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Gustafson, S

A conjecture: pi changes with time

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A CONJECTURE: PI CHANGES WITH TIME

 

 

Steven C. Gustafson, 14 February 2007

 

 

The following is conjectured:

 

The measured value of pi changes with time. At the big bang it was 2, but now it is very slightly smaller than the calculated value of 3.14159... . The current measured value of pi, if it could be measured accurately enough, would be found to be smaller than the calculated value by about one part in 10 to the 121st power.

 

The difference in the measured and calculated values of pi will be indicated by a significant change in the randomness of the digits of the calculated value of pi before and after about the 121st decimal place. Thus determining randomness will constitute measurement, and the significant change will be robust in that it will be found using digits generated in various bases and using various measures of randomness.

 

 

A Justification is as follows

 

As indicated in reference 1 and its citations, at the big bang the universe can be modeled on the surface of a sphere of approximately the Planck radius r = (hG/cE3)E1/2 = 4.05 10E-35 meters (where r is obtained by eliminating m from mcE2 = GmE2/r and mcr = h). The smallest circle that could have been drawn on the surface of this sphere would have had a radius of r, and since the diameter of this circle must have been measured along the surface of the sphere, the measured diameter would have been half of the measured circumference, and thus the measured value of pi would have been 2. Any method of measuring pi (e.g., summing the angles of a triangle drawn on the surface of the sphere) would have produced this value of pi.

 

Currently the universe can be modeled on the surface of a sphere of radius R = 13.7 10E9 light years, but the smallest circle that can be drawn on the surface of a sphere of this radius still has the Plank radius (see reference 1 and its citations). Thus the diameter of this circle, which again must be measured along the surface of the sphere, is very slightly curved and is therefore longer than it would be in a flat or Euclidian space (i. e., a space in which the radius of the universe is infinite). Accordingly, the measured value of pi, which is the ratio of measured circumference to measured diameter, is very slightly less than the calculated value, which is currently known to more than a trillion decimal places. Again, any method of measuring pi, including summing the angles of a triangle, using Buffon’s needle, etc., would yield a value of pi slightly less than 3.14159… . As is easily shown (see reference 3) this line of reasoning leads to a measured value of pi that is smaller than the calculated value by about one part in 10 to the 121st power, which is three times the square of the ratio of R to r.

 

The calculated value of pi found using any of many possible formulas is of course accurate to the calculated number of decimal digits, binary bits, etc. However, the calculation may be considered to be a measurement of pi in that the sequence of digits or bits obtained has a degree of randomness which is not predetermined. Thus it is conjectured that the degree of randomness of about the first 121 decimal digits (or about the first 404 binary bits) is significantly different than the degree of randomness of the following digits (or bits). The degree of randomness might be quantified using various metrics, including the approximate entropy considered in reference 2, where this metric was used to determine, for example, that the digits of √2 are more random than the digits of e. If the change in the degree of randomness is sufficiently sharp, then the digit or bit (or digit or bit region) at which the change occurs might be used to specify a more accurate value for the radius or age of the universe.

 

 

1. A. Ashtekar, T. Pawlowski, P. Singh, “Quantum Nature of the Big Bang”, Physical Review Letters, vol. 96, pp. 1413011- 1413014, 14 April 2006.

 

2. S. Pincus and R. E. Kalman, “Not all (possibly) “random” sequences are created equal”, Proc. Nat. Acad. Sci., vol. 94, pp. 3513 – 3518, April 1997.

 

3. S. C. Gustafson, “The Planck radius, the randomness of pi, and the age of the universe”, http://www.scienceforums.net, 28 May 2006.

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Is it just me, or did that last paragraph take this post from interesting to nonsensical?

 

The statements of the first two paragraphs at least are backed up by some reasoning. But the 3rd paragraph just seems to come out of nowhere. Why are some of the digits more random than others? Why is it to 121 decimal digits yet 404 to binary digits?

 

Interesting either way, I await the comentary of people who know more geometry than I.

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The third paragraph is intended to be entirely rational. It states, AS A CONJECTURE, that digits before a precision of about one part in 10 to the 121st power in a listing of calcualted digits of pi will be found to have a different degree of randomness than the following digits. It also states that this conjecture can be tested using various metrics for the degree of randomness (a particular entrpy-based metric is suggesed as an example), and it indicates that pi can be expressed in any base, where base 10 or base 2 would be common choices. In base 10 a precision of one part in 10 to the 121st power corresponds to the 121st decimal place, whereas in base 2 (binary) this precision corresponds to the 404 binary digit (i.e., more binary than decimal digits are needed to express a given precision).

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More evidence of a variable value for pi

 

The value of pi is clearly increasing with time. In biblical times, the value of pi was 3.

 

 

1 Kings 7:23 (New International Version)

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.

 

If the circumference was 30 cubits and the diameter was 10 cubits, then pi was 3.

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Ok I see what you're saying now. ( I initially took your post as a joke, sorry about that)

 

In curved space, a curved diameter will be longer than a straight diameter in flat space. So depending on the radius of curvature of space, the ratio of the circumference to the curved diameter will be somewhere between 2 and our currently accepted value for pi.

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Still, it demonstrates a misunderstanding of what pi is. It is essentially calculated in a completely fictional universe. Usually it's worked out as if all we were dealing with were a flat plane, but the same number works out to have the same properties in whatever model we come up with, be it based on reality or not.

 

I think what Gustafson is trying to say is that the Universe's proportion of diameter to circumference changes, which I suppose it does, or at least I see no reason why it shouldn't. This doesn't mean that pi itself changes, but it is quite interesting.

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the bit I can`t get my head around is the "Degree of randomness"?

what is that meant to mean?

 

Random is Random, you can`t have More random, random just IS or it is NOT, there`s no "Degrees".

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;323690']Is it just me' date=' or did that last paragraph take this post from interesting to nonsensical?

 

The statements of the first two paragraphs at least are backed up by some reasoning.[/quote']

 

Interesting? The whole thing starts with this ludricrous lack of reasoning:

 

The measured value of pi changes with time.

 

Pi is not a measured quantity. It is a mathematical invention.

 

Tree almost had it right:

Still, it demonstrates a misunderstanding of what pi is. It is essentially calculated in a completely fictional universe.

 

This article demonstrates a complete misunderstanding of what pi is.

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Gaaah, so much for my mega fantastic Detective work:-p

 

anyway, that makes it all more (how shall we say..) Not very Good, if you have to cite yourself as a reference!:rolleyes:

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Still, it demonstrates a misunderstanding of what pi is. It is essentially calculated in a completely fictional universe. Usually it's worked out as if all we were dealing with were a flat plane, but the same number works out to have the same properties in whatever model we come up with, be it based on reality or not.

 

I think what Gustafson is trying to say is that the Universe's proportion of diameter to circumference changes, which I suppose it does, or at least I see no reason why it shouldn't. This doesn't mean that pi itself changes, but it is quite interesting.

 

If space is curved (which WMAP measurments indicate it may be) then the ratio of diam to circumference is DIFFERENT for differentsize circles.

 

Ned Wright who worked on both COBE and WMAP and is a worldclass cosmologist just came out with a paper where he gives a "best fit" case and in that paper the best fit universe has overall aveage positive curvature and the bestfit radius of curvature is 130 billion LY.

 

From our point of view that is such a large radius of curvature that for all practical, we can consider space to be flat. And for all practical, the value of pi is exactly correct.

 

However mathematically speaking pi is only correct for small circles and as you make the circle larger and larger the ratio of circumf to diameter deviates from ideal ratio.

 

In Ned Wright best fit universe, with its very slight positive curvature, a very very big circle has a circumference which is not quite as big as one expects.

 

and a very very big triangle the sum of the angles is slightly more than 180.

 

Some readers might like to have a look at the Ned Wright paper

http://arxiv.org/abs/astro-ph/0701584

Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant

Edward L. Wright (UCLA)

17 pages Latex with 8 Postscript figure files. One new Table, one new Figure, and several new references added. Submitted to the ApJ

 

"The luminosity distance vs. redshift law is now measured using supernovae and gamma ray bursts, and the angular size distance is measured at the surface of last scattering by the CMB and at z = 0.35 by baryon acoustic oscillations. In this paper this data is fit to models for the equation of state with w = -1, w = const, and w(z) = w_0+w_a(1-a). The last model is poorly constrained by the distance data, leading to unphysical solutions where the dark energy dominates at early times unless the large scale structure and acoustic scale constraints are modified to allow for early time dark energy effects. A flat LambdaCDM model is consistent with all the data."

 

Note that the flat LambdaCDM model (which is mainly what cosmologists use these days) is CONSISTENT with the data (within uncertainty bounds) but the flat model as it happens is not the BEST fit. One should know about the best fit but one should not necessarily believe either way because more work to be done narrowing down the uncertainty.

 

Look at page15 where he says

The best fit model is slightly closed with

Omega_tot = 1.011 ...

The best fit flat LambdaCDM model has less than one more unit of chi-square

than the best fit non-flat LambdaCDM model so there is no evidence for spatial curvature from these fits.

 

what he is saying is that flat is worse, but it is worse by LESS THAN ONE CHI-SQUARE, so therefore it is NOT STATISTICALLY SIGNIFICANT that the bestfit has positive spatial curvature.

 

Also be careful about the word "closed". It does not mean universe expects a big crunch. In modern cosmo models a closed universe can expand forever. Closed just refers to spatial closure. A spatially closed positive spatial curvature universe is like a sphere but in higher dimension (so called 3-sphere).

That is the thing which has radius of curvature 130 billion LY and it is space at this moment in time, however the 3sphere is expanding and as far as anyone knows will keep on doing so.

 

The flat LambdaCDM model, by contrast, has space being infinite. It is more simple to calculate with and generally preferred by the pros at present.

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Thanks to all of you for your comments on my “A CONJECTURE: PI CHANGES WITH TIME”. I completely agree with comments to the effect that pi is a mathematical construct and not a measured quantity. Some less philosophical observations are as follows.

 

1. The conjecture is not idle---it can be verified by computer experiments. These experiments would be similar to experiments which used an approximate entropy metric for randomness to verify that the digits of √2 are more random than the digits of e (reference: S. Pincus and R. E. Kalman, “Not all (possibly) ‘random’ sequences are created equal”, Proc. Nat. Acad. Sci., vol. 94, pp. 3513 – 3518, April, 1997). However, the conjecture-verifying experiments should be conducted by persons more familiar with Matlab, Mathematica, etc., than I am.

 

2. Here is an initial recipe for the computer experiments. Find the randomness of the sequence of the digits of pi from digit 1 to digit n and from digit n + 1 to digit 2n using approximate entropy, find the fractional change in randomness, repeat for n = 2, …, 242, and plot the fractional change in randomness versus n. Examine the plot for changes in level, slope, curvature, etc., near n = 121. Generate plots using other measures of randomness and using pi expressed in other bases to characterize the changes (if any). Note that if changes in randomness are found for many measures of randomness and for many bases and if they can be interpolated to the same fractional digit location (e.g., n = 121.327), then a more accurate value for the radius of the universe (currently estimated at 13.7 billion light years) might be proposed.

 

3. Is anyone out there interested in performing the conjecture-verifying experiments?

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