Fourth Dimension Calculation?

[lizzy]

1.) Can anyone show me the formula or method used to calculate the fourth dimension?

2.) Input on how to apply it would also help.

Thanx!
Lizzy :?

[mghtymoop]

calculate what, you want to be taught to integrate? calculous needs a little more hands on approach than someone showing you the formula in a web forum

[PWrong]

There's formulas for the surface volume and bulk of many 4D objects here: http://tetraspace.alkaline.org/shapes/formulas.htm

4D objects also have equations to define them, which are here:
http://tetraspace.alkaline.org/shapes/rotatopeclass.htm

Other than that there isn't a single formula for the fourth dimension, any more than there is a formula for the third dimension. It's not hard to understand all the concepts though, once you get into it. Usually you can answer a question just by saying, "this happens in 2D, and this happens in 3D, so this must happen in 4D". Read some more posts and you'll see what I mean.

The actual maths in the fourth dimension is usually insane, and most of us here can't do it The general forum is mostly about speculation and stuff. The geometry forum is a bit more maths orientated, but it doesn't go into the really tricky stuff

[mghtymoop]

speak for youself, the maths is simple calc and geometry with an extra axis, i daresay most of us can do it, at least everyone who is capable of understanding any text written on dimensional physics, if you can't understand the maths you can't understand the concepts, especially the concepts of the string theory nazis who seem to be the majority in this forum. I would hate to think anyone without an understanding of maths would subscribe to a purely math based theorum.

[PWrong]

if you can't understand the maths you can't understand the concepts

Well, actually I understand most of the maths on the the tetraspace site, although it's more difficult than what I've learnt in school (I'm in year 12, doing applicable maths, calculus and physics). I've probably learnt more maths from the internet than I have at school.

I would hate to think anyone without an understanding of maths would subscribe to a purely math based theorum.

It's not purely math based. Read the introduction to the site, a 12 year-old could understand it to some extent. There's a wide diversity in the age and level of education in the members. That's what's so good about the forum.

speak for youself, the maths is simple calc and geometry with an extra axis, i daresay most of us can do it, at least everyone who is capable of understanding any text written on dimensional physics.

If it's so easy, why didn't you explain it? Most texts written on dimensional physics are like textbooks. They don't make much sense if you're just interested in the fourth dimension and not studying it seriously. Anyway, it's not all calculus and geometry.

[mghtymoop]

alkalines intro is aimed at 12 year olds as far as i can tell, there are far more informative and accurate sites available on the net than the tetraspace one, sites that don't contain made up terms and mathematical failures. not all of the books read like textbooks but a textbook understanding really does help you to understand even these. its no more difficult than calc (provided they are still teaching what they were teaching a few decades ago) you just need to know how to apply the calculous, anyone with a decent highschool education. besides my statment of you subscribing to a purely maths based theorum was based upon your belief in string theory, which is entirely complicated maths but is by no means the only theory or even the one with the most (or any for that matter) evidence. i have no need to explain anything, explanation is for people with little to no understanding who will not have views that i am interested in so do not wish to converse with.[/quote]

[jinydu]

In any case, here is a derivation of Kepler's First Law (planets travel in an ellipse with the Sun at one focus) using Newton's Laws:

http://usna.edu/Users/physics/mungan/Sc ... rstLaw.pdf

Unfortunately, its a bit to advanced for me to handle. But my question is, can anyone try to derive an analogue of Kepler's First Law in tetraspace? You could try to follow the same sequence of steps as the proof above, except change Newton's Gravitational Law to an inverse-cube instead of an inverse-square law.

[PWrong]

lol, what does Kepler's first law have to do with anything? Sounds interesting, but I can't open it for some reason

Mightymoop, I only mentioned string theory in one thread. If you read some of my other posts, most of them never mention string theory. In fact my first post was about whether 4 dimensional creatures might be vegetarian. Vegetarianism is more important to me than string theory, but it doesn't mean I was trying to corrupt anyone with my evil beliefs.

Sorry about the mixup with theorum and forum though, my mistake.

[jinydu]

I'm wondering what path a planet would follow around a "sun" in tetraspace.

[elpenmaster]

a planet zooming around a sun in tetraspace. . .

by the laws of gravity, the planet would simply follow a circular orbit, only moving in two dimensions

but it might be more volatile to minute gravitatioal changes around it and do an electron thing.

[pat]

by the laws of gravity, the planet would simply follow a circular orbit, only moving in two dimensions

First, the laws of gravity allow elliptical orbits in three-space. Second, part of the assumption of this thread is that gravity would be an inverse-cube law rather than an inverse-square law.

[pat]

First, the laws of gravity allow elliptical orbits in three-space. Second, part of the assumption of this thread is that gravity would be an inverse-cube law rather than an inverse-square law.

Going through the derivations on this page about orbital mechanics and bumping gravity up to an inverse-cube law, I come up with the following....

Assuming that conservation of momentum holds, the radius times the linear velocity of the planet is a constant. This implies that the angular velocity of the planet around the sun is a constant.

Assuming further that conservation of energy holds, the linear velocity of the planet is constant.

A circular orbit is the only one supported then assuming the conservation laws and gravity being an inverse-cube law.

If we flop down a dimension and assume gravity is only an inverse-linear law, then my calculations indicate that conservation of momentum requires that the linear velocity of the planet be constant. And, conservation of energy requires that the radius be constant. So, again, I'm stuck with a circular orbit.

[jinydu]

Hmm. Then why do elliptical orbits work in 3D, but not in 2D or 4D?

[mghtymoop]

pats calculations are lacking in one very important part. you cannot only integrate or differentiate the force of gravity, you must also integrate or differentiate the vector quantity of velocity as in 4d velocity is 3d acceleration and in 2d velocity is 3d displacement. but i'm not going to plug the numbers because i have my own numbers to plug, but if pat would like to post the calculations that she used i'm sure i and others would be able to assess there correctness.

[Geosphere]

that she used

Pat's a girl?

[mghtymoop]

pats always been a girls name as long as i've known it, is it a guys name in the US, if so i apologise to pat but if so thats also surprising, i thought that names would be pretty similar in two english speaking countries.

[Geosphere]

Pat can be the short form of Patrick here.

[PWrong]

pats calculations are lacking in one very important part. you cannot only integrate or differentiate the force of gravity, you must also integrate or differentiate the vector quantity of velocity as in 4d velocity is 3d acceleration and in 2d velocity is 3d displacement.

I didn't see any mention of differentiation or integration in the article. I know it can be done with calculus, but I thought they tend to avoid calc when they can find a simpler formula. My physics class never uses it, anyway.

Why is 4D velocity 3D acceleration? The object will only move in a plane anyway. :?

We haven't done anything on elliptical orbits in physics, but there's about half a page on Kepler's laws in my textbook. So I'll have a go at proving that orbits are circular. I know my working out process isn't very good, I got 48% in my physics exam (in my defense, it was a really hard exam; I actually got better than the average mark). Just see if you can work out what I mean and if I'm right or not.

This is what I understood from Kepler's Laws
In 3D:
F=GMm/r^2 = 4*pi^2*R*m/T^2

i.e. GM/(4*pi^2)= R^3/T^2

This makes R^3/T^2 a constant, with the mass given.

I'll do 2D first, using the inverse linear law, and giving G in Nmkg^-2, rather than Nm^2kg^-2, to make the units cancel out properly

F=GMm/r (gravity)
F=4*pi^2*R*m/T^2 (centripetal force)

i.e. GM/(4*pi^2)= R^2/T^2

So in 4D, given mass,

R^2/T^2=constant
Therefore R/T= sqrt(GM/(4*pi^2))

Now we know that
v=2*pi*R/T
i.e. velocity is R/T multiplied by a constant

Substituting for R/T:
v=2*pi*sqrt(GM/(4*pi^2))= a constant
So in 2D, velocity is constant, meaning the orbit must be a circle.


It doesn't seem to work like that for 4D though. Again, assume the only difference to relevant formulas is the inverse cube law, and G is measured in Nm^3kg^-2.

F=GMm/r^3 (gravity)
F=4*pi^2*R*m/T^2 (centripetal force)

i.e. GM/(4*pi^2)= R^4/T^2

So R^4/T^2 is constant
Therefore R^2/T is constant

now v= 2*pi*R/T
and R*R/T=sqrt(GM/(4*pi^2))

so Rv/(2*pi)=sqrt(GM/(4*pi^2))

The pi stuff cancels out, so
(Rv)^2=GM

v^2=GM/R^2
So velocity is inversly proportional to the radius. I'm not sure what implications this has for the orbit. I know I'm really close to an answer, but it's late and I can't think properly.

[mghtymoop]

moving between dimensions needs calculous simply because if you need to increase a power to give a new dimension there is no other simpler way. it also doesn't matter how the object is moving the object itself is made up of the integral of 3d matter so any motion equations must be integrated, 3d acceleration is 4d velocity, this is why we have gravity that remains constant for a constant mass or inertia that remains constant for a constant rate of acceleration.

[PWrong]

moving between dimensions needs calculous simply because if you need to increase a power to give a new dimension there is no other simpler way. it also doesn't matter how the object is moving the object itself is made up of the integral of 3d matter so any motion equations must be integrated, 3d acceleration is 4d velocity, this is why we have gravity that remains constant for a constant mass or inertia that remains constant for a constant rate of acceleration.

3D acceleration is 4D velocity? That doesn't make sense.

Velocity is measured in ms^-1
Acceleration is measured in ms^-2
You add the power to time, not to any spatial dimension.
Acceleration is just the derivative with respect to time, not distance or velocity. I'm not sure if you're assuming time is the fourth dimension, or if you're confusing the motion equations with the extra power in the gravity formula.

I know you can use calc to go from a velocity equation to an acceleration equation, but that's nothing to do with dimensions, and it's easy to do simple physics like this without using calc. Actually, I wish I could use calc in physics, I might get a better mark on exams. But unfortunately most physics students don't know calc, so the teacher can't teach us the method.

[mghtymoop]

exactly, time IS a spatial dimension, it is just our perception of it as 3d beings that is temporal. so you must integrate 3d values to get their 4d equivalents, oh, and calculous is an important part of physics from anyones point of view, if you go on to do any physics or engineering untis at uni you will need to use it, however i fear that you will also discover that your grasp of calculous as you would currently have it is probably far below what you would need to use it on the calculations you would come across at a uni level.

[PWrong]

So you're saying I failed physics because I don't know enough calculus, despite my 83% in the calc exam, and my rank of 5th? Ok, whatever, but that doesn't matter.

It's generally accepted on this forum that time is not the fourth dimension, nor is it a spatial dimension. Even if it is the 4th dimension here, it would be the fifth dimension in tetraspace. Time is clearly different from an ordinary spacial dimension because entropy increases through time. There is no analogue of this in the 3 dimensions. The website explains this in the introduction.

Anyway, relativity is particularly irrelavent in this case. We're discussing a 4 dimensional version of Newton's gravity and Kepler's Laws, which were established long before Einstein.

[jinydu]

Back to the original question, there is one problem with circular orbits: they are unstable. That is, the slightest nudge will send a planet spiraling into the sun or escaping into space. Elliptical orbits, however, can survive small perturbations, as far as I know. Is there another possible orbit in 4D that is stable?

[mghtymoop]

you failed physics? perhaps i should have a look at the new syllabus but i know that the undergrads i have working for me reckon that with the new calculators physics is now one of the easier TEE, i don't care what you were ranked in a TEE subject really, their only there to assess your capability to undergo study at a university and nothing to do with actually learning anything.
anyway whats generally accepted is not what is generally right, if that was so how many wars do you think we would have had, i'm betting on none. if you wish to take alkalines words as the word of god you go right ahead but, with all due respect to alkalines views, i don't beleive that the science they are based upon is even approaching realistic.
and your also right that relativity plays no part in newtons laws, but it doesn't even enter into the debate because you are integrating into the fourth dimension, which is most probably the last dimension and also is in fact a spatial dimension so you must integrate the equations of motion not only over the spatial components but also the temporal components, and if the 5th dimension was time in tetraspace that would only be because the 4d beings interpreted it that way, it would still have no effect upon our interpretation of the 4th dimension as time. i still don't see why you brought up relativity in this sence, relativity is the evidence for the 4th being both temporal and spatial but it is not the fabric of what the debate concerns, and i beleive that if the integration is done properly you will infact discover that hyperelliptical orbits can and do exist.

[PWrong]

you failed physics? perhaps i should have a look at the new syllabus but i know that the undergrads i have working for me reckon that with the new calculators physics is now one of the easier TEE, i don't care what you were ranked in a TEE subject really, their only there to assess your capability to undergo study at a university and nothing to do with actually learning anything.

I should rephrase that. I haven't actually failed physics yet. I just got 48% on the first semester exam, which most of the class failed. Including tests and other work, which are included in the TEE now, I'm still passing overall. The Ranking is actually more important than the TEE score in determining who gets into uni. Perhaps you should look at the new syllabus, because most of my friends agree that physics is one of the hardest TEE subjects. Either that or physics teachers just hate kids.

and your also right that relativity plays no part in newtons laws, but it doesn't even enter into the debate because you are integrating into the fourth dimension, which is most probably the last dimension and also is in fact a spatial dimension so you must integrate the equations of motion not only over the spatial components but also the temporal components, and if the 5th dimension was time in tetraspace that would only be because the 4d beings interpreted it that way, it would still have no effect upon our interpretation of the 4th dimension as time.

The fourth dimension is the last dimension? So which is the second last, length, width or height? Or is it latitude, longitude or altitude? The names are arbitrary. The only dimension that isn't arbitrary is time, because it's the direction in which entropy increases. That's why time doesn't count as a spatial dimension. It doesn't matter how we interpret it, we're doing calculations in a universe with four space and one time dimensions, and assuming that gravity works with an inverse cube law.
Sorry to bring all that back up again.

Anyway, I've looked over what I did before, and I think it's right, although I didn't show the reasoning very well. It seems to agree with Pat's reasoning, that you can't get an elliptical orbit in 2D or 4D, but you can in 3D.

Ok, back to the original question. Elliptical orbits are stable in 3D because if you nudge the planet, it just goes into a different elliptical orbit. But perhaps if you start out in an elliptical orbit in 4D, you simply fall into a stable circular orbit.

[swirl gyro]

I don't know the maths for orbits, so I shouldn't even be talking... but perhaps it's an even-odd thing? Check out 5D orbits, see if they are more similar to 3D orbits than 2D or 4D.

[RQ]

As polyhedron Dude mentioned in the ol' archives in the 10th or 11th page of this forum, in 4D electrons are somewhat ringshaped around their atoms. Interpret this as you like about the planets.

[PWrong]

I'm not sure about the even odd thing. I'm sure my working out is right, but I don't know how to interpret it with respect to elliptical orbits. I think it would help if we Pat's calculations.

Actually, I worked something else out with the gravitational formula. Regardless of whether elliptical orbits can exist or not, there's no need to change the gravitational constant.

On Earth, you can calculate the gravitational acceleration at the surface as follows.

Force = G*Mass of Earth*Mass of object/Radius^2
F=GMm/R^2
since F=ma
a = GM/R^2

I can't remember what the values are, but it all comes to 9.8 ms^-2

Then I did it in 4D. I took a planet with the same radius as the earth, but rather than using the same mass, I multiplied the mass by the radius to get the mass of the 4D planet.

So 4Dmass = 3Dmass * radius
So with the 4D gravitational formula,
a = G*4Dmass/R^3
a = G*3Dmass*R/R^3
a = G*3Dmass/R^2

So there's no difference in the formula anyway. I don't get it at all.

[RQ]

There might be more mass in the higher dimension.

[PWrong]

You're right, there is, assuming same radius and density. A gongyl will always have a greater mass than a sphere with the same radius and the same density. But the mass of any particular planet is a constant, in that it doesn't change much throughout an experiment, so it doesn't matter as much. The point of the calculations was to show that the extra radius cancels itself out, so there's no reason to say that elliptical orbits can't exist in 4D. I could have done the same thing using only 1 for all the measurements. At least the units cancel out.

I was trying to get a planet with the same density as earth, as well as the radius. I see now that it was irrelevent, and I forgot to use the correct volume/bulk formulas anyway.

But just out of interest, I'll do them properly. It seems to me that if we ever want to discuss Emily, we should probably decide what kind of planet she lives on. We like to think of her as being similar to humans, so it wouldn't do any good to say that she lives on a giant 4D gas planet. I reckon it should be as similar to earth as possible, except with the extra dimension. Of course, Emily could live anywhere, but she is a fictional character, and we can say what we want about her.

So anyway, I'll work out the mass of a 4D planet with the same radius and the same density as the earth. Sorry I go kind of slowly, I'm thinking as I type.

It turns out that the mass is equal to:
mass of earth * radius of earth * pi * 3/8
mass of earth = 6*10^24 kg
radius of earth = 6.4*10^6 meters

So the mass of the 4D planet is 4.52*10^31
This is huge, but it's not surprising really, with the extra dimension.

This gives an acceleration at the surface of 11.5 ms^-2
(Here we have 9.8 ms^-2)
So it's a perfect place for Emily to live, with almost the same amount of gravity as earth, as well as the same density and radius.
The difference in gravity is just a factor of pi*3/8, about 1.178

The tangent velocity of a satellite at height d, differs by a factor of sqrt(pi*3/8) when calculated in 4D.

It seems whenever you calculate something to do with gravity, you can make it work in 4D by multiplying by pi*3/8. I reckon I could show that the gravitational constant, G, should be multiplied by this amount for the 4th dimension.

So, in 3D, G = 6.67 * 10^-11 Nm^2kg^-2
in 4D, G = 7.858*10^-11 Nm^3kg-2
Note that the 4D value for G has a different unit, to account for the inverse cube law.

Apart from G, and the gravitational formula, all formulae should now work the same in 4D as in 3D. Wow, I can't believe I did that. I'll keep checking and trying to prove that result, but I'd still like to hear from someone who knows physics a bit better than I do.