Pi

[Tres Juicy]

Just to reiterate - this is pure speculation.

 

Pi has always annoyed me, 3.14159265358979323846264338327950288419716939937510 is not sensible to my mind.

 

Surely Pi should be 3? A nice rational number.

 

Which made me wonder, with the expansion of the universe essentially stretching the fabric of space/time, could Pi be moving?

 

What I maen is: in the early universe could Pi have been closer to my ideal 3?

 

Any comments welcome as always...

[granpa]

pi is determined by the metric of space

 

as long as c² = a² + b² pi will not change

 

http://en.wikipedia.org/wiki/Lp_space

 

for the taxicab metric pi = 4, tau = 8

Edited December 12, 2011 by granpa

[Tres Juicy]

pi is determined by the metric of space

 

as long as c² = a² + b² pi will not change

 

http://en.wikipedia.org/wiki/Lp_space

 

 

"pi is determined by the metric of space"

 

Exactly my point, a circle drawn on an irregular surface will have a slightly different value of pi.

 

What if the expansion of the universe is changing the "metric of space"?

[ajb]

[math]\pi[/math] can be understood as a geometric thing tied down to the fact that Euclidean space is flat. For instance, let us take the sum of the interior angles of a triangle as defining [math]\pi[/math]. One can in a way change "[math]\pi[/math]" by changing the geometry.

 

For instance, it is true that in Euclidean space the sum of the angles of a triangle equal to [math]\pi[/math]. In a hyperbolic geometry the sum of the angles of a triangle is always less than [math]\pi[/math]. One could then define a new constant in this geometry say [math]\pi'[/math] that is always less than [math]\pi[/math].

 

On a spherical geometry sum of the angles of a triangle is always greater than [math]\pi[/math], so in a similar way one could define a new constant for that geometry [math] \pi''[/math] which is always greater than [math]\pi[/math].

[Schrödinger's hat]

You'd fit in well in Ancient Greece

 

Spacetime is locally euclidean, so you can always get the same 3.141.... number by drawing smaller and smaller circles and seeing what values you get for the ratio of the circumference and diameter. Unless your circle included a singularity, in which case you'd have a bit of difficulty measuring the radius.

[Tres Juicy]

You'd fit in well in Ancient Greece

 

Spacetime is locally euclidean, so you can always get the same 3.141.... number by drawing smaller and smaller circles and seeing what values you get for the ratio of the circumference and diameter. Unless your circle included a singularity, in which case you'd have a bit of difficulty measuring the radius.

 

 

"You'd fit in well in Ancient Greece"

 

I'm choosing to take this as a compliment!

 

 

Edit: Having now read the link I realise its not... Stupid Hippasus get off my boat

Edited December 12, 2011 by Tres Juicy

[md65536]

What I maen is: in the early universe could Pi have been closer to my ideal 3?

A hexagon inscribed in a circle (so that the hexagon's side length = the circle's radius) will have a perimeter of 6.

I've tried to reason that if you can get the hexagon's perimeter to equal the circle's, then tau (=2pi) would be 6, so pi would be 3. What does that mean though, I dunno! Could you curve space so that a hexagon and circle are the same? Could you pinch one side of an equilateral triangle so that the hexagon's perimeter becomes smaller, and the circle's becomes 6r?

 

 

 

pi is determined by the metric of space

http://en.wikipedia.org/wiki/Lp_space

There's a link on that page to: http://en.wikipedia.org/wiki/Astroid

Surprising to me, "An astroid created ... inside a circle of radius a will have ... a perimeter of 6a."

 

Does this mean that if pi were 3, then a unit circle would be an astroid?

 

So in Tres Juicy's ideal universe, an astroid (not a hexagon) and a circle would be identical?

 

 

 

 

Edit: Oh wait, there must be some value of p > 1 (unlike an astroid where p=2/3) but p < 2 where the perimeter of the unit circle is 6r. Maybe p=3/2??? That would be an "ideal universe" that is closer to our own than the space where p=2/3.

 

Edit2: I was curious and tried to look it up... the best I found was http://www.procato.com/superellipse/

If you use a value for n of 1.582 (not 1.5 = 3/2 as I'd guessed), and a=b=2 you will get a plot of a superellipse with a perimeter of approximately 6.00. This would be a unit circle if pi were 3. Unfortunately I can't find anything that is more precise. With a=b=1, a value of 1.57 is good enough, so I thought at first that maybe an L^p space with p=pi/2 would give a unit circle of perimeter 6. Does anyone know if there's any significance to this value (about 1.582)?

Edited December 13, 2011 by md65536

[Tres Juicy]

The thing is... without any maths or measuring you just look at it and think "it's going to be 3" then it turns out it's off by .14, it's annoying....

 

 

But my point is: can the value of pi be altered by the bending/warping of space through expansion?

 

The universe is roughly 14 billion years old and pi is roughly 14 points off 3 - could these be related?

 

Again I am speculating my ass off here...

[insane_alien]

thats clutching at straws.

 

besides, pi being an irrational, random string of numbers, you can do better than roughly.

 

If you look long enough you will be able to find a string of numbers that gives the precise age of the universe in plank units.

 

but pi is always equal to 3.14159 etc.. in flat space by definition. whether the universe is flat or not would not change the value of euclidean pi

[Bignose]

The universe is roughly 14 billion years old and pi is roughly 14 points off 3 - could these be related?

 

I think it is hardly certain that the Universe is roughly 14 billion years old. Every few years, that estimate gets longer and longer. I think it is pretty clear that 14 billion and 3.14.... is just coincidence. Especially when you think that of all the bases we could have used, 3.14 is only the value in base 10.

Edited December 13, 2011 by Bignose

[md65536]

I think it is hardly certain that the Universe is roughly 14 billion years old. Every few years, that estimate gets longer and longer. I think it is pretty clear that 14 billion and 3.14.... is just coincidence. Especially when you think that of all the bases we could have used, 3.14 is only the value in base 10.

Coincidence?! Or... ?

 

...not!

 

Consider this: Every two or three years the estimate of the age of the universe increases by two or three years. AND, every few years they find that there are even a few more digits to pie than they previously thought! Mostly due to faster computers with bigger hard drives, is my guess. Just a couple years ago, they thought that pie had only about a million digits, but then they found a whole bunch more.

 

Maybe, just like the universe, pie ages, and it grows. What is this mysterious stuff that grows on pie as it gets really old?

 

 

[Tres Juicy]

If the nature of space changes so will the value of pi

 

What I'm asking is can the geometry of space alter pi?

[Schrödinger's hat]

If the nature of space changes so will the value of pi

 

What I'm asking is can the geometry of space alter pi?

 

The geometry of space can alter pi in that space.

As insane alien said, 3.14159... is pi in euclidean space.

Our space is not globally euclidean. If you were to draw a big enough circle in a gravitational field, you'd notice the ratio between its circumference and diameter was a tiiiny bit different.

Our space is locally euclidean¹, so for a small enough circle you get 3.14159....

 

We use the euclidean pi in our mathematics, so curvature of space cannot change this.

 

 

 

 

 

 

¹Possible exception of singularities -- if they exist. I do not know if they'd count as locally euclidean.

[ewmon]

If you used a "unit" radius (that is, equaling 1) and scribed a circle at an angle of about 12½° to the drawing surface, the circle's circumference will equal 3, and effectively, "pi" will equal 3/1, ie 3.

[michel123456]

If you draw a circle on the curved surface of a sphere, it will remain a circle. If you draw a square on a sphere, it will not remain exactly a square because the sides will be curved. A circle inscribed into this square will have a curved radius: the radius will follow the curve of the sphere. Some other geometer will tell you what happen to Pi in this case, but intuitively I think you can make a configuration where Pi=3.

Edited December 13, 2011 by michel123456

[md65536]

As insane alien said, 3.14159... is pi in euclidean space.

Our space is not globally euclidean. If you were to draw a big enough circle in a gravitational field, you'd notice the ratio between its circumference and diameter was a tiiiny bit different.

 

I'm still confused, with questions like "Is flatness subjective?; what defines euclidean space and could any of those properties be modified?"

 

But I think that a big part of the confusion is that since pi is irrational it can appear to be just an arbitrary sequence of apparently random digits.

It can be calculated by fairly simple infinite series that corresponds to some geometric interpretation...

Such as:

[http://en.wikipedia.org/wiki/Pi]

 

 

This is a series of very simple rational numbers.

From this, I would think that pi would not change by small amounts over time.

If you imagined that pi changed somewhere in say the billionth decimal place, over the past few million years, then what formula could be used to calculate that different value?

What small change in geometry could yield a different pi, but still allow it to be calculated with a simple series?

[Tres Juicy]

I'm still confused, with questions like "Is flatness subjective?; what defines euclidean space and could any of those properties be modified?"

 

But I think that a big part of the confusion is that since pi is irrational it can appear to be just an arbitrary sequence of apparently random digits.

It can be calculated by fairly simple infinite series that corresponds to some geometric interpretation...

Such as:

[http://en.wikipedia.org/wiki/Pi]

 

 

This is a series of very simple rational numbers.

From this, I would think that pi would not change by small amounts over time.

If you imagined that pi changed somewhere in say the billionth decimal place, over the past few million years, then what formula could be used to calculate that different value?

What small change in geometry could yield a different pi, but still allow it to be calculated with a simple series?

 

 

It obviously wouldn't be visible through calculation - only through measurement (why would you change the maths without observng a reason to?)

[md65536]

It obviously wouldn't be visible through calculation - only through measurement (why would you change the maths without observng a reason to?)

Oh. Well, in that case what you'd be measuring wouldn't be euclidean space. As mentioned previously in this thread, if you draw a big enough circle in curved space, it won't have the same ratio of circumference to radius of 2pi.

 

So then the question might be, "Has local space always been euclidean?" Again I don't think it's possible that it can vary by small amounts, and yet we just happen to be in a situation where it is, by chance.

 

 

http://news.discovery.com/space/once-upon-a-time-the-universe-was-really-weird-110321.html

This article suggests that the 3 spatial dimensions of euclidean geometry were not always "there" in the early universe.

Pure speculation based on very weak understanding of all this: Perhaps curved space can somehow have fractional dimension to it, but it can always be observed consistently with an integer number of dimensions, so that "local space" is always flat no matter the number of dimensions. Perhaps if space can curve so much that it would fold over on itself, we would instead experience 4 spatial dimensions??? So, it may have been the case that local space was non-euclidean in the early universe (since euclidean space is defined as having 3 spatial dimensions), and may not be in the distant future? I don't know if pi or an analog would be defined for such spaces, and if so what its value would be. But I would still assume that if it changed, it would not do so gradually.

 

 

[Bignose]

Just a couple years ago, they thought that pie had only about a million digits, but then they found a whole bunch more.

 

Ummmm, maybe this is a joke, but no. The proof pi is irrational has been known for over 200 years. And it has been known to be transcendental since 1882 (per Wikipedia). That means infinite number of digits after the decimal point. Just because we can't ever calculate all of them, doesn't mean that we didn't know they are there.

Edited December 13, 2011 by Bignose