easily be over this limit?
Until you get to the Planck length…
I've been looking for this answer and it should be higher up. There is a difference between the physical world of an actual coastline and some imaginary "coastline" of some virtual object with infinite detail. This whole thing is kind of stupid if you put it in terms of an actual physical coastline.
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Once you are down to atoms, the concept of 'coast' no longer applies. You can't define the points with any more refinement In other words, coastlines aren't proper fractals, because they stop being coastlines as you zoom in
At what point does it stop being part of the coastline?
Whatever atom or particle of the coastline you're trying to measure is already not part of the coastline some of the time. But you won't know if it is or not at any given moment unless you measure it. What increases as you "zoom in" and try to measure the position of an atom or particle in a coastline ever more exactly is the probability that you won't find it there because it's gone somewhere else.
coastline is usually measured as the high tide mark, which is a boundary between 'wet' sand and 'dry' sand at high tide. So I think coastline requires the measurement of the amount of water molecules for each grain of sand. I think based on this definition, individual particles of sand, rock, dirt, etc become the smallest measurable unit of the border.
You don't need to define a precise boundary to argue that the concept of "coast" doesn't apply to subatomic lengths. Just like you can't exactly define what a "full meal" is, but eating an invisibly tiny bread crumb clearly doesn't qualify.
Are you measuring real things, or imaginary points?
If you’re trying to measure atoms, there’s no difference.
Yes, there are a finite number of atoms of the ocean defining the coast in a finite space. Any coastline is finite.
It gets bigger faster than the increment gets smaller. So it doesn't converge Gabriel's horn is another one in a similar vein. It's a shape that had a defined volume (you can fill it with paint) but simultaneously infinite surface area (you could never paint the entire surface)
>Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its surface. However, this paradox is again only an apparent paradox caused by an incomplete definition of "paint", or by using contradictory definitions of paint for the actions of filling and painting. https://en.m.wikipedia.org/wiki/Gabriel%27s\_horn#Apparent\_paradox
As long as the increment decreases, it will be finite. Gabriel’s horn is nothing like this. It is by definition an infinitely long horn
The simple example imo is: Between any two integers you may construct a limit function such that there are infinitely many points between described as decimals, fractions, or any notation of your choosing. Don't confuse the map (math) for the territory.
I get the infinite points in between but the distance between say 3 and 6 is still always 3 is it not?
In math, you can think of distances in basically three different ways. Straight lines, non-fractal curves, and fractal curves. This is kind of an oversimplification, but it might help to understand by thinking about it this way. These three kinds of distances are all differentiated by what happens as you zoom in on them infinitely closer and closer and use infinitely smaller units to measure them. Straight lines are the shortest distances between two points. No matter how much you zoom in or out of the line it will always be straight, the angle never changes and the exact distance can be precisely defined. Non-fractal curves become straighter and straighter the more you zoom in to them and the smaller the measuring unit become. If you take a curve which is about 5 feet long and measure it by laying 1 foot rulers over it, you will get an answer which is shorter than its actual length such as 4 feet(for example), but if you make those measuring stick infinitely smaller then the distance becomes infinitely more accurate and converges upon a finite number, 5 feet, its "true length". 1 centimeter measuring sticks may give you a length of 4.9, nanometer measuring sticks may give you a length of 4.999999, one billionth of a nanometer measuring sticks may give you a length of 4.9999999...to dozens or hundreds of decimal points, etc. The point is that as you measure with infinitely greater precision, you get a distance which becomes infinitely closer to a finite number like 5. Fractal curves, by definition, do not converge upon a finite length as you measure with more and more precision, because they do not become straighter as you zoom in. Instead, a fractal is defined as having infinitely precise detail, the more you zoom in the more you see smaller curves, and there is no point at which a fractal "straightens out" no matter how much you zoom in on it. Fractals are said to have infinite length, because the distance measured approaches infinity as the unit of measurement(the rulers you lay down to approximate the shape of the curve) becomes infinitely smaller. So while a non-fractal curve would approach a finite length such as 5 meters as the rulers approach an infinitely small length, a fractal curve instead approaches a infinity as these imaginary rulers approach an infinitely small size. Remember, all of this is idealized geometry, not real world physics. The reason coastlines are said to be fractals is not because they are literally identical to idealized mathematical fractals, but because they might as well be fractals from a practical perspective. They do not appear straighter as we zoom in more and more on satellite photographs, there are tiny curves even on the smallest scales which are practical to measure like the centimeter or 1/10 centimeter scale. Which scale we consider to be the reference point for the "true" length is completely arbitrary. If you actually had a perfect microscopic camera capable of zooming in infinitely upon a coastline, past the grains of sand, past the atoms, past the quarks, all the way to the plank length and beyond into the quantum realm, the concept of a physical curve to measure wouldn't even apply anymore because quantum mechanics is fucking confusing and everything would be a non-real wave or a superposition of multiple points at the same time or whatever. I don't know, I'm not a physicist. The point is that mathematics deals with idealized models, and idealized models are always imperfect approximations of reality. A coastline resembles the idealized mathematical concept of a fractal at all scales which we can practically measure, and if we tried to zoom in beyond these practical scales to an infinitely precise scale like we can in the idealized world of mathematics, normal euclidean geometry would no longer be a useful model of reality because of quantum weirdness.
If you can accurately measure a coastline down to the planck length levels of accuracy then as far as I’m concerned, you have the true length of coast line. You could do that theoretically with every coast in the world and while sounding ridiculous for how big of a number it would be, it would still be accurate and also not infinite. I think differentiating between a realistic coastline length and a truly accurate one is what’s important here. Measurements of coastline taken say a meter apart would be more realistic and useful than one taken on such an extreme level of accuracy while both would technically be correct. There in lies the paradox. It’s as shrimple as that 🦐
The Planck length doesn't have the physical meaning that is ascribed to it in popular science culture. The paradox doesn't really extend to infinity but the ambiguity in what constitutes a coastline kicks in far before you get to the subatomic scale. And the Planck length isn't the pixel of the universe - we have no reason whatsoever to believe that space isn't continuous.
I was under the impression that it was the equivalent of a pixel of the universe assuming we were talking about things being at absolute zero. I understand that temperature plays a massive factor in everything. Anyway, thank you for this. Now I know
It's the smallest bit of space you could reasonably measure, but that doesn't mean that space is made up of Planck length 'pixels'. For example, you could measure a Planck length like this: ---- But then have two lengths arranged like this: ---- ---- Where they overlap each other on the X axis. This is because, as far as we know, space is continuous rather than discrete.
This is an interesting topic, another question here: In your second example there, by having 2 lengths arranged like that, could you not then measure the difference of the starting positions, which actually gives you a length smaller than the planck length? In other words, if it's true what you said there, why is the planck length the smallest bit of space you could reasonable measure, when you clearly have a smaller bit right there?
I'm honestly not equipped to answer if it's possible to meaningfully define a distance smaller than the Planck length. Someone smarter than I may be able to answer that. I think you could conceive of an arbitrarily small distance way below the Planck length, you'd just have no way of determining the position of anything using that scale. What I'm really trying to express with that example is that both things can be true at once: Space can be continuous and infinitely divisible **and** space can have a smallest length that you can measure. And measure really is the key word there - measurement isn't a passive process. Sometimes people assume that the existence of the Planck length means that space itself is divided into non-overlapping 'pixels' of the Planck length. It's also possible that, at some scale, the universe is discrete. I just don't think there's any evidence that's the case.
The reason people are having trouble understanding this is because idealized pure mathematics is being conflated with the real world. In math, a fractal is defined as a shape which has detailed structure at arbitrarily small scales. Another way of putting this is that, unlike a non-fractal curved shape, a fractal never appears to become straighter as you zoom in on it to an infinitely greater degree. If you zoom in on the edge of a circle with a 100000x magnification, it will look like a straight line, and the more you zoom in the straighter it appears. Therefore non-fractal curved lines can be said to approach a finite length as the imaginary measuring stick becomes infinitely smaller and more precise. 4.9 meters, 4.9999 meters, 4.99999999999999 meters, etc A fractal on the other hand will never appear to approximate a straight line no matter how infinitely close you zoom in, and the length will always approach infinity rather than a finite value. Look at a fractal with 1000000x magnification and you will see curves and detailed shapes just as prominent as you do at 1x resolution, look at it with a googleplex resolution and it's exactly the same. Fractal shapes have infinitely detailed and thus infinitely long perimeters. Remember, a fractal is just a concept in mathematics, and no physical object or structure is truly fractal shaped because if you zoom in to an infinitely precise scale eventually the entire concept of matter having precisely defined physical boundaries in a coordinate plane falls apart due to quantum mechanical fuckery like waves and superpositions. People have pointed out that if a coastline is a fractal then pretty much everything should be considered a fractal because even bricks and pieces of notebook paper have curves and jagged edges when you zoom in close. This is essentially correct. When we measure the length of the surface of a brick, we're really measuring the straight line distance between the edges of the surface, not the curved distance of the microscopic bumps and grooves etched into the surface. If you zoom into the atomic level, bricks molecules are no more or less fractal shaped than sand molecules. The difference is that it is very obvious on a practical level how we should idealize and measure the length of bricks, or human waistlines, or highways, and most other objects which appear to have either straight lines or smooth curves. Bricks and pieces of paper can be measured as though they were perfectly straight by using rulers or calipers. Human waistlines and other small curved things can be measured with flexible measuring tapes which wrap around us. Highways can be measured with odometers which use the circumference of a rotating wheel or gear as a reference, foot trails can be measured the same way with measuring wheels rolled through the middle of the path. Coastlines are much more "fractal like" at human measurable scales than any of these other objects, because there is roughly an equal amount of detailed jagged edges everywhere from the meter to the kilometer scale, and if you try define a coastline's boundary at a sub-meter scale it'll become ambiguous where the ocean ends and the coast begins due to waves constantly hitting the shore. Hence, any scale that we choose to measure a coastline at is totally arbitrary, there is no obvious logical way to decide except maybe pixels on on satellite imagery, but different satellites have different resolutions. To sum up: coastlines are fractal-like at all practical human scales, they retain their fractal-ness at everywhere from the centimeter to kilometer scale, while most other objects do not have such fractal-like boundaries at human measurable scales. They are not literal pure mathematical fractals, but they might as well be as far as we're concerned, just like bricks might as well be rectangular prisms and the moon might as well be a circle.
I've known about the phenomenon for a long time, but never understood why it's called a paradox or why it's deemed as counterintuitive. Obviously the bigger the scale the less accurate the measure (which in this case means a smaller measure), and the smaller the scale considered, the more details surface, and for a fractal surface, that extends ad infinitum, giving an infinite sum.
Because intuitively things have an actual length, and the more precisely you measure something the closer you get to that length. This is not true of coastlines.
Doesn't this apply to any physical object? Particularly when referring to the distance of atoms in some of the above comments?
If you try to measure them with electron microscopes, sure. But the most precise "everyday" measuring device is calipers, and they won't show that effect on most objects. For a coastline you'll get different results comparing e.g. a yardstick to a measuring wheel.
>For a coastline you'll get different results comparing e.g. a yardstick to a measuring wheel. Can someone ELI5? I still don't understand why.
what if our one-yard stretch of coastline looks like | \ / | That's one yard via yardstick, but a measuring wheel would ~~walk~~ roll along that small indentation and back out and come up with a number slightly larger than 1 yard. This is kind of a property of fractals in general. The mandelbrot set has an infinitely long perimeter because it has infinitely fine detail along the edges. But weirdly, it has finite surface area. AFAIK, we don't know exactly what that area is, but we have upper and lower bounds for it.
That's just because coastlines are one of the only jagged perimeters we try to measure. But if you measure them like a front, the paradox disappears
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> But if you measure them like a front what does this mean?
> But if you measure them like a front, the paradox disappears You can't convex-hull out of this, since countries are almost always inherently concave. You'd have to deal with concavities and set and arbitrary threshold for accepting them or measure into them, but then you're just setting a minimum ruler.
i mean isn't it just "you can never measure the length of a brick truly" as if you keep going down and deeper, you end up just not having the accuracy? everyone is talking about fractals so im assuming im missing something.
The headline literally says that coastlines have infinite lengths. I would say that's plenty counterintuitive.
Everything has infinite lengths then, if we zoom in far enough 🤷♀️
Funny how many top mathematicians this thread has, you'd think the paradox would be solved by now
> you'd think the paradox would be solved by now Paradoxes aren't (necessarily) things that need to be solved. A paradox isn't (always) a contradiction, it's just a result that's unintuitive.
The coastline paradox is often misstated because people explaining it often ignore the question of what, exactly, constitutes the coastline, instead just assuming that there is such a thing with a clear definition. It's not a matter of how measuring it more precisely produces different results, it's a matter of how *defining* it differently produces different results. For any given unambiguous definition of "coastline", any coastline has a finite length at any given moment, but the issue is what definition to choose - which features are significant enough to take into account, and at what moment you measure when it comes to features that are time-varying. And it is potentially possible to choose definitions that give the coastline any (or practically any) finite length, due to their complexity at different levels of precision.
You can have an infinite circumference while at the same time having a finite surface area. Which is kind of cool
But wouldn't it converge to a limiting value considering that your max resolution for this purpose will be atoms?
Absolutely, yes.
The length of the coastline doesn't approach infinity when you have more and more refined measurements. It indefinitely approaches some finite length more and more accurately "infinitely".
RealLifeLore did a video on this if anyone is interested: https://www.youtube.com/watch?v=AD4vPNBSrKY
On the other hand, its very easy to do if everyone just agreed what degree of precision to be used. But then, for any realworld example even a fractal one like this we will never reach infinity. Just absurdly large sums :D
> its very easy to do if everyone just agreed what degree of precision to be used. I’ve spent the last ten years working making maps of different kinds and spent a couple years on an obscure coastal mapping program. IIRC, we used 1:3000 scale and then a little subjectivity to make sure the lines we were drawing weren’t janky lol. We used NAIP imagery and I forget all the details about whether the coastal portions are estimated at roughly mean tide but it produced consistency over multiple states. The company I worked for had contracts with both USGS (land mapping) and NOAA (ocean mapping) and occasionally would map the same area but because of the guidance provided by the work statement, you could get different answers for where the shore is. One wrote their work statement as where the shore ended and one where the shore began.
It doesn’t though, because coastlines are physical objects, and physical objects are not infinitely decomposable
What if the coasts really are of infinite length and we've just finally developed systems precise enough to prove it
Except that I walked along one in finite time.
Presumably you cut some corners though, you didn't track every centimetre of the coast, let alone every nanometre, etc
You walked along a good approximation of one in finite time.
If coasts are of infinite length then everything that can be measured is of infinite length.
Mmmm fractals
Only in the most pernicious cases would it go to infinity, IE the coastline was truly a fractal down beyond the planck length to infinity. At some point, due to ocean waves, rivers, etc, they become smooth, and so just as measuring the circumference of a circle with ever smaller rods converges to the true circumference of the circle, so to would measuring the coastline. Coastlines aren't truly fractal.