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Can pi be reduced to a rational number?


Dennisg
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If space is not continuous, then, physically, there is no such thing as a circle. Circle is just an approximation of a polygon. It could be said that equilateral triangle, square, pentagon..., are just approximations of a circle. Therefore, what we know as "pi" is a ratio of perimeter of a polygon with infinite number of sides to the distance between the opposite sides (in case of even number of sides), or diameter of hypothetical circle that is inscribed in such polygon.

 

But from the physics we know that there is a limit. Somewhere around the Plank's constant. Therefore, the minimum size of a polygon's side is defined. Now, the question is how many sides such polygon has? Is it it is infinity?

 

In quantum realm threre are such numbers as 1, 2, 3, 4..., that we call integers. If we take a polygon that would be chacterized by integers only in all directions? Perhaps, 90, 22 and 7 are those integers. Just a thought.

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What's your point?

 

Probably that the ratio approaches pi as the number of sides increase?

 

Edit: Oh, Vts apparently answered already (sort of). But as noted before, the discussed subject concerns the mathematical concept of a circle rather than any physical body that actually exists.

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If space is not continuous, then, physically, there is no such thing as a circle. Circle is just an approximation of a polygon. It could be said that equilateral triangle, square, pentagon..., are just approximations of a circle. Therefore, what we know as "pi" is a ratio of perimeter of a polygon with infinite number of sides to the distance between the opposite sides (in case of even number of sides), or diameter of hypothetical circle that is inscribed in such polygon.

 

But from the physics we know that there is a limit. Somewhere around the Plank's constant. Therefore, the minimum size of a polygon's side is defined. Now, the question is how many sides such polygon has? Is it it is infinity?

 

It seems like you're also mixing up math and physics. Physically, there's no such thing as a circle, and there's no such thing as a polygon, either. There is no minimum size of a polygon's side in the "real world," because polygons aren't in the real world. Planck's constant has nothing to do with it. You can still describe a circle of any size, but you could never "make" one anyway.

 

In quantum realm threre are such numbers as 1, 2, 3, 4..., that we call integers. If we take a polygon that would be chacterized by integers only in all directions? Perhaps, 90, 22 and 7 are those integers. Just a thought.

 

I can't tell what you're trying to say, here. Especially since, according to your above post, the ratio you're talking about is just an approximation. You said the real ratio is 7:22.00008478, not 7:22. But even if it were exactly 7:22, I don't understand what significance you're claiming that would have.

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If space is not continuous, then, physically, there is no such thing as a circle.

Mathematics is not bound by reality. If space is curved (which it is, at very long scales of distance and near large concentrations of mass), this does not invalidate Cartesian geometry. It merely invalidates the use of Cartesian geometry for modeling the shape of the universe at those long distance scales or near large concentrations of mass. If space is quantized (which it might be, at very scales of distance), this does not invalidate continuum mathematics. It merely invalidates using continuum mathematics to model space at those short scales of distance.

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I can't tell what you're trying to say, here. Especially since, according to your above post, the ratio you're talking about is just an approximation. You said the real ratio is 7:22.00008478, not 7:22. But even if it were exactly 7:22, I don't understand what significance you're claiming that would have.

 

I agree with you, guys. I am just stating the fact that polygon with 90 sides is the closest to 22:7 perimeter to diameter ratio than any other polygon. I have no idea if it means anything in physical reality. Perhaps it does. I just recently found direct connection of regular tetrahedron and the tetrahedral close sphere packing to the quantum numbers n, l and ml. Sometimes it pays off to look closely at coincidences.You can check it out at my web site.

Edited by Vts
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Any of them in particular? None are that easy to explain or prove. Or is it simply that you'd like to know more precisely what the notation means?

 

Thanks, any will do - how about one that you are most comfortable with?

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Well I'm afk until Monday so I'm not going to sketch up any proofs right now but the famous Leibniz formula comes down to a basic deduction from the power series of a trig function.

So long as your okay with Taylor's theorem then it shouldn't be hard to appreciate the Leibniz formula.

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Well I'm afk until Monday so I'm not going to sketch up any proofs right now but the famous Leibniz formula comes down to a basic deduction from the power series of a trig function.

So long as your okay with Taylor's theorem then it shouldn't be hard to appreciate the Leibniz formula.

 

I had a quick look at Taylors theorem - I think I could cope if there was a bit of verbage explaining things. It look more of a way of estimating a value rather than a way to directly calculate one.

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A point by definition has zero size, and if you want to say otherwise you are neither doing Euclidean geometry nor are you working with real numbers. You will have to start from the ground up starting with a set of axioms that describe your new mathematics.

 

When you do so, please do not use terms like "pi" in your new mathematics. The term "pi" is already reserved to mean the ratio of the area and radius of a circle on a Euclidean plane. Mathematicians have developed several non-Euclidean geometries in which the ratio of the area of a circle to the radius of a circle is not pi -- and they do not redefine pi to be that ratio.

 

Throwing out Euclidean geometry is one thing; mathematicians have done exactly that already. Throwing out the real numbers is quite another thing. I wish you luck in your endeavors.

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Right, and what could be better than a plank length? It is as close as one can get to zero without losing the plot.

 

You can do "pointless geometry", which is definitely "non-Euclidean" as we are outside set theory. (I.e. the spaces are not point sets!) Generically this comes under the name of noncommutative geometry (NCG). I work with a mild form of noncommutative geometry called supergeometry. It is pointless and noncommutative but in the nicest possible way.

 

If NCG (more specifically noncommutative differential geometry) is to be a generalisation of classical differential geometry then we must be able to formulate geometric objects on these noncommutative spaces. To do this half of the work is in reformulating classical things in a way independent of the notion of a point.

 

I don't know the full status of general NCG, but in supergeometry almost everything that can be done classically carries over. I expect on large classes of NGC's the same is true.

 

From physical ideas it is reasonable that space-time should consist of Bohr-cells with Planck length radius. Compare this to the cell structure on quantum phase-space. Ok, I agree with that.

 

However, we are dealing with classical modern geometry in this thread and so we have no issues with the notion of a point. The spaces such as [math]\mathbb{R}^{n}[/math] are set theoretical objects. Moreover, we can give them a natural topology which is equivalent to thinking of [math]\mathbb{R}^{n}[/math] as [math]n[/math] copies of [math]\mathbb{R}[/math] with its standard topology.

 

What you are doing is mixing mathematical constructions with physical ideas. Just be careful doing this.

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What you are doing is mixing mathematical constructions with physical ideas. Just be careful doing this.

 

So true and I am way over my head already. Thanks for talking me seriously- being a non expert allows me to think outside the square and ask silly questions that get people thinking. For me there are two ways of looking at this problem. First, one could say that a point has a dual nature or second, one could say that a point is defined as the bottom point on a hyperbolic curve and that measurements smaller than plank length are less than zero.

Edited by Dennisg
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Hehe...this reminds me of the first debate of my Debate class last semester. The topic was "Is pi better than pie?", and I argued the pro-pie side. Basically I argued that because perfect circles do not exist in real life, pi does not exist, and therefore cannot be better than anything. Hey, we won. :D

 

--

 

Part of our opening case:

 

Contention #1

a. Examples of mathematical concepts that do not actually exist – only approximations

i. Square

ii. Circle

iii. Pi

 

b. No such thing as a perfect mathematical circle

i. Irregularities

ii. On computer: collection of square pixels

iii. Borderline has thickness

 

c. Pi based on nonexistent circle

i. Ratio of circumference to diameter

ii. Pi itself does not exist

iii. Cannot be better than pie, which does exist

iv. Cannot be better than anything

 

--

 

Of course, I conveniently forgot to mention that mathematical concepts that do not represent the tangible world precisely can still be highly useful, and that most of our technology would fall apart without them.

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Isn't a bubble when viewed in 2D a perfect circle? I am not too ready to accept your basic premise that a perfect circle doesn't exist, as all it takes is one example to prove it completely wrong.

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Isn't a bubble when viewed in 2D a perfect circle?

No, for several reasons:

  • The edge of a bubble is at least one molecule wide. A circle has zero thickness.
  • The bubble comprises a finite number of molecules. A circle comprises an uncountably infinite number of points.
  • Bubbles (literally) do not exist in a vacuum. At the macroscopic scales, wind, sound, and turbulence will make the bubble deviate from a perfect sphere.
  • Bubbles are not at absolute zero. At multi-atomic scales temperature variations will also make the bubble deviate from a perfect sphere.
  • Things get bizarre at the atomic scales, and yet more bizarre at sub-atomic scales. The atoms and molecules that form the bubble dance all over the place. At this scale, the bubble doesn't look anything like a sphere.

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There is that word again exist. Mathematics and physics (generally) have a different notion of existence. This seems to be a source of a lot of confusion in many threads on here.

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Yeah, pretty much every part of that argument is complete bull (as I'm sure you realize). Congrats on making it work anyway. If I was debating you, I probably wouldn't bother picking it apart, though, and instead argue that pie doesn't exist, either...

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