Is Pi significant because of efficiency

[Jimmy Higgins]

I was wondering well outside of my comfort level and thinking whether pi is like e, where it actually has a viable physical meaning. e being about growth, and pi being about the most efficient area for a 2D object. IE, it is the lowest possible ratio of perimeter to average width (diameter) of a 2D object.

1) is "the lowest possible ratio of perimeter to average width (diameter) of a 2D object" an accurate reflection
2) is there a physical significance of this aspect (if true).

 

[Shadowy Man]

I think the relevant ratio is perimeter to area, not width.

 

[Jarhyn]

I'll note that all this works because space is as far as we can tell "flat".

Interestingly enough there are definitions of PI, based on E, significances in math, which derive from set theory.

It would be visible in discrepancies between the scaling of space and the scaling of discrete counts of objects that exist in that space.

It would imply existing in 3.x dimensions in 4d spatial system.

Still, eventually PI would be derived, not as the ratio of dimensions of a "circle" but as a number derived from e(count) rather than e(space).

e^(i*pi)+1=0 (re: Euler)

Thus it would be apparent there was a discrepancy, as math for positions in space would be different than math for counts of stuff.

Edit: assuming, however, that the two numbers are close enough, or at least are apparently close enough, the solvers in the system would probably isolate the more mathematically simple "flat versions", and then ask some hard questions about why the universe was slightly "wrong" spatially.

 

[Swammerdami]

The "Basel Identity" — ranked #5 in a survey of "Most Beautiful Theorems" — seems to be unrelated to a circle or any question of curvature:
. . . . . . . . . π²/6 = 1 + 1/4 + 1/9 + 1/16 + 1/5² + 1/6² + ...
(However there is an excellent YouTube explaining how to derive the Basel Identity by imagining lamps spaced evenly around a circle.)


Another famous identity involving π is
. . . . . . . . . e^π·√163 = 262...744
The left-side looks like it should be transcendental , but the right-side is a perfect integer. Here's a bc script showing the convergence:

scale = 20; e(sqrt(163)*a(1)*4)​

262537412640768743.91859144793972132518​

scale = 25; e(sqrt(163)*a(1)*4)​

262537412640768743.9999993638962660662393686​

scale = 30; e(sqrt(163)*a(1)*4)​

262537412640768743.999999999986611507401764628792​

scale = 35; e(sqrt(163)*a(1)*4)​

262537412640768743.99999999999925004794479473049741408​

We might as well stop here. The odds of getting this close to an integer without actually being an integer are trillions-to-one.

That integer (262 quadrillion 537 trillion ...... and 744) is called Martin Gardner's April Fool's Integer.

 

[Jarhyn]

seems to be unrelated to a circle or any question of curvature

To be fair it still relies on the spatial idea of "flat squareness". As noted in situations of hyperbolic spatial modes, this would in fact be different.

Angularity of curvature is a product of spatial function.

Looking at hyperbolic math in my head is really fucking hard, especially on triangles. I have at least a few compartments in my brain  crying over the fact I tried to think about that.

Seriously, look at it from the perspective of a hyperbolic or even spherical universe. For reference see: Hyperbolica.

Non-euclidean space is fun.

Assuming a Non-euclidean space, how do you find euclidean space in math?

 

[Jimmy Higgins]

Still, eventually PI would be derived, not as the ratio of dimensions of a "circle" but as a number derived from e(count) rather than e(space).

e^(i*pi)+1=0 (re: Euler)

This is kind of my point, that Pi means something a bit larger, but we see it physically in a circle. That the most efficient ratio of the Perimeter to an Area has a meaningful importance to math, physics, and the universe, not merely regarding petty geometry. And it is the efficiency that makes it critical, not anything else.

 

[Jarhyn]

Still, eventually PI would be derived, not as the ratio of dimensions of a "circle" but as a number derived from e(count) rather than e(space).

e^(i*pi)+1=0 (re: Euler)

This is kind of my point, that Pi means something a bit larger, but we see it physically in a circle. That the most efficient ratio of the Perimeter to an Area has a meaningful importance to math, physics, and the universe, not merely regarding petty geometry. And it is the efficiency that makes it critical, not anything else.

But that's the thing: while we would eventually find out there was some underlying wrongness in euclidean math, the fundamental rate of growth of space vs the fundamental growth of mass and the ratios that exist in between the two in their modifications...

Essentially, one would have to put themselves in a hyperbolic space and attempt to find discrepancies between the apparent behavior of space and mass, and again, thinking about this too hard makes my brain cry.

Then, the universe isn't exactly euclidean, and it's only "flat" on average. This locally inconsistent shape, even among globally consistent shape, in fact would point to "flat math" being accessible even if space weren't flat. Just seeing relative flatness would eventually yield pi.

 

[Swammerdami]

Topological spaces are abstractions independent of the "real world." Manifolds with zero curvature are noteworthy cases whether they relate to any specific physics problem or not.

 

[Jarhyn]

Topological spaces are abstractions independent of the "real world." Manifolds with zero curvature are noteworthy cases whether they relate to any specific physics problem or not.

My point is more... When we were first asking the question "why circles" or "why squares" or "what is the ratio between these two apparently linear relationships", there would be no basis to suspect anything was "wrong" with the way "four sided objects" functioned.

We got all our math in the first place from people observing physical puzzles, one of which is a discussion of quantities of MASS and the other of which is a discussion of SPACE and for us these happen to be directly 1:1 linear, or at least as best we can tell, on most of the scales important to us.

Trigonometry came from the geometric relationships of triangles, and so most of our understanding of pi. Assuming a globally hyperbolic apparent spatial manifold, the "universal" answers would be different and there would be an additional set of constants linking spatial squaring to "mass mechanics", the "curve constant".

This is the third time now that I've forced parts of my brain to jog that track, and if I go back I think I'll be signing up for the Migraneathelon.

I think it's a pretty valid question to ask
"if I lived in Hyperbolica, when would I first see Pi in my construction of math?"
"What would "the ratio of a circle's diameter to it's radius" look like?"
"What would the coefficient relationship between between 'flat pi' and 'hPi' be, such that it acts as an absolute description of how hyperbolic the system is?"

Oof... I haven't done a number on my brain like this since I sat and sat and sat until I managed to grok 4 spatial dimensions. No I am not up to 3+, merely 1, 2, 3, and 4, the natural numbers. Fractals still throw me.

I wonder if I keep doing it, whether I will vomit.

 

[lpetrich]

Efficiency?

It can be shown that the circle is the closed curve with the shortest length for its area. geometry - Among all shapes with the same area, a circle has the shortest perimeter - Mathematics Stack Exchange

Here is the proof. I will work in rectangular coordinates because the proof is the clearest in them. I will use coordinates x and y with parameter t.

Area: \( \displaystyle{ A = \frac12 \int \left( x \frac{dy}{dt} - y \frac{dx}{dt} \right) dt = \frac12 \int (x {\dot y} - y {\dot x} ) dx } \)

Length: \( \displaystyle{ L = \int \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } dt = \int \sqrt{ (\dot x)^2 + (\dot y)^2 } dt } \)

It seems difficult to solve this minimization problem, but a common trick is the method of Lagrange multipliers. One must minimize \( L - \lambda A \) for Lagrange multiplier λ. Using the Euler-Lagrange equations, one finds

\( \displaystyle{ \frac{d}{dt} \frac{\dot x}{\sqrt{(\dot x)^2 + (\dot y)^2}} + \lambda \frac{dy}{dt} = 0 } \\ \displaystyle{ \frac{d}{dt} \frac{\dot y}{\sqrt{(\dot x)^2 + (\dot y)^2}} - \lambda \frac{dx}{dt} = 0 } \)

Now rescale t into a new parameter variable s, representing the distance along the curve: \( ds = dt \sqrt{ (\dot x)^2 + (\dot y)^2 } \)

This gives us
\( \displaystyle{ \frac{d^2x}{ds^2} + \lambda \frac{dy}{ds} = 0 } \\ \displaystyle{ \frac{d^2y}{ds^2} - \lambda \frac{dx}{ds} = 0 } \)

It's easy to solve these two equations with trigonometric functions:
\( \displaystyle{ x = x_0 + r_1 \cos(\lambda s) + r_2 \sin(\lambda s) } \\ \displaystyle{ y = y_0 + r_1 \sin(\lambda s) - r_2 \cos(\lambda s) } \)

One finds the circle equation from them:
\( \displaystyle{ (x - x_0)^2 + (y - y_0)^2 = (r_1)^2 + (r_2)^2 } \)

 

[steve_bank]

I was wondering well outside of my comfort level and thinking whether pi is like e, where it actually has a viable physical meaning. e being about growth, and pi being about the most efficient area for a 2D object. IE, it is the lowest possible ratio of perimeter to average width (diameter) of a 2D object.

1) is "the lowest possible ratio of perimeter to average width (diameter) of a 2D object" an accurate reflection
2) is there a physical significance of this aspect (if true).

The is a lot of mechanical design theory. I don't know anything spefific. I can imagine it having an application related to mecanical strength or minimizing materials.

 

[lpetrich]

Let's say that one knew nothing about trigonometric functions cos(t) and sin(t) for parameter t other than their being related to the solutions of the differential equations that I mentioned in my previous post. That is,

\( \displaystyle{ \frac{d}{dt} \cos t = - \sin t ,\ \frac{d}{dt} \sin t = \cos t ,\ \cos 0 = 1 ,\ \sin 0 = 0 } \)

One can easily derive a well-known trigonometric identity.

\( \displaystyle{ \frac{d}{dt} ( \cos^2 t + \sin^2 t ) = - 2 \cos t \sin t + 2 \sin t \cos t = 0 } \)

Meaning that \( (\cos^2 t + \sin^2 t) \) is a constant. To find that constant, we evaluate the expression for t = 0, giving 1² + 0² = 1. Thus, \( \cos^2 t + \sin^2 t = 1 \)

Now take those defining differential equations and rearrange them, using \( i = \sqrt{-1} \):

\( \displaystyle{ \frac{d}{dt} (\cos t + i \sin t) = - \sin t + i \cos t = i (\cos t + i \sin t) } \)

This shows us that \( e^{i t} = \cos t + i \sin t \) since \( \frac{d}{dt} e^t = e^t \) and \( e^ 0 = 1 \) where e is the base of the natural logarithms, 2.7182818...

From this result, one can drive negation and addition identities:

\( cos (-t) = \cos t \\ \sin (-t) = - \sin t \\ \cos (t+u) = \cos t \cos u - \sin t \sin u \\ \sin (t+u) = \sin t \cos u + \cos t \sin u \)

 

[Jarhyn]

Efficiency?

It can be shown that the circle is the closed curve with the shortest length for its area. geometry - Among all shapes with the same area, a circle has the shortest perimeter - Mathematics Stack Exchange

Here is the proof. I will work in rectangular coordinates because the proof is the clearest in them. I will use coordinates x and y with parameter t.

Area: \( \displaystyle{ A = \frac12 \int \left( x \frac{dy}{dt} - y \frac{dx}{dt} \right) dt = \frac12 \int (x {\dot y} - y {\dot x} ) dx } \)

Length: \( \displaystyle{ L = \int \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } dt = \int \sqrt{ (\dot x)^2 + (\dot y)^2 } dt } \)

It seems difficult to solve this minimization problem, but a common trick is the method of Lagrange multipliers. One must minimize \( L - \lambda A \) for Lagrange multiplier λ. Using the Euler-Lagrange equations, one finds

\( \displaystyle{ \frac{d}{dt} \frac{\dot x}{\sqrt{(\dot x)^2 + (\dot y)^2}} + \lambda \frac{dy}{dt} = 0 } \\ \displaystyle{ \frac{d}{dt} \frac{\dot y}{\sqrt{(\dot x)^2 + (\dot y)^2}} - \lambda \frac{dx}{dt} = 0 } \)

Now rescale t into a new parameter variable s, representing the distance along the curve: \( ds = dt \sqrt{ (\dot x)^2 + (\dot y)^2 } \)

This gives us
\( \displaystyle{ \frac{d^2x}{ds^2} + \lambda \frac{dy}{ds} = 0 } \\ \displaystyle{ \frac{d^2y}{ds^2} - \lambda \frac{dx}{ds} = 0 } \)

It's easy to solve these two equations with trigonometric functions:
\( \displaystyle{ x = x_0 + r_1 \cos(\lambda s) + r_2 \sin(\lambda s) } \\ \displaystyle{ y = y_0 + r_1 \sin(\lambda s) - r_2 \cos(\lambda s) } \)

One finds the circle equation from them:
\( \displaystyle{ (x - x_0)^2 + (y - y_0)^2 = (r_1)^2 + (r_2)^2 } \)

Hello sir Abstract Algebra, I think what is being asked is fairly important.

The point that was made is whether the relationships here being referenced are purely spatial. You reference the relationships WE see between squares, triangles, and circles. It's right there in your discussion of r^2.

I'm asking specifically about where someone is going to find Pi if they lack access to a flat "space".

How do you derive Pi in Hyperbolica?

I would recognize that eventually, one would find, even existing in a hyperbolic space, a value of Pi, but this would not touch on area except in "flat math"

I'm genuinely curious where pi appears in hyperbolic geometry.

 

[lpetrich]

One can find e with \( \displaystyle{ e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n } \)

The exponential function is \( \displaystyle{ e^x = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^{n x} = \lim_{m \to \infty} \left( 1 + \frac{x}{m} \right)^m } \)

where m = n*x. From that last result, one gets the derivative of e^x by using (d/dx) x^a = a*x^a-1.

This result can be proved using the binomial theorem for a a nonnegative integer, and with a proof for reciprocals and the chain rule, one gets that result for all integers. Using the chain rule for reciprocal powers, one gets that result for all rational numbers. But nonnegative integers are enough to prove the exponential function's derivative value.

But once one has that derivative value, one can prove the power-function derivative value for all complex numbers a. One does that with the logarithm function, defined as the inverse of the exponential function:

\( e^{\log x} = x ,\ \log e^x = x \)

With the chain rule, one finds

\( \displaystyle{ \frac{d}{dx} \log x = \frac{1}{x} } \)

and

\( \displaystyle{ \frac{d}{dx} x^a = \frac{d}{dx} e^{a \log x} = e^{a \log x} \frac{d}{dx} (a \log x) = a x^a \frac{1}{x} = a x^{a-1} } \)

 

[lpetrich]

Now for integrals for π. We start with defining a third trigonometric function, tan(t) = sin(t)/cos(t). For inverse trigonometric functions, we prefix their names with arc: arcsin(sin(t)) = t and sin(arcsin(t)) = t. Using the chain rule, one finds

\( \displaystyle{ \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1-x^2}} } \\ \displaystyle{ \frac{d}{dx} \arctan x = \frac{1}{1 + x^2} } \)

With \( \displaystyle{ \arcsin x = \int_0^x \frac{dt}{\sqrt{1 - t^2}} } \) let us define M = arcsin(1). We know from that that cos(M) and that e^i*M = i. Taking powers 2, 3, and 4, we find cos(2M) = -1, sin(2M) = 0, cos(3M) = 0, sin(3M) = -1, cos(4M) = 1, sin(4M) = 0.

Using the addition identities, we find sin(x+4M) = sin(x) and cos(x+4M) = cos(x) -- so these functions have period 4M. Likewise, the tangent function has period 2M. Thus, M = π/2.

 

[Swammerdami]

... , the "curve constant".

This is the third time now that I've forced parts of my brain to jog that track, and if I go back I think I'll be signing up for the Migraneathelon.

I think it's a pretty valid question to ask
"if I lived in Hyperbolica, when would I first see Pi in my construction of math?"

Newton's series for exp(x) leads to trig and eventually pi. At some point mathematicians would invent this even in Hyperbolica. Conversion to polar coordinates might be useful in integration.

BTW, am I wrong that any constant-curvature manifold is homeomorphic over curvatures? If an ordinary geometry is imposed, it can be obviated with an affine mapping.

"What would "the ratio of a circle's diameter to it's radius" look like?"

Uhh. Two-to-one?

 

[Jarhyn]

... , the "curve constant".

This is the third time now that I've forced parts of my brain to jog that track, and if I go back I think I'll be signing up for the Migraneathelon.

I think it's a pretty valid question to ask
"if I lived in Hyperbolica, when would I first see Pi in my construction of math?"

Newton's series for exp(x) leads to trig and eventually pi. At some point mathematicians would invent this even in Hyperbolica. Conversion to polar coordinates might be useful in integration.

BTW, am I wrong that any constant-curvature manifold is homeomorphic over curvatures? If an ordinary geometry is imposed, it can be obviated with an affine mapping.

"What would "the ratio of a circle's diameter to it's radius" look like?"

Uhh. Two-to-one?

Shut yes. Radius to circumference.

The question isn't how you make the curvature go away, the problem is deriving the existence of the curvature, and getting to "flat squares".

My question is, how would this manifest as a measurable distortion, beyond only first spotting the discrepancy when the CMWBR has the wrong granularity or whatever.

 

[lpetrich]

How do you derive Pi in Hyperbolica?

Simple. You use an arbitrarily small bit of it. The angle excess of a polygon is (curvature)*(area), and by making it arbitrarily small, you make it arbitrarily close to planar.

 

[Jarhyn]

How do you derive Pi in Hyperbolica?

Simple. You use an arbitrarily small bit of it. The angle excess of a polygon is (curvature)*(area), and by making it arbitrarily small, you make it arbitrarily close to planar.

And so the indication is that at any scale in which the curvature is so extreme as to be noticable at normal scales, we would still notice it in smaller things, and there would be a visible "curvature constant" on the area (and volume, and mass), yielding pi.

Thanks, now I can see it without getting such a headache. I still get a headache.

This does have a relationship to fractional dimensionality yes? I'm curious about that. I've never seen it properly. Is there a maximal curvature expressible of a hyperbolic plane?

How does this interact with gravity since the presence of volume increases mass, which curves spacetime differentially?

And how does that work with the vacuum energy interactions across the average of the gravitationally interactive universe? That would imply a curve but the analysis I've seen touts that we're cosmologically and not merely locally flat, and I understand why there would be an apparent discrepancy only when looking at the stars with respect curves at that scale.

But this is how folks figured out mass distorted space, yeah? Because paths through space shortened in proximity to mass?

 

[Bomb#20]

I was wondering well outside of my comfort level and thinking whether pi is like e, where it actually has a viable physical meaning. e being about growth, and pi being about the most efficient area for a 2D object. IE, it is the lowest possible ratio of perimeter to average width (diameter) of a 2D object.

1) is "the lowest possible ratio of perimeter to average width (diameter) of a 2D object" an accurate reflection
2) is there a physical significance of this aspect (if true).

I'm not sure how you'd define "average width" in general, but it's clear what it means at least in the case of curves with constant width -- which doesn't just mean circles:

Funny story about that -- according to Barbier's theorem, pi isn't just the ratio of a circle's perimeter to diameter, but the ratio of all of their perimeters to diameters.