This got me thinking as to whether is actually a true statement, and whether there are physical laws, equations or theories for which extension to infinity is required. That is where the substitution of any finite estimate will give an incorrect answer.
The example that springs to mind is a black hole which may involve a point of infinite density at the center and which infinite energy is required to escape. Which brings up the next question and get us back to the question of the OP. Following the no-hair conjecture of the black hole, to what degree can the event horizon of a black hole differ from a perfect circle.
A black hole that exists alone in empty space will probably have an event horizon that is close to a perfect circle (sphere) – if there is a precise way to define an unobservable boundary. If you cannot observe it, it becomes profoundly hard to measure it to find out whether it is mathematically perfect.
Of course, if there is any stuff nearby, there will be tidal forces that affect the shape of the horizon, albeit minutely, but still enough to make it imperfect (I think).
I look at it like this: throughout history, pi was approximated based on level of technology at the time. When we progressed to rulers and tape measures, the value of pi was refined until the next technological leap.
Trigonometry enables us to calculate a single area that has straight sides, but a circle has none. The circle can be broken up into squares and triangles, and then areas can be calculated for each piece and added together, but it’s still an approximation. If we were to insert an equilateral 8-sided polygon (octagon) inside a circle with radius 1 and calculate its area, we’d get 2.828. (visual aid)
If we increased the number of sides to the inner polygon to 18, its area would be 3.078, a bit closer. Let’s go to 36 sides: the area adds up to 3.126. How about 180 sides? We get 3.1409. 360 sides? 3.1414.
These values are determined by using the equation Area = ½(N)r²(sin Θ), where N is the number of sides of the inner polygon, and Θ is the angle of the triangle that originates from the center of the circle to meet the ends of the side that touches the perimeter of the circle. (36 degrees in this example) As Θ approaches 0, so does sin Θ, but neither value can be 0. The closer Θ gets to 0, the closer we get to the exact value of pi, but it will never be attained.
So, even though pi’s value looks arbitrary, it can be approximated to the tiniest degree possible.
Pi is not infinite. It is a perfectly finite number, between 3.1 and 3.2. 1/3 requires an infinite number of digits to express in decimal form, but nobody thinks “one third is impossible because infinities don’t exist in the real world”. I mean, cutting a cake into exact thirds is impossible, but so is cutting it exactly in half, and 0.5 only has one digit. Infinity has nothing to do with it.
Pi is a beautiful and interesting number, just not for the reasons most people seem to think it is. It’s definitely not infinite in any sense.
That is kind of specious, though. You can define a circle as an infinite ring of points, but a point is and infinitely small entity, so you would be using an infinity to define an infinity. It may work well in the abstract realm, but that is not where we abide.
At the risk of asking the obvious, how does the human mind know the perfect? How does it conceive of such ideals; conceive of the perfect form? Where does this notion of perfection come from? Is it just an approximation?
By imagining something with no imperfections. Ideas don’t need referents outside of the mind, and it’s impossible for a purely imaginary thing to not be what it was defined to be, as if it weren’t, it must have been defined to be something else.
Therefore, the fact there’s no perfect physical circle doesn’t mean we cannot imagine one, and our imaginations don’t refer to some circle outside of our imagining which may be imperfect after all.
While I agree with your point I have a minor nitpick. Instrumentation to measure those atomic or sub atomic differences exist.
Atomically smooth surfaces are created regularly.
Quantum stabilized flat surfaces and spheres have been created In which the electrons arrange to smooth the surface far below atomic levels. Though they still aren’t currently perfect.
And the kicker here, which does involve quantum physics is that you can actually fill that space with part of a subatomic particle.
Why, because electrons can exist in more than one place at the same time.
So theoretically you could have a perfect whatever shape you want.
Let’s say circle, or sphere. If it’s atomically perfect, and it’s electrons arrange themselves to smooth out the edge, given they can exist in more than one place at once, they could exist in slight arc segments that form a perfect circle.
It would be represented by a path.of motion typically, but since electrons can exist in more than one place simultaneously, the ascribed interconnected arc segments could be a perfect circle at any given point in time.
Then you get integer number of atoms for both c and d but as others have pointed out the pi ratio would still exist if you used some unit of measure less than an atom. So like they’ve said, it doesn’t mean you can’t have a perfect circle, it just means you can’t express it’s c/d ratio as a rational number.
Here is the thing with math: numbers are inherently inaccurate. You have a problem that you can calculate in, say, 37 steps. Each of those steps introduces inaccuracy into the calculation. Then you take a look at the structure of your problem, move some things around aggressively and get it down to 5 steps. You have reduced your error factor by sevenfold, because xs and ys are agnostic with respect to precision. As abstract concepts, they are essentially perfect.
An electron has a probability distribution given by the Schrödinger equation. This does not mean that the election can exist in more than one place at a time.
Nor is it true that we can measure to fifty decimal places, let alone fifty million.
And that means it is not true that “theoretically you could have a perfect whatever shape you want.” That is meaningless. At the sub-atomic levels there are no shapes in the first place.
Well firstly it’s been proven electrons can exist in more than one place simultaneously.
Second
As I stated, quantum stabilized flat films and quantum stabilized spheres have been made wherein their shape is perfectly flat or perfectly spherical with far less variance than the size of an atom.
As Exapno Mapcase accurately noted, an ‘electron’ isn’t like a basketball or a brick. It’s not a distinct ‘thing’ whose location is perfectly defined. It’s ‘location’ can be described by a probability distribution, and, yes, the center that distribution can, under certain frames of reference, can be in multiple places at the same time.
Now, how relevant or helpful is that for defining a ‘perfect’ circle/sphere in the real world? I’d say not much at all.
Again, so what if that’s the case? That’s still infinitely less precision than required by the OP.
Yes, I read the article. It’s using “electrons can be in two places at once” as layman’s talk for the electron being a probability distribution until its wave function collapses. It can emerge at any point within that distribution, explaining phenomena like quantum tunneling.
Moreover, other experiments show that even atoms have a probability distribution. A trapped, single atom, however, has a different one from an atom in a material. And forces within that material will work on it so that the shape is also different, with the electron probability cloud varying with the forces opposing it.
This is all saying that atoms are not perfect little balls, and that quantum descriptions make perfection impossible.
Could you provide a cite for these perfections?
[Got called away and I see that Great Antibob said more or less the same thing in the interim. That’s two strikes against you.]
