# OT: Basel problem in math

• posted

Hi,

For the Basel problem:

where the sum of 1/n^2 = Pi^2/6, for n=1 to n=infinity, there doesn't seem like there would be a relationship between Pi and 1/n^2, I checked on this page for the Basel problem:

from the page: " Sum_{m = 1..inf } 1/m^2.

"In 1736 he [Leonard Euler, 1707-1783] discovered the limit to the infinite series, Sum 1/n^2. He did it by doing some rather ingenious mathematics using trigonometric functions that proved the series summed to exactly Pi^2/6. How can this be? ... This demonstrates one of the most startling characteristics of mathematics - the interconnectedness of, seemingly, unrelated ideas." - Clawson

Pi^2/6 is also the length of the circumference of a circle whose diameter equals the ratio of volume of an ellipsoid to the circumscribed cuboid. Pi^2/6 is also the length of the circumference of a circle whose diameter equals the ratio of surface area of a sphere to the circumscribed cube. - Omar E. Pol, Oct 07 2011 "

Does that make sense as a intuitively as a geometric "proof" to relate these two things:

1. length of the circumference of a circle whose diameter equals the ratio of volume of an ellipsoid to the circumscribed cuboid

1. length of the circumference of a circle whose diameter equals the ratio of surface area of a sphere to the circumscribed cube

to both Pi^2/6 and to sum of 1/n^2, for n=1 to n=infinity?

Can things like the Basel problem have equivalent geometric proofs to the standard mathematical proofs?

cheers, Jamie

• posted

These things can be simplified if you just ask yourself why were they made so complicated? First, the ratio of diameter to circumference is just Pi, no? So if you take that whole circle part out you get, Pi/6 is "the ratio of volume of an ellipsoid to the circumscribed cuboid" and "the ratio of surface area of a sphere to the circumscribed cube". Doesn't sound so mysterious when you say it this way. BTW, the second one is already covered by the first since a sphere is just a special case of an ellipse. So now we just have the one statement establishing the ratio of the volume of an ellipsoid to the volume of the circumscribed cuboid.

What is so special about that?

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• posted

No- those are trivial and dumb observations.

• posted

Try:

The Basel problem is a specific case of evaluating the Riemann zeta function Z(s) for a general complex number s, where |s| != 1. The function evaluated at the even integers can all be written in closed form in the form C(2n)pi^(2n), where C(2n) is a function of the Bernoulli numbers, and it follows immediately that they are all transcendental.

The analysis of the zeta function at the odd integers is considerably harder; as far as we know they cannot be expressed in a form other than a series or an integral. Only zeta(3), Apery's constant, is known to be irrational - it isn't known if all of the others are irrational as well or transcendental, though I belive there have been certain constraints derived about how many of the rest must be irrational as well. I don't know of any geometric argument that could be applied to the odd zeta function values.

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• posted

Hi,

Thanks that's an interesting paper, shows a geometric proof for the Basel problem, I thought there would have to be one since at first glance it isn't solvable algebraically, and Pi is associated with geometry :D

I think that all mathematical equations should be able to be proved geometrically, it is just a matter of finding the right geometric objects that match the terms in the math, ie for the zeta function with imaginary numbers, maybe some 4D geometry would be required.

If the various geometric objects that could be made from the function are drawn ie maybe derived from things like this:

Then finding patterns between geometry should be able to prove whatever can be proved algebraically as well I think.

There is probably some underlying equivalence, if all math was proven geometrically and superimposed/correlated more patterns could emerge perhaps. Math that can't be proven geometrically just needs more geometric dimensions I think.

cheers, Jamie

• posted

I don't think it's possible - it has been proved that a formalization of Euclidean geometry is decidable:

The mathematics of any formal set of axioms large enough to encompass what we consider ordinary mathematics is not decidable:

irst_incompleteness_theorem

Roughly speaking, the axioms of Euclidean geometry as defined above are not even sophisticated enough to encompass what is expressible with the axioms of Peano arithmetic.

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• posted

I guess it would be possible to "define" the axioms of Euclidean geometry in such a way that they contained some set of axioms isomorphic to the Peano axioms; but then by saying that you could express your proof geometrically would be the same as saying "I can prove this statement in a formal system consisting of the Peano axioms in another formal system containing the Peano axioms" which would not be saying much.

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• posted

Hi,

Apparently that only applies to the "substantial fragment of Euclidean geometry, called "elementary,""

ie I think the proof that Euclidean geometry is decidable applies to "The first-order theory of Euclidean geometry" only

But if Euclidean geometry doesn't suffice then there should be another geometry that can be equivalent to any algebraic or other type of math I think, or an extension to make it undecidable I think.

From that page:

" For each consistent formal theory T having the required small amount of

proved within the theory T". This interpretation of G leads to the following informal analysis. If G were provable under the axioms and rules of inference of T, then T would have a theorem, G, which effectively contradicts itself, and thus the theory T would be inconsistent. "

I think that basically is saying that math is infinite, and whatever set of axioms is used they can never be infinite and thus is always incomplete, this is kind of irrelevant though to deciding if a geometrical proof can correlate 1:1 with an algebraic or other type of proof, neither are complete but it is possible they could be equivalent still I think, but it would be cool if there is a geometrical structure that can describe all mathematics and is consistent and complete, but if that exists I think there should be a matching algebra as well (probably both are far in the future to be discovered if they are even possible)

Maybe that just means there is a primitive notion missing in the geometry used if it can't geometrically describe something:

"Euclidean geometry, under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness and incidence."

If a primitive notion is added it will be more complex geometry, ie higher dimensional etc maybe. If a more complex geometry is used it makes sense it will require more primitive notions to describe it, and also I don't think there is any limit to geometrical complexity, ie it can be infinitely complex and should be able to describe all types of math.

cheers, Jamie

• posted

Hi,

If you give the geometry a coordinate space then that implicitly is using axioms from algebra too, and for more complex geometries more complex coordinate spaces are required.

cheers, Jamie

• posted

That is far away from an electrical engineering issue, we engineers are users of math and no mathematicians.

; n(integer)->inf) is a particular issue on convergence of functions zeta (poles and zeros)... I am suffering and i cry when i remember on my past studies on the math analysis (Taylor, Mc Laurin, Bertrand series, Dirichlet series ...etc ...) I cry !

Habib.

• posted

Hi,

I know what you mean, I would cry too if I knew how to do that stuff, I think it is possible that all the math could be made more common sense and accessible with geometric (+time dimension) explanations and get rid of all the high level algebra stuff.

At that point if math was fully geometrical (including calculus with a geometric "time" dimension) then it could be useful for more things maybe, ie put in a geometric model that normally wouldn't be considered mathematical and then have it simplified.

assume all math can be described with geometry of a sufficient dimension including time and a sufficiently complicated coordinate system

Heres a list of crazy stuff that might be good to add to it:

have defined axioms in the geometric space (ie for mathematical concepts, sets, sums, volume, equations etc)

derive the algebra equations for a given geometry (including time) automatically

derive the geometry (including time) for a given algebra equation automatically

analyze geometries (including time) such as a neural network to derive algebraic equations

analyze geometries (including time) such as a neural network to derive simplified geometric (including time) models

cheers, Jamie

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