Is Pi a Rational Number in Discrete Space?

[amppatel]

I've been reading that it is, there is a smallest volume of space, if this is so then there is also a smallest length.

So what i was wondering is that if there is a smallest length than any length could be measures exactly, like the circumfrence of a circle and the diameter, so if:

\piD=Circumfrence
then
\pi=circumfrence/D

If circumfrence and D are whole numbers, doesn't that mean that pie would be a rational number?! I am sooo confused!

 

[atyy]

I've been reading that it is, there is a smallest volume of space, if this is so then there is also a smallest length.

So what i was wondering is that if there is a smallest length than any length could be measures exactly, like the circumfrence of a circle and the diameter, so if:

\piD=Circumfrence
then
\pi=circumfrence/D

If circumfrence and D are whole numbers, doesn't that mean that pie would be a rational number?! I am sooo confused!

Yes, if space is discrete then we could only have polygons, or something. The discreteness of space is right at the frontiers of research, so everyone doing it is confused. Anyway, if space is discrete (which is only a conjecture at the moment), the little bits of which it is made of are very small, and we can't see them even with our best current experiments. So we are not making any sort of big mistake if we pretend space is smooth and circles exist in the real world. Just like your TV screen is discrete, but if the resolution is very high, then the image still looks smooth.

 

[amppatel]

so if it is discrete then pie would be rational?

 

[atyy]

so if it is discrete then pie would be rational?

Ha, ha. To be honest, I don't know. I guess that circles won't even exist in the real world in that case.

There will always be the perfect imaginary world of Euclidean geometry, where pi will always be irrational. It's just that we won't be able to use that perfect imaginary world to describe the real world.

 

[cristo]

It's just that we won't be able to use that perfect imaginary world to describe the real world...

... on very small scales.

 

[atyy]

... on very small scales.

Yeah! very,very,very,very,very,very,very ... small scales

 

[atyy]

Yeah! very,very,very,very,very,very,very ... small scales

Oh, I forgot, and already on very, very large scales - not because space is discrete - but because spacetime is curved, and Euclidean geometry is flat.

 

[wolram]

If space is discrete what stops the discrete (packages) merging, i guess individual (packages) of space would have to be attracting other wise holes could occur.

 

[atyy]

If space is discrete what stops the discrete (packages) merging, i guess individual (packages) of space would have to be attracting other wise holes could occur.

This stuff is really right at the edge of research, so we have no idea what's right or wrong. All current attempts to make discrete theory of spacetime may eventually fail, but I'll give some links to what seem to be promising leads:
http://arxiv.org/abs/hep-th/0408048
http://arxiv.org/abs/gr-qc/0601121
http://arxiv.org/abs/gr-qc/0606100

I should add this cautionary statement from John Baez: A lot of people read pop books about quantum mechanics, black holes, or Gödel's theorem, and immediately want to study those subjects. Without the necessary background, they soon become frustrated - or worse, flaky.

http://math.ucr.edu/home/baez/books.html

 

[amppatel]

ooo i get it now, if space is discrete there is no such thing as a circle so the formula doesn't apply! Cheers for the help! So space being discrete hasn't been proven? I read the Three Roads to Quantum Gravity and the author put across that is was definitely right.

 

[atyy]

So space being discrete hasn't been proven? I read the Three Roads to Quantum Gravity and the author put across that is was definitely right.

Ha, ha! I have never read that book myself. But Smolin wrote a notorious book "The Trouble With Physics", which accused string theorists of misleading the public that string theory was destined to be right! Yeah, as far as I know, neither string theory, nor Smolin's own theory is known to be right at the moment.

Marcus has a very good thread on this forum with updates of the latest theories of discrete spacetime - what I like about his posts is that there's tons of nonsense in this area, and he chooses stuff that has at least some promise.
https://www.physicsforums.com/showthread.php?t=7245

 

[granpa]

if space is discrete (which is only a conjecture at the moment), the little bits of which it is made of are very small, and we can't see them even with our best current experiments.

and probably not ever since elementary particles are spread out over a large area by the uncertainty principle. which is convenient since if they weren't then we would expect weird quantum effects to altar their interactions.

 

[granpa]

If space is discrete what stops the discrete (packages) merging, i guess individual (packages) of space would have to be attracting other wise holes could occur.

attracting? things attract one another through space. I'm not sure the same idea can be applied to space itself.

I tend to think of the smallest units of space as just numbers. (or vectors. or maybe tensors).

 

[Nick89]

If space is indeed discrete, that would not automatically mean pi was rational! It would only mean that, in practice, there can be no perfect circle. Indeed, even if space is not discrete, (let's assume for the moment it's not) any circle you would construct (by building one, drawing one, whatever) would be made up out of matter whose atoms cannot possibly form a perfect circle.

 

[amppatel]

If space is indeed discrete, that would not automatically mean pi was rational! It would only mean that, in practice, there can be no perfect circle. Indeed, even if space is not discrete, (let's assume for the moment it's not) any circle you would construct (by building one, drawing one, whatever) would be made up out of matter whose atoms cannot possibly form a perfect circle.

surely in continuous space you CAN draw a perfect circle. i think haha

 

[Chronos]

If space is discrete would photons emitted by extremely remote objects [like GRB's] be more diffracted than photons emitted by nearby objects?

 

[Nick89]

surely in continuous space you CAN draw a perfect circle. i think haha

Not with any kind of matter, since the particles making up the matter will cause the circle to be 'pixelated' if you look from close by enough. Compare it to zooming in on a circle you drew on your computer. If you don't zoom in it looks nearly perfect, but once you zoom in and see the individual pixels you see it's actually just squares aligned in a circle: not a perfect circle.