New Formula for Pi?

[JMCx4]

Score 1 for physicists (accidentally)!

From: Scientific American
August 29, 2024

String Theorists Accidentally Find a New Formula for Pi

Two physicists have come across infinitely many novel equations for pi while trying to develop a unifying theory of the fundamental forces

By Manon Bischoff

The number pi (π) appears in the most unlikely places. It can be found in circles, of course—as well as in pendulums, springs and river bends. This everyday number is linked to transcendental mysteries. It has inspired Shakespearean thought puzzles, baking challenges and even an original song. And pi keeps the surprises coming—most recently in January 2024, when physicists Arnab Priya Saha and Aninda Sinha of the Indian Institute of Science presented a completely new formula for calculating it, which they later published in Physical Review Letters.

Saha and Sinha are not mathematicians. They were not even looking for a novel pi equation. Rather, these two string theorists were working on a unifying theory of fundamental forces, one that could reconcile electromagnetism, gravity and the strong and weak nuclear forces. In string theory, the basic building blocks of the universe are not particles, such as electrons or photons, but rather tiny threads that vibrate like the strings of a guitar and in so doing cause all visible phenomena. In their work, Saha and Sinha have investigated how these strings could interact with each other—and accidentally discovered new formulas that are related to important mathematical quantities. ...

Read more at: String Theorists Accidentally Find a New Formula for Pi | SciAm

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[Hippasus]

Look at this simple series:

4(1 -1/3 +1/5 - 1/7 + 1/9 . . . ) = Pi.

This seems to have a straightforward pattern for computation. I was looking into Taylor series expansions of Pi.

 

[Fourier]

Look at this simple series:

4(1 -1/3 +1/5 - 1/7 + 1/9 . . . ) = Pi.

This seems to have a straightforward pattern for computation. I was looking into Taylor series expansions of Pi.

For those who are curious this comes from the Taylor Series for arctan(x) centered at 0 evaluated at x=1. To prove that this is in fact valid you do need a bit of analysis since 1 is on the boundary of the interval of convergence for this series. It can also be obtained from elementary aspects of Fourier series. In particular by looking at the Fourier series for f(x)=x. In fact Fourier series can generate other series representations for Pi that are more rapidly convergent.

 

[Hippasus]

For those who are curious this comes from the Taylor Series for arctan(x) centered at 0 evaluated at x=1. To prove that this is in fact valid you do need a bit of analysis since 1 is on the boundary of the interval of convergence for this series. It can also be obtained from elementary aspects of Fourier series. In particular by looking at the Fourier series for f(x)=x. In fact Fourier series can generate other series representations for Pi that are more rapidly convergent.

If we can express arctan from x=0 to x=1, involving real analysis at the latter, as an infinite Taylor series, could we not say that continuous functions are based on discrete functions (like summations of series with discrete "events")? If it is expressible in this way under all circumstances, perhaps discreteness is really at the bottom of the matter. I mentioned something like this in class but the professor said no. Real numbers are not based on Taylor series, or something. He mentioned Dedekind cuts as a way to define, but that's not the way we tried to define real numbers. I think we took the approach of the denseness of the rational numbers, the epsilon-N (delta) method, the Compactness theorem, Archimedes' Property, etc.

 

[Fourier]

If we can express arctan from x=0 to x=1, involving real analysis at the latter, as an infinite Taylor series, could we not say that continuous functions are based on discrete functions (like summations of series with discrete "events")? If it is expressible in this way under all circumstances, perhaps discreteness is really at the bottom of the matter. I mentioned something like this in class but the professor said no. Real numbers are not based on Taylor series, or something. He mentioned Dedekind cuts as a way to define, but that's not the way we tried to define real numbers. I think we took the approach of the denseness of the rational numbers, the Compactness theorem, Archimedes' Property, etc.

It really depends on what you mean by discreteness since this is a matter of context. From a topological perspective for example even the rationals are not discrete when viewed as they would be in analysis. In contrast any set including the reals could be viewed simply as a discrete set.

As for the reals, there are several ways to define them from the rationals. Each way involves denseness of the rationals and each has its pluses and its minuses. Dedikind cuts has the advantage that the Least Upper Bound Property, which is equivalent to completeness, is an almost trivial consequence. I am not sure what you mean by defining it through the Compactness Theorem or even precisely what theorem you are referencing here. Perhaps you can clarify this for me and from that I may be able to see what your method might be.

As to your first bolded statement, you may need to clarify what you mean by every continuous function being based on discrete functions. If you are referencing Taylor series specifically, then it is not true that every continuous function is representable via it Taylor Series. In fact most continuous functions are nowhere differentiable and hence do not even have a Taylor series in the first place. Some functions have Taylor series that only converge on parts of their domain. Moreover, even if a function has a Taylor series that is everywhere convergent the value of the series may only agree with that of the Taylor series at the center of the series and no where else. The function

f(x) = e^(-1/x^2) if x is not 0

=0 if x=0
is such an example. For every n, if f^[n](x) is the n-th derivative, then f^[n](0)=0. Hence the Taylor series is

f ~ 0+0x+0x^2+0x^3+....

which of course converges to 0 for every x. But f(x) is only 0 for x=0.

Fourier series are defined for every continuous function but it is again not the case that the series will reproduce the function everywhere. To get that

f(x) = S(f)(x) = a_0 + sum a_n cos(nx) + b_n sin(nx)

for sure you actually need the original continuous function to "behave well". To get around this you can average the partial sums of the series and convergence improves substantially. In fact it will be uniform on any interval (-d,d) where d<Pi. But unless the original function is 2Pi periodic you have an issue at the points -Pi and Pi.

There are different variants of Fourier series that behave better...(see Wavlets) but continuous functions as a whole are too complicated to be represented fully by any reasonable series format. What is true however is that on any interval [a,b] a continuous function f is indeed the limit of a sequence of polynomials. But generally speaking the sequence will not be the sequence of partial sums of a specific series.

 

[Hippasus]

It really depends on what you mean by discreteness since this is a matter of context. From a topological perspective for example even the rationals are not discrete when viewed as they would be in analysis. In contrast any set including the reals could be viewed simply as a discrete set.

As for the reals, there are several ways to define them from the rationals. Each way involves denseness of the rationals and each has its pluses and its minuses. Dedikind cuts has the advantage that the Least Upper Bound Property, which is equivalent to completeness, is an almost trivial consequence. I am not sure what you mean by defining it through the Compactness Theorem or even precisely what theorem you are referencing here. Perhaps you can clarify this for me and from that I may be able to see what your method might be.

As to your first bolded statement, you may need to clarify what you mean by every continuous function being based on discrete functions. If you are referencing Taylor series specifically, then it is not true that every continuous function is representable via it Taylor Series. In fact most continuous functions are nowhere differentiable and hence do not even have a Taylor series in the first place. Some functions have Taylor series that only converge on parts of their domain. Moreover, even if a function has a Taylor series that is everywhere convergent the value of the series may only agree with that of the Taylor series at the center of the series and no where else. The function

f(x) = e^(-1/x^2) if x is not 0

=0 if x=0
is such an example. For every n, if f^[n](x) is the n-th derivative, then f^[n](0)=0. Hence the Taylor series is

f ~ 0+0x+0x^2+0x^3+....

which of course converges to 0 for every x. But f(x) is only 0 for x=0.

Fourier series are defined for every continuous function but it is again not the case that the series will reproduce the function everywhere. To get that

f(x) = S(f)(x) = a_0 + sum a_n cos(nx) + b_n sin(nx)

for sure you actually need the original continuous function to "behave well". To get around this you can average the partial sums of the series and convergence improves substantially. In fact it will be uniform on any interval (-d,d) where d<Pi. But unless the original function is 2Pi periodic you have an issue at the points -Pi and Pi.

There are different variants of Fourier series that behave better...(see Wavlets) but continuous functions as a whole are too complicated to be represented fully by any reasonable series format. What is true however is that on any interval [a,b] a continuous function f is indeed the limit of a sequence of polynomials. But generally speaking the sequence will not be the sequence of partial sums of a specific series.

Sorry for the late reply.

Dedekind cuts use the following:

The four arithmetic axioms and the first three ordering properties.

Archimedean property: "If a>0 and b>0, then for some positive integer n, we have na>b."

The Completeness axiom: "Every nonempty subset S of the real numbers that is bounded above has a least upper bound. In other words, sup S exists and is a real number." apparently can be proved from the construction of the real number field based on the rational number field. I am not sure how, though. It just says it in my text.

Then this can be shown to follow as well: Corollary: "Every nonempty subset of the real numbers that is bounded below has a greatest lower bound inf S."

Cauchy sequences, are more mathematically general, according to my professor. Dedekind cuts were more of an isolated answer to a specific (philosophical?) problem.

Now I see why my professor disagreed that real numbers are based on Taylor series: I was confusing Taylor series with Riemann sums.

As for the Compactness theorem, I also confused this with the Completeness axiom. My mistake. Hopefully I keep learning.

As for the rest of what you wrote: very interesting details on the types of numbers that can spring up within the real number field.

 

[Fourier]

Sorry for the late reply.

Dedekind cuts use the following:

The four arithmetic axioms and the first three ordering properties.

Archimedean property: "If a>0 and b>0, then for some positive integer n, we have na>b."

The Completeness axiom: "Every nonempty subset S of the real numbers that is bounded above has a least upper bound. In other words, sup S exists and is a real number." apparently can be proved from the construction of the real number field based on the rational number field. I am not sure how, though. It just says it in my text.

Then this can be shown to follow as well: Corollary: "Every nonempty subset of the real numbers that is bounded below has a greatest lower bound inf S."

Cauchy sequences, are more mathematically general, according to my professor. Dedekind cuts were more of an isolated answer to a specific (philosophical?) problem.

Now I see why my professor disagreed that real numbers are based on Taylor series: I was confusing Taylor series with Riemann sums.

As for the Compactness theorem, I also confused this with the Completeness axiom. My mistake. Hopefully I keep learning.

As for the rest of what you wrote: very interesting details on the types of numbers that can spring up within the real number field.

One of the key advantages of the Dedekind cut is that the proof of the LUB Property is almost trivial. You just take the union of the rationals determining the cut associated with each element.

The notion of a Cauchy sequence is of course much more general as it applies to any metric space. But you can show the equivalence of the LUB Property and the convergence of Cauchy sequences quite easily. In fact this is the simplest way to obtain the equivalence of the key versions of "completeness for R.

LUB P => Monotone Convergence Theorem => Bolzano-Weierstrass Property (ie every bounded sequence has a convergent subsequence) => Every Cauchy sequence converges.

Even though the MCT makes no sense in R^n One can actually deduce the completeness of R^n quite easily from that of R using these ideas.