Sisyphus Posted September 16, 2008 Posted September 16, 2008 at the plank lenght scale pi is close to 3.0 For the love of Zeus, enough already. This is false. Very very false.
Dennisg Posted September 16, 2008 Author Posted September 16, 2008 For the love of Zeus, enough already. This is false. Very very false. You keep making that assertion but - for the love of Godel - based on what?
ydoaPs Posted September 16, 2008 Posted September 16, 2008 You keep making that assertion but - for the love of Godel - based on what? What's the basis for claiming that pi is 3 at plank length?
I_Pwn_Crackpots Posted September 16, 2008 Posted September 16, 2008 I wouldn't say that pi is "just one number" because at the plank lenght scale pi is close to 3.0 Having an absolute limit on how small things can get does change things - don't you think so? What????
Sisyphus Posted September 16, 2008 Posted September 16, 2008 What's the basis for claiming that pi is 3 at plank length? What???? He thinks that the world is made up of planck-length-sized balls, and furthermore that mathematical entities are somehow dependent on our ability to physically make something that shape. From there, I get the impression he's thinking of arranging seven of those balls in tightest packing, i.e. a hexagon, which would have a diameter of two planck lengths, and a circumference (measured from the center of each ball) of six, making a ratio of three to one. Since, in that universe, this would be the most circle-like thing you could physically construct at that scale, he's decided it is a circle, therefore making "pi" equal to three. He seems pretty adamant about the whole thing, too.
Dennisg Posted September 16, 2008 Author Posted September 16, 2008 What's the basis for claiming that pi is 3 at plank length? A circle with a diameter = 2plank lengths can have only 6 points that define its circumance. A straight line drawn through any of these points (tangent) will only intersect one point – therefore the line between any two points is curved. One can then can calculate the area of the “circle” by calculating the area of the 6 “triangles” and then calculate pi = A/r squared. He seems pretty adamant about the whole thing, too. not really - just pushing things.
I_Pwn_Crackpots Posted September 16, 2008 Posted September 16, 2008 But a circle is a mathematical construct, not a physical one. Perfect circles don't exist in nature. They don't have any meaning beyond the mathematical "realm". You can therefore make a circle arbitrarily small, and pi will always be irrational.
Bignose Posted September 17, 2008 Posted September 17, 2008 RE: therefore the line between any two points is curved. and calculating the area of the 6 “triangles” How can these two statements be compatible? If the "line" between any two points is curved, how can this be a triangle? A triangle has straight sides. By definition. So, either the sides aren't curved or they aren't triangles.
Kyrisch Posted September 17, 2008 Posted September 17, 2008 Look, [math]\pi=3.1415927...[/math]. However, given the constraint that the smallest length possible is the Planck Length, then a significantly small circle may have a circumference-to-diameter of nearly 3.0. Pi is defined as that ratio in constraint-less space; thus, even though you can theoretically construct a "circle" whose ratio is nearly 3.0, this does not change the value of the constant, pi.
I_Pwn_Crackpots Posted September 17, 2008 Posted September 17, 2008 You cannot construct a circle whose ratio is nearly 3.0, not even theoretically. For example, I can construct a circle whose diameter is the plank length, and the ratio will be pi. Or rather, since a circle can be ANY arbitrary size, I can theoretically construct a circle whose diameter is 10^-1000 m, and the ratio will still be pi.
the tree Posted September 17, 2008 Posted September 17, 2008 Incidentally, does Dennisg believe that [math]\sqrt{2}[/math] also becomes a rational number for really tiny triangles?
Sisyphus Posted September 17, 2008 Posted September 17, 2008 Look, [math]\pi=3.1415927...[/math]. However, given the constraint that the smallest length possible is the Planck Length, then a significantly small circle may have a circumference-to-diameter of nearly 3.0. Pi is defined as that ratio in constraint-less space; thus, even though you can theoretically construct a "circle" whose ratio is nearly 3.0, this does not change the value of the constant, pi. Given those constraints, it might have a ratio of 3.0, but it wouldn't be a circle. It would be a hexagon. And even a hexagon has incommensurable ratios in it, so the whole exercise is pretty much a reductio ad absurdum.
Dennisg Posted September 17, 2008 Author Posted September 17, 2008 Given those constraints, it might have a ratio of 3.0, but it wouldn't be a circle. It would be a hexagon. And even a hexagon has incommensurable ratios in it, so the whole exercise is pretty much a reductio ad absurdum. Because of uncertainty at the plank scale it definitely couldn't be defined as a hexagon - a proto circle would be more apt. What this shows is that pi in not one mumber but depends an scale. At our scale the differences are so small the we cannot measure them. Incidentally, does Dennisg believe that also becomes a rational number for really tiny triangles? Hey don't rock the boat - you might fall off.
I_Pwn_Crackpots Posted September 17, 2008 Posted September 17, 2008 (edited) Given those constraints, it might have a ratio of 3.0, but it wouldn't be a circle. Exactly. It wouldn't be a circle. I think we can all put this to rest now. . What this shows is that pi in not one mumber but depends an scale. No, it does not! Edited September 17, 2008 by I_Pwn_Crackpots multiple post merged
Dennisg Posted September 17, 2008 Author Posted September 17, 2008 No, it does not! Please back this assertion with some evidence.
Sisyphus Posted September 17, 2008 Posted September 17, 2008 Please back this assertion with some evidence. Ok, pi is defined as the ratio of a circle's circumference to its diameter in Euclidian geometry. Euclidian geometry is not dependent on scale. Of course, four or five people have already said that, so I don't know what else you could possibly be looking for.
ydoaPs Posted September 17, 2008 Posted September 17, 2008 You cannot construct a circle whose ratio is nearly 3.0, not even theoretically. Yes you can. Spin it.
Kyrisch Posted September 17, 2008 Posted September 17, 2008 If you spin it at relativistic speeds, the circumference (and thus the ratio) increases.
ydoaPs Posted September 18, 2008 Posted September 18, 2008 The circumference will decrease at relativistic speeds.
Dennisg Posted September 18, 2008 Author Posted September 18, 2008 Ok, pi is defined as the ratio of a circle's circumference to its diameter in Euclidian geometry. Euclidian geometry is not dependent on scale. Of course, four or five people have already said that, so I don't know what else you could possibly be looking for. This answer is based on the assumption that there is no limit to "smallness" Since the plank length is that limit then Euclidian geometry needs to be updated.
I_Pwn_Crackpots Posted September 18, 2008 Posted September 18, 2008 No it doesn't. If you don't come up with a ratio of pi, then what you have is not a circle. It's as simple as that.
Kyrisch Posted September 19, 2008 Posted September 19, 2008 The circumference will decrease at relativistic speeds. Say this circle is a merry-go-round. You are on the edge, spinning at relativistic speed, and you are trying to measure the circumference with a ruler (the circle is large enough that this isn't completely foolish). The ruler would contract, and therefore more ruler lengths would fit around the outside. You would measure the ratio to be larger than pi. Maybe this is a question of reference frames, though. In a rest frame, the circumference would contract. No such specification was made in the original post, however.
Dave Posted September 19, 2008 Posted September 19, 2008 Guys, I would like to remind you that this is a mathematics, not relativity, forum. Bearing this in mind, the answer to your question is: no, [imath]\pi[/imath] is irrational because it is a mathematical definition. You can debate 'relativistic circles' or whatever until the cows come home, but that is not mathematics. Unless someone can convince me to keep this thread open, I will close it later on this evening.
insane_alien Posted September 19, 2008 Posted September 19, 2008 dave, you should know that relativity is a highly mathematical theory and it does bear some relationship to the question at hand. although i do agree with you that pi is not subject to change as it is for a hypothetical perfect circle in flat space.
I_Pwn_Crackpots Posted September 19, 2008 Posted September 19, 2008 (edited) dave, you should know that relativity is a highly mathematical theory and it does bear some relationship to the question at hand. Yeah, but so are all the other physical theories. But ultimately they are all imperfect models of the universe. So I'm going to disagree with you here. The circumference will decrease at relativistic speeds. And so will the radius. As such, pi is always irrational. Bearing this in mind, the answer to your question is: no, [imath]\pi[/imath] is irrational [b]because it is a mathematical definition[/b]. Quoted for emphasis. Edited September 19, 2008 by I_Pwn_Crackpots multiple post merged
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