Idea for using pi approximations to get close to 1
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Idea for using pi approximations to get close to 1
One thing that I never liked in physics was setting constants equal to 1 so they disappear from the equation. It also causes problems for things like c and c^2 because they are only equal when c is one which is kind of unrealistic (for example e = mc and e = mc^2 would mean the same thing). What if instead you take the actual value of pi divided by the pi approximation (or vice versa) to get a value that is close to 1. This seems to provide a nice and mathematically elegant application for pi approximations.
Re: Idea for using pi approximations to get close to 1
Sorry, I don't think I understand what you're trying to do here. What do physical constants have to do with approximations of pi?
Re: Idea for using pi approximations to get close to 1
When you're doing it in physics, you are actually just picking units so that the constants come out to 1 in those units. So e=mc^2 is not actually the same as e=mc, as they have different units (one is mass*distance/time, the other is mass*distance^2/time^2)
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Re: Idea for using pi approximations to get close to 1
mikel wrote:When you're doing it in physics, you are actually just picking units so that the constants come out to 1 in those units. So e=mc^2 is not actually the same as e=mc, as they have different units (one is mass*distance/time, the other is mass*distance^2/time^2)
Example: when you set c = 1, your unit is a lightyear/year (or lightsecond/second, etc.).
Re: Idea for using pi approximations to get close to 1
quarkcosh1 wrote:What if instead you take the actual value of pi divided by the pi approximation (or vice versa) to get a value that is close to 1.
I'm not quite sure what you're trying to do here. However, it might be worthwhile to point out that you cannot play with the value of pi. (We had a student who thought that if you can rescale units so that c is 1 and hbar is also one, you could do the same with pi. Which caused much hilarity.) Changing the value of pi means that you're adding curvature to space, which is nontrivial. While setting c to 1 is simply a choice of units, as others have stated.
Re: Idea for using pi approximations to get close to 1
As others have alluded to, is that pi is unitless, so there is nothing you can do to change that. There is nothing you can do so that the a circle of radius r has a circumference of 2r. If you have a specific equation involving both a constant with units and pi, you may be able to change your units to make that equation nicer, but it's probably not going to be useful unless it is found in a lot of different equations.
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 sinnxpionL
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Re: Idea for using pi approximations to get close to 1
Defining pi to be 1 results in some pretty interesting geometry, but I'm sure if that's what OP intended.

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Re: Idea for using pi approximations to get close to 1
Thesh wrote:As others have alluded to, is that pi is unitless, so there is nothing you can do to change that. There is nothing you can do so that the a circle of radius r has a circumference of 2r. If you have a specific equation involving both a constant with units and pi, you may be able to change your units to make that equation nicer, but it's probably not going to be useful unless it is found in a lot of different equations.
Well you could always give it units of periods per hertz.
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Re: Idea for using pi approximations to get close to 1
You could always work in base pi instead of base 10.
But yeah, you can't set pi = 1 for the same reason that you can't set 1 = 7.
But yeah, you can't set pi = 1 for the same reason that you can't set 1 = 7.
 gmalivuk
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Re: Idea for using pi approximations to get close to 1
No, it doesn't. There is no geometry where the circumference:diameter ratio is constant but not equal to the usual value.sinnxpionL wrote:Defining pi to be 1 results in some pretty interesting geometry, but I'm sure if that's what OP intended.

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Re: Idea for using pi approximations to get close to 1
gmalivuk wrote:There is no geometry where the circumference:diameter ratio is constant but not equal to the usual value.
Yes there is. There are geometries that aren't hyperbolic, eulcidean, or elliptic. For example the 1norm on R^{2} gives pi = 4.
pi=1 violates the triangle inequality though. It'd be interesting to see what the possible values of pi are. Say in the class of 2D metric spaces.
Re: Idea for using pi approximations to get close to 1
TwistedBraid wrote:gmalivuk wrote:There is no geometry where the circumference:diameter ratio is constant but not equal to the usual value.
Yes there is. There are geometries that aren't hyperbolic, eulcidean, or elliptic. For example the 1norm on R^{2} gives pi = 4.
pi=1 violates the triangle inequality though. It'd be interesting to see what the possible values of pi are. Say in the class of 2D metric spaces.
Pi is a constant equal to 3. 141... That's true in every conceivable geometry. Because the real numbers are the real numbers, regardless.
It would be helpful if there were a separate word for the ratio of a circle's circumference to its diameter. Let's call it Tuna. Then you could say: In Euclidean space, the value of Tuna is pi; but in nonEuclidean space, the value of Tuna might be different, perhaps 4.
That would help avoid a lot of confusion in these types of threads. Pi is a real number, its value is not contingent on the ambient geometrical space.
 gmalivuk
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Re: Idea for using pi approximations to get close to 1
Right you are. I forgot a word in that statement. "Smooth", perhaps?TwistedBraid wrote:gmalivuk wrote:There is no geometry where the circumference:diameter ratio is constant but not equal to the usual value.
Yes there is. There are geometries that aren't hyperbolic, eulcidean, or elliptic. For example the 1norm on R^{2} gives pi = 4.
 NathanielJ
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Re: Idea for using pi approximations to get close to 1
gmalivuk wrote:Right you are. I forgot a word in that statement. "Smooth", perhaps?TwistedBraid wrote:gmalivuk wrote:There is no geometry where the circumference:diameter ratio is constant but not equal to the usual value.
Yes there is. There are geometries that aren't hyperbolic, eulcidean, or elliptic. For example the 1norm on R^{2} gives pi = 4.
Well this depends a bit one what we mean by "pi" in the new geometry. If you don't just talk about circles, but rather define pi to be (circumference of the unit disc)/2, then any value between 3 and 4 is possible (4 is attained by the 1norm, 3 is attained by the norm whose unit ball is a regular hexagon, and you can continuously deform between the two). The values *strictly* between 3 and 4 are attainable by norms that have unit balls that are "smooth".
Re: Idea for using pi approximations to get close to 1
You can stick a scaling factor into any of those norms without it violating the definition of a norm. So you can have any value you want. (well, not negative ones or 0 ...)
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 gmalivuk
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Re: Idea for using pi approximations to get close to 1
How would such a scaling factor affect distance from the center but not distance along the circumference, or vice versa?lorb wrote:You can stick a scaling factor into any of those norms without it violating the definition of a norm. So you can have any value you want. (well, not negative ones or 0 ...)
Re: Idea for using pi approximations to get close to 1
gmalivuk wrote:How would such a scaling factor affect distance from the center but not distance along the circumference, or vice versa?lorb wrote:You can stick a scaling factor into any of those norms without it violating the definition of a norm. So you can have any value you want. (well, not negative ones or 0 ...)
Apparently I did not think that through
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 MartianInvader
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Re: Idea for using pi approximations to get close to 1
I don't think the term "geometry" is welldefined in mathematics.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
Re: Idea for using pi approximations to get close to 1
Hogwash. I would give a number of well formed definitions for different uses of geometry
1) A (smooth) geometry is a complete, connected, Riemannian manifold (or Lorentzian in GR). We usually consider only homogeonous manifolds otherwise that would kick us out to "differential geometry" (Examples, R^n with the standard metric, the nsphere with the standard metric, the upperhalf space with the Poincare metric, the upper half space with the Rindler metric.)
2) A (synthetic) geometry is any set (often finite) where the notions of "point" and "line" make sense, and satisfy some, but not possibly all, of Euclids five axioms (examples, Euclidean Geometry, Projective geometry, the Fano plane, the game Set. )
'
3) Any other field concerned with "shape." Main example are differential geometry and algebraic geometry.
In most cases when people talk about "a geometry" they mean an example of 1 or 2.
In this thread since we are talking about distances, we probably mean 1. In that case, you need a definition of pi. Three possible definitions seem reasonable to me.
1) Pi is half the limit of the ratio of the circumference of a disk to its radius as you make that radius small. Then Pi is the same for all spaces. Call this value Pi_0
2) Pi is half the circumference of a unit disk. Then different spheres will have different Pi's. So for example the sphere of radius R>1/Pi_0 will have Pi = Pi_0*R*sin(1/R)
3) Pi is a function R>R that gives the semicircumference of a circle of that radius. So for flat space Pi(r) = Pi_0 * r, and for a sphere of radius R, Pi(r) = Pi_0*R*sin(r/R)
In 2 and 3 you can get the Pi's for hyperbolic space of radius R by changing sin's to sinh's.
Note that if in both 2 and 3 if you take R to infinity (for either spherical or hyperbolic space) you the flat space Pi. Also if you set r to one in 3 you get 2 and if you take the limit as r to zero in 3 you get 1, (for either spherical or hyperbolic space).
If you wanted to deal with ndimensional spaces you could stick with Pi defined on 2dimenensional planes or use surface area of the nball. (you would get different, but related Pi's)
1) A (smooth) geometry is a complete, connected, Riemannian manifold (or Lorentzian in GR). We usually consider only homogeonous manifolds otherwise that would kick us out to "differential geometry" (Examples, R^n with the standard metric, the nsphere with the standard metric, the upperhalf space with the Poincare metric, the upper half space with the Rindler metric.)
2) A (synthetic) geometry is any set (often finite) where the notions of "point" and "line" make sense, and satisfy some, but not possibly all, of Euclids five axioms (examples, Euclidean Geometry, Projective geometry, the Fano plane, the game Set. )
'
3) Any other field concerned with "shape." Main example are differential geometry and algebraic geometry.
In most cases when people talk about "a geometry" they mean an example of 1 or 2.
In this thread since we are talking about distances, we probably mean 1. In that case, you need a definition of pi. Three possible definitions seem reasonable to me.
1) Pi is half the limit of the ratio of the circumference of a disk to its radius as you make that radius small. Then Pi is the same for all spaces. Call this value Pi_0
2) Pi is half the circumference of a unit disk. Then different spheres will have different Pi's. So for example the sphere of radius R>1/Pi_0 will have Pi = Pi_0*R*sin(1/R)
3) Pi is a function R>R that gives the semicircumference of a circle of that radius. So for flat space Pi(r) = Pi_0 * r, and for a sphere of radius R, Pi(r) = Pi_0*R*sin(r/R)
In 2 and 3 you can get the Pi's for hyperbolic space of radius R by changing sin's to sinh's.
Note that if in both 2 and 3 if you take R to infinity (for either spherical or hyperbolic space) you the flat space Pi. Also if you set r to one in 3 you get 2 and if you take the limit as r to zero in 3 you get 1, (for either spherical or hyperbolic space).
If you wanted to deal with ndimensional spaces you could stick with Pi defined on 2dimenensional planes or use surface area of the nball. (you would get different, but related Pi's)
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Re: Idea for using pi approximations to get close to 1
Nicias wrote:3) Any other field concerned with "shape." Main example are differential geometry and algebraic geometry.
In most cases when people talk about "a geometry" they mean an example of 1 or 2.
I know we're talking about specific mathematical definitions that mathematicians use. But when most people use the word "geometry" in relation to maths, they mean the study of shapes.
Specifically, the study of shapes they were required to do as part of their school curriculum (shape names and properties, measurement, perimeter/area/volume, angles, etc...)
When they say "a geometry", it is usually followed by the word "test".

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Re: Idea for using pi approximations to get close to 1
I guess trying to generate numbers that are arbitrarily close to a number is too obscure of a problem for anyone to care about. There are obvious ways to do it but I was looking for a non obvious way.
 gmalivuk
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Re: Idea for using pi approximations to get close to 1
The discussion never moved onto approximations because your underlying question (about setting pi equal to something other than 3.14...) was a non starter.
Apart from that deeply flawed premise, there's really nothing here that's new or interesting. We've been approximating pi for millennia and there are already approximations provably better (given certain requirements) than any other possibilities.
Apart from that deeply flawed premise, there's really nothing here that's new or interesting. We've been approximating pi for millennia and there are already approximations provably better (given certain requirements) than any other possibilities.

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Re: Idea for using pi approximations to get close to 1
I guess too many people misunderstood what the question was about. I was never trying to approximate pi but actually use the pi approximations to generate a number close to 1. I guess I can give you more details as to why I wanted to do this. Even though things like cosh^2(1) approximate the down/up quark mass ratio they don't get it exactly but if you can find an elegant way to vary 1 slightly you might be able to get a better approximation. Connecting this process to pi makes things even better.
Re: Idea for using pi approximations to get close to 1
I was about to compare you to scratch123, but I see someone beat me to it over in the linguistics thread. Is there compelling reason to believe you're not the same person?
Anyway, there's infinitely many mathematical constants, so being able to find some to relate to physical ones isn't surprising. We've given special symbols to a handful of those mathematical constants, but we could just as well do that with pretty much any number we can define, including ones that are identical or near identical to physical constants. It doesn't imbue those mathematical constants with physical significance though, it'd just make some people feel compelled to bark up the wrong tree trying to explain the connection when there isn't one.
Anyway, there's infinitely many mathematical constants, so being able to find some to relate to physical ones isn't surprising. We've given special symbols to a handful of those mathematical constants, but we could just as well do that with pretty much any number we can define, including ones that are identical or near identical to physical constants. It doesn't imbue those mathematical constants with physical significance though, it'd just make some people feel compelled to bark up the wrong tree trying to explain the connection when there isn't one.
 gmalivuk
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Re: Idea for using pi approximations to get close to 1
Like scratch123, who is probably the same person, I'm going to go ahead and lock most of these threads. If you insist on starting new discussions about random patterns you think you see between unrelated numbers, quarkcosh1, you can start one thread for all of them.
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