Cosmological Pi Day Question

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  • Member since
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Cosmological Pi Day Question
Posted by Bullfox on Wednesday, March 14, 2012 9:49 PM

Was the value of pi different in the very early universe?

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Posted by TeleNoob on Thursday, March 15, 2012 2:57 PM

I doubt it, unless you think that whatever affects the shape of a circle would not equally affect a square.

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Posted by Primordial on Tuesday, March 20, 2012 12:33 PM

Bullfox : Interesting question, being Pi, is or is not, an infinite repeating decimal.

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Posted by Bullfox on Saturday, March 31, 2012 9:51 PM

Primordial,

I see Pi, as a mathematical number, to be unchanging regardless of any possible universe at any time.  However,  if you drew an actual circle large enough in a universe with changing curvature, then tried to measure Pi with actual instruments, then I think you would get a measured value of Pi that would change over time. 

I wonder if anyone has ever tried to actually measure Pi with real instruments on a real circle to the highest possible accuracy, and  if so, to how many decimal places they were able to get agreement with the mathematically computed value.

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Posted by TeleNoob on Sunday, April 01, 2012 10:28 AM

Bullfox

I see Pi, as a mathematical number, to be unchanging regardless of any possible universe at any time.  However,  if you drew an actual circle large enough in a universe with changing curvature, then tried to measure Pi with actual instruments, then I think you would get a measured value of Pi that would change over time. 

I doubt this because, what pi actually represents is the ratio between the circumference of a circle and the perimeter of a square. If a square has 4 sides of length "1", then its perimeter is 4. Now take a circle with diameter = 1, and its circumference is pi. In doing so we must mathematically cut the corners off the square an infinite number of times, to get it to be a circle. Essentially pi is trying to approximate a circle by saying it is equivalent to a square with an infinite number of tangential straight lines. Hence, pi cannot be resolved absolutely.

In your analogy, if one drew an actual circle large enough in a universe with changing curvature, the square within its reference frame must also be in the same universe with the same curvature. It would not be fair to compare with a square in a region of the universe, with a different curvature.

Primordial's post does make me think of something... there could be a finite limit to the value of pi, due to the proposed quantum nature of physical reality. If we accept that there is a planck length, it would not be possible to have a circle with curvature or tangential lines smaller than this. There would be an inherent "graininess" to the structure of all matter in the universe. Then there is a physical limit to the resolution of pi.

NOW, if the value of planck's constant changes in the early part of the universe...

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Posted by Primordial on Sunday, April 01, 2012 11:25 AM

Bullfox : I have serious doubt about that measurement being carried out. I see your logic in establishing a cosmological Pi, if Planck is correct, within the electromagnetic domain. I think the depth of reality depends on the oservers choice of interaction. That is to say Planck length is based on the electromagnetic, it is possible for the length to be much lesser within the strong interaction and possibly even lesser within the geodesics of space-time (gravitational-dark interaction). Just my opinion. Just think about it.

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    December, 2013
Posted by Ultra Denominator on Sunday, December 08, 2013 2:13 PM

TeleNoob:

"If we accept that there is a planck length, it would not be possible to have a circle with curvature or tangential lines smaller than this. There would be an inherent "graininess" to the structure of all matter in the universe. Then there is a physical limit to the resolution of pi."

This is a very exciting idea! It implies that the value of "pi" in the physical universe must be a ratio of integers (or more accurately, that all ratios of physical circumferences to physical diameters must be, since mathematical pi is a fixed abstraction independent of physics).  Specifically because all circumferences and all diameters would have to be integral multiples of the Planck length (if we assume the not-at-all-given concept that space is discrete and that the Planck length is the smallest possible segment length and the smallest possible separation between locations).  All "circles" would resolve to polygons with an integral number of sides when examined at the Planck scale.  Futher, for very small circles approaching the Planck scale, one would expect dramatic variations in the value of "physical pi" as the "nano-polygonality" of the circle became more and more significant.

Exciting question: could such a difference in "physical vs. mathematical" pi be detectable in an experiment?  Could it account for any small unresolved observed variances from theoretical predictions of subatomic particle motion for example? i.e. in electron equations of motion? Orbital shapes/sizes?  Such effects might show more with fast particles orbiting relativistically at atomic/subatomic scale, as the tiny "deltas" from polygonality would add up even though they would be infinitesimal.  Also, I wonder if any terms in existing equations, could be reinterpreted as such variances due to discrete vs. continuous space and path length.

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Posted by Primordial on Sunday, December 08, 2013 2:52 PM

Ultra Denominator Good point, I'll wait for tele noob's reply, but I do see a problem. Planck is quantum dependent.

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Posted by Ultra Denominator on Sunday, December 08, 2013 4:11 PM

Specifically, at the Planck scale, the smallest approximation to a circle would presumably involve 6 discrete, Planck-size "locations" or "space bits" ("metrobits"?) arranged around one central bit--thinking of them as blobs in a hexagonal close-packing arrangement.  The diameter "center" to "center" would be 2.  The circumference would be 6.  Your "physical pi" would converge to an exact vallue of 3 at the ultimate limit.

Now I'm thinking - at one extreme "physical pi" approaches mathematical, ideal pi at the Hubble scale, while at the Planck scale it might well be exactly 3 if the above admittedly simplistic model is meaningful.  So the difference (pi - 3, or .1416...) is potentially an interesting number.  Like, could it relate to the fine structure constant somehow?

 


(Update) So after playing around with that last question I discovered the following tantalizing facts, using the fine structure approximation 1/137 (abbreviated FSA):

(pi - 3) / FSA = 19.3981...

Cube root of FSA = .193981... Surprise

Now... can that be a coincidence?  OK. I know 1/137 isn't the real fine structure constant.  But still... 

Continuing the madness:

((pi - 3) / 100 FSA)  [approximately =]  cube root of FSA

FSA [approx =] fourth root of [(pi - 3) cubed / 1,000,000]


Hmmm.... fine structure constant just in terms of pi vs. 3... purely a geometric factor relating to the difference in actual Planck scale quantized path lengths vs. ideal continuous mathematical curves?

 

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Posted by Primordial on Sunday, December 08, 2013 8:10 PM

Ultra Denomitor : Like I said, you have a point, in the quanta, but field lines which permeate space-time from its origin are not quanta, but interact with quanta, math must apply to both. Would you consider this as a possibility. I like your approach. I'll think more on this. Thank you.

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Posted by Bullfox on Monday, December 09, 2013 11:11 AM

This dicussion seems to point to the possibility that physical pi is scale dependent.  That maybe has some ramifications for something somewhere at sometime.

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Posted by Primordial on Monday, December 09, 2013 12:14 PM

Bullfox : Yes, and he has a transition I've wondered about, how the singularity; from being a dimensionless point, forms a four dimensionl space-time with mass and fields. I would encourage him to take it on. There are a lot of questions to resolve in determining the initial process. Just think about it.

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    December, 2013
Posted by Ultra Denominator on Tuesday, December 10, 2013 10:09 AM

Bullfox

This dicussion seems to point to the possibility that physical pi is scale dependent. 

Exactly!  I wonder if one could devise an experiment to test that hypothesis, perhaps using very large vs. very small laser interferometers, etc.   Also "physical pi" inside the nucleus should be just slightly different than at the electron orbital scale.  Of course there are a lot of orders of magnitude between the nuclear scale and the Planck scale, and it's probably only in the tiniest 6 orders or so where you would be able to see the effects easily.  Still, all light and all particles have to traverse the same "pixellated" space.  I think someone at Stanford is doing a laser experiment to detect such graininess, though not with regard to pi-scaling effects.

Bullfox

That maybe has some ramifications for something somewhere at sometime.

I couldn't have said it better myself! Big Smile

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Posted by Primordial on Sunday, December 15, 2013 11:44 PM

Ultra Denominator : If Pi is the proportion of the measured diameter relative to the measured circumference. Do we use the Planck space or not use the Planck space in our measurement? Also in this microscale do we negotiate the interactions through the use of the virtual photon? Do virtual photons,  used to negotiate the below wavelength range, where energy exchange is extreme or do they only negotiate through the expanding space-time in this microscale? You do have a point. Thank you for your idea.

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