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Happy Pi Day

bilby said:
Malintent said:
Perhaps my thought experiment is flawed... but here is what I am thinking...

an arc has a length that is calculated based on the radius of the arc and Pi.

If the radius of that arc is 27.4 bly, how many digits of Pi is needed for the accuracy of the calculation of the length of that arc to be within 1 plank distance?


You have an infinitely powerful laser that emits a 1 particle wide beam of light that is unaffected by gravity or matter, and arrives at the end of the universe instantaneously. A magic laser.

You stand on one side of the universe and shine the laser in a direction. The laser instantly hits the other side of the universe.
At what degree of precision must you be able to rotate the laser pointer such that the endpoint of the laser moves only 1 plank length from its original point? Disregard uncertainty and relativity... this is an Euclidean geometry problem at unreasonable scale.

It is starting to sound paradoxical.. How can you move less than 1 plank distance (1 plank radian - is that a thing?) such that the other end so far away only moves less than 1 plank distance? This is sort of what I mean by an upper limit of the useful number of digits of Pi... calculating fractional units of Plank is meaningless.
Click to expand...

Reality isn't granular.

As your thought experiment implies, the Planck length is not the smallest possible subdivision of space; you can in principle produce a measurable change by moving a distance which despite being literally immeasurably small, is calculable and far smaller than your proposed minimum distance. Angles (and therefore arcs) can be arbitrarily small, and an infinite number of possible angles exist between any two positions separated by a finite angle, just as an infinite number of reals exist between any chosen pair of reals. There's no number so small that it cannot be divided by two.
Click to expand...

Yea, I get that... Math, as a model, can do more than the universe can do. While precision can reach into infinity, there is an upper limit of what is USEFUL.
I am a wood worker. Precise cuts (especially in very hard woods) is very important. Do I need an electron microscope attached to my table saw to get my cuts "perfect". Of course not. I ARBITRARILY choose an upper limit of precision... down to maybe a 32nd of an inch, at best.

1/32 of an inch, in my thought experiment, would cause an error greater than the width of our solar system (galaxy?, sector?,... a lot).

I arbitrarily chose the plank length as the smallest distance... the chosen degree of precision for this experiment. I also arbitrarily chose the diameter of the universe for the chosen magnitude.

I understand that using a mathematical model allows us to divide angles and distances infinitely, but lets just say we are galactic woodworkers that only really care about useful measurements... forget my statement of "the most" precise, or any notion of "maximum describable using any tool", imagined or real. I apologize if that caused confusion. For this thought experiment, 27.4 billion is the largest number and 1 plank is the smallest. That is the range of numbers that exist in this imaginary universe.
 
Malintent said:
Yea, I get that... Math, as a model, can do more than the universe can do. While precision can reach into infinity, there is an upper limit of what is USEFUL.
I am a wood worker. Precise cuts (especially in very hard woods) is very important. Do I need an electron microscope attached to my table saw to get my cuts "perfect". Of course not. I ARBITRARILY choose an upper limit of precision... down to maybe a 32nd of an inch, at best.

1/32 of an inch, in my thought experiment, would cause an error greater than the width of our solar system (galaxy?, sector?,... a lot).

I arbitrarily chose the plank length as the smallest distance... the chosen degree of precision for this experiment. I also arbitrarily chose the diameter of the universe for the chosen magnitude.

I understand that using a mathematical model allows us to divide angles and distances infinitely, but lets just say we are galactic woodworkers that only really care about useful measurements... forget my statement of "the most" precise, or any notion of "maximum describable using any tool", imagined or real. I apologize if that caused confusion. For this thought experiment, 27.4 billion is the largest number and 1 plank is the smallest. That is the range of numbers that exist in this imaginary universe.
Click to expand...

Quantum mechanics and general relativity both require spacetime to be modeled as continuous and no theory has been found to replace them that successfully uses a discretized model of spacetime. So while your idea might seem intuitively plausible, once you get down to the nuts and bolts of predicting reality it doesn't work and does not match experiments. Unless you can come up with a theory like that that works (where the best physicists have failed for a century), that is pretty much the end of the story.
 
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