The Wikipedia list of natrual fractals lists, among other things, broccoli (dimension 2.7), the surface of the human brain (dimension 2.79), cauliflower (2.8), lung surface (2.97). One might add many things like the fracture surface of steel (2.28), etc.
You might argue that, ok, the steel surface area is not really infinite if you look at it under a good microscope, but the exponent is applicable over a considerable range of length scales.
Especially this one.
That article brings up another one that I don’t think has been mentioned, clouds.
Clouds arguably gets you to turbulence, and that is another area where there is a legitimate range of self similarity.
That does break down at the largest sizes though, since there are features that only show up then. Central peaks and multiple rims are two I can think of, but there may be others.
Okay, I will concede that if you’re measuring any physical object in the real world at ever increasing levels of detail, you will eventually reach the Planck Length limit.
But if it is never zero then it is never infinite. “Bigger and bigger” is not infinite. It may be a giga-ginormous number but not infinite. Whatever you want to say about reality at the Planck length we know it is not zero in size.
I don’t know. Is it really any different than saying that a black hole has infinite density?
I should have been more clear, the infinite zoom business properly applies to mathematical fractals, which can be examined at arbitrarily small length scales so in that case it really makes mathematical sense to discuss limits as the size tends to 0. If you are taking real-world objects, then, sure, you will run into problems at really small scales, like what counts as the coastline, and eventually perhaps space-time is not modelled by Euclidean space…
ETA also this:
not to mention that the coastline will be constantly fluctuating due to waves and such long before we are dealing with atoms
The Planck length is irrelevant, because no physical object has anything that can meaningfully be called “shape” at a level smaller than about the size of an atom.
Ok.
Then we can use that as the lower size increment of our measuring stick.
We’re waaay better off now since we are much, much larger than Planck length. Our measured value may still be stupidly large but not quite as stupidly large.
My point being, eventually you cannot shrink your measuring stick anymore and still be said to be measuring the length of a real world object and thus no coast is ever “infinitely long”.
For any useful measurements we would probably define our scale as something that runs over what we could define as a physical surface, which for most uses is perhaps defined by the electron cloud that stops objects passing through one another. Define our stylus as a neutral hydrogen atom and roll it over our object with some sensibility defined force, perhaps one that does not deform the electron cloud field by more that a set amount.
Any smaller scale and it becomes unclear what a boundary actually means.
[quote=“Senegoid, post:28, topic:944844”]
To demonstrate (more simply than the arcane Mandelbrot Biscuit ) how region of finite area can have an infinite perimeter, consider the simple snowflake curve (which seems to appear in the video that @Joey_P linked above).[/quote]
The snowflake provides a simple and compelling visual and and conceptual (i.e. no math required) example of how the perimeter can be infinite. However, a similar nonmathematical way of clearly illustrating why the area is finite is not apparent to me. It seems to my simple mind that with every iteration of adding a new set of triangles to the perimeter, you’d also be adding a tiny bit of area. In that way of construing it, if the perimeter is infinite the area is too. What am I not getting?
Every time you add to it, you are only adding a tiny bit of area, the amount of area that you add each time actually decreases. It will approach a certain area asymptotically, but never reach it.
At the same time, every time you add to it, you are increasing the perimeter by an increasing amount, much more than doubling it each time.
The area will approach a certain finite number, and the perimeter will grow without bound.
This is correct, however the amount of area you add at each step grows smaller and smaller with each step. The total area cannot be infinite, as is easily seen since it all fits into a bounded region.
The perimeter gets larger and larger by adding fractal convolutions. Furthermore, simple shapes can have perimeter equal to 1, e.g., but arbitrarily small area, so it’s not like there is a minimum area-to-perimeter ratio. There is a maximum, of course.
Circumscribe a circle around the original triangle. No part of the snowflake will ever extend outside of that circle, and the area of the circle is finite. The area of the snowflake is smaller than the circle, so the snowflake’s area must be finite, too.
Of course, you could also come up with other shapes that the snowflake curve fits inside, and so get even tighter bounds on the snowflake’s area. You could even use a succession of tighter and tighter bounds to calculate the snowflake’s area exactly, like Archimedes might have done. But the circle (or any other one boundary) is enough to prove that it’s finite.
Actually, multiplying it by 4/3 each time. But yes, (4/3)*(4/3)*(4/3)*(4/3)*… clearly tends towards infinity.
Just because it is bounded, does that mean it can’t be infinite? If you start with 1.0 and add 0.1, then 0.01, then 0.001, and so forth, your number will be getting infinitely bigger, but you will never get to 1.2.
ETA: Back to the snowflake: the perimeter is getting bigger by bigger increments, and the area is getting bigger by smaller increments, but they are both getting infinity bigger.
If 1.2 is an upper bound at each step, then your sequence of numbers will converge to a number no bigger then 1.2, therefore not infinite.
The area does not; just write down the area explicitly and you can check for yourself. The perimeter does get arbitrarily large, though, for reasons already discussed (it is too convoluted to have a finite length)
Why?
It is only infinite if you never stop doing it.
Seems like circular reasoning.
" (4/3) * (4/3) * (4/3) * (4/3) *… " is infinite if you do it infinitely many times.
