Important Math Question - Page 5

XenoTheMorph · #41 November 3rd, 2003, 02:07 AM

Quote:

Originally posted by Grandpa Kim:

quote:
Originally posted by atari_eric:

Since we're on the subject of math, does anyone know the formula for finding the area of a 4-dimensional sphere? Assuming an equal radius for all 4 axes...

Sorry can't help you there but how about an additional question:

the formula for the 3D volume of a 4 dimensional sphere? This would be analogous to the area of a 3D sphere.

Well in anwser to these little questions:
wwwDOTbright.net/~mrf/hierarchy(1).html
(Yikes can't use links with brackets!!!! DOT is a ".")

1. A hypersphere (4D sphere) has a 4D "Area" of
V⁴ = π²r⁴/2

2. A hypersphere (4D sphere) has a 3D "Surface" of
S³ = 2π²r³

XenoTheMorph · #42 November 3rd, 2003, 02:43 AM

O.K. now onto the subject of infinite energy in an infinite universe.

Well Space is expanding due to the cosmological constant (vacuum energy, quintessance, so on whatever you call it or think it is, it all makes up the value of Lambda. If you followed that tell me how!

). This means that there is an edge to even an infinite universe, at least from our viewpoint, it is where the 'speed' of the expansion between the viewer and another point in space (whatever it happens to be) exceeds the speed of light.

This means that even though the both viewer and the other point are 'stationary' light cannot travel from one to the other (even over infinite time). The fact that nothing can travel faster than the speed of light (don't even say Tachyons

)is not a problem since nothing is, only the distance between the two points itself is changing.

Therefore even in an infinite universe you only get to see a finite volume of space. So only the Photons from objects in this volume can reach you, so no superheating problem! This also means that the cooling of space does not mean energy is being 'lost' only that the density of the energy is decreasing, same amount of energy larger volume.

Now i'm sure you are all wondering why I wrote all this, I don't know, no idea whatsoever.
Good Luck I hope you can follow this guys I'm not sure I can

[Edited for some clarity

]

[ November 02, 2003, 12:48: Message edited by: XenoTheMorph ]

Ran-Taro · #43 November 4th, 2003, 12:23 PM

Quote:

Originally posted by Phoenix-D:

quote:
Originally posted by Grandpa Kim:
If the universe is infinite, then there are an infinite number of earth-like worlds, and infinite number of alien civilizations, and the sky would be not be blue but the color, brightness and intensity of your average sun 24/7. Half of infinity is still infinity!

Frankly, we could not survive an infinite universe. Of necessity the temperature would be far beyond what our bodies could stand. I mean, we are talking thermonuclear temperatures.

Since this is not true, the universe is not infinite.

Yep, centillion is huge, but it's pretty miniscule next to infinity.

Not true at all. Look at Hubble's Deep Field photographs sometime- you see entire galaxies that aren't even visible to the naked eye. Infititely sized means infinitely far away, as well. And heat? Do you feel any heat from the stars at night? Didn't think so, and those are the closest ones. Heat does trail off the further you get from an object.

Plus an infinitely large universe does not mean infinite numbers of everything in it. You could have an infinite amount of space, with a finite amount of planets and suns, for example.

[edit] Oops - I responded to something on the first page thinking it was the whole conversation. Now I see the converstion has covered this ground. Ignore me.

[ November 04, 2003, 10:31: Message edited by: Ran-Taro ]

geoschmo · #44 November 4th, 2003, 03:20 PM

Let me preface this by saying I don't believe the universe is infinite. But the arguments given here against an infinite universe don't hold up. This is somewhat restating what others have already said, but maybe it will help to think of it this way.

If the universe is infinite, there are an infinite number of photons, but each photon is still a finite particle. It can only be in one point in the universe at any particular time. And you are finite as well. You only exsist in one point in space. Being hit with an infinite number of photons all at once would be bad. But for that to be possible one of these has to be true:

1. The infinite number of photons in the universe have to all be in the same place; just where you happen to be standing. Besides being an incredible piece of bad luck, this would be quite difficult to time. And if you managed it somehow, you'd be left with an infinite sized universe empty of photons, except where you are.

2. Every one of the infinite number of photons would have to exsist in every point in space simultaneously. Impossible since they are finite particles.

3. You would have to exsist in every point in space simultaneously.

The point is that over time, an infinite amount of time actually, you would be hit by an infinite number of photons. But at any particular point in time you are being hit by a very small, and very finite fraction of the total number of photons. Your perceived temperature is not dependant on the total number of photons to ever hit you, but on the total number of Photons hitting you at any particular point in time.

Geoschmo

[ November 04, 2003, 13:22: Message edited by: geoschmo ]

geoschmo · #45 November 4th, 2003, 03:49 PM

Hmmmm, I seem to have hit a problem here. My previous post started out as a statement showing why Phoenix-D was right, that an infinite universe would not be infinitly hot.

However, in my post I stated that any particular point in in infinite universe would contain not an infinite number of photons, but very small and finite fraction of the total number of photons in the universe. However, according to those that are supposed to know about this stuff, any fraction of infinity is still infinity. Of course mathematics also tells us that any number divided by itself is 1. Therefore infinity divided by infinity is simultaneously equal to infinity, and 1. (By the way, if noone has thought of that before I am going to name it the Geoschmo paradox.

)

So I guess Phoenix-D and Grandpa Kim are both right. Glad we cleared that up.

[ November 04, 2003, 13:51: Message edited by: geoschmo ]

Cyrien · #46 November 4th, 2003, 04:58 PM

Quote:

Originally posted by geoschmo:
Hmmmm, I seem to have hit a problem here. My previous post started out as a statement showing why Phoenix-D was right, that an infinite universe would not be infinitly hot.

However, in my post I stated that any particular point in in infinite universe would contain not an infinite number of photons, but very small and finite fraction of the total number of photons in the universe. However, according to those that are supposed to know about this stuff, any fraction of infinity is still infinity. Of course mathematics also tells us that any number divided by itself is 1. Therefore infinity divided by infinity is simultaneously equal to infinity, and 1. (By the way, if noone has thought of that before I am going to name it the Geoschmo paradox. )

So I guess Phoenix-D and Grandpa Kim are both right. Glad we cleared that up.

Any fraction of infinity is indeed infinity so in any given piece of space in an infinite universe there are an infinite number of particles. But what is overlooked is they don't have to be photons or any particle we know of. One could say that in an infinite universe things advance to the infinitely large and infinitely small. Thus in any given point of space there exists an entire infinite universe composed of the infinitely smaller and smaller particles that make up that space.

The infinite universe would be infinetly large and infinitely small.

That is more than my mind wants to take in atm. Sooo....

Jack Simth · #47 November 4th, 2003, 07:21 PM

Geo - perhaps instead of fraction you should have used "finite portion of" - which is doable without having infinities in the portion. E.g., the set of whole numbers is an infinite set, while the set of whole numbers between 0 and 10 inclusive is a finite portion of the set of whole numbers.

geoschmo · #48 November 4th, 2003, 08:22 PM

Yes, finite portion might have been more correct. However, you still have problems trying to calculate the possible number of photons at any one point in space at any particular point in time.

The whole issue of infinity is probbably beyond our reach as a species anyway, except in a very theoretical discussion. Calculations involving infinity tend to break down into paradoxes. Perhaps the question shouldn't be whether the universe is infinite, but whether infinity even exsists.

Jack Simth · #49 November 4th, 2003, 09:24 PM

Quote:

Originally posted by geoschmo:
Yes, finite portion might have been more correct. However, you still have problems trying to calculate the possible number of photons at any one point in space at any particular point in time.

Once you have it limited to a finite volume, the task becomes much easier; the energy of a single photon is hv, where h is a particular constant (and depends on units) and v is the frequency of the photon. With the energy density of the light in that volume, and the freqency distribution in that volume (both measureable), the total number of photons can be calculated fairly readily.

If you are talking about an absolute point, rather than a volume, then the answer is trivial: 0. Photons don't seem to exist as points; they seem to exist as waves and particles.

Ran-Taro · #50 November 4th, 2003, 10:52 PM

Quote:

Originally posted by geoschmo:
Hmmmm, I seem to have hit a problem here. My previous post started out as a statement showing why Phoenix-D was right, that an infinite universe would not be infinitly hot.

However, in my post I stated that any particular point in in infinite universe would contain not an infinite number of photons, but very small and finite fraction of the total number of photons in the universe. However, according to those that are supposed to know about this stuff, any fraction of infinity is still infinity. Of course mathematics also tells us that any number divided by itself is 1. Therefore infinity divided by infinity is simultaneously equal to infinity, and 1. (By the way, if noone has thought of that before I am going to name it the Geoschmo paradox. )

So I guess Phoenix-D and Grandpa Kim are both right. Glad we cleared that up.

Infinity is not a number. It is a concept.

Dividing Infinity by Infinity is like dividing an apple by an apple.

[edit] However dividing infinity in half is not like dividing an apple in half. You would get two infinities, not two halves of infinity (because you cannot reduce something that is limitlessly big in size by dividing it, since the half would still, by definition, be limitlessly big).

Imagine trying to halve all the possible idea's in the world - both halves would still be limitlessly big, hence infinite.

My head hurts.

[ November 04, 2003, 20:59: Message edited by: Ran-Taro ]