PDA

View Full Version : 4d volume?



tommac
2008-Sep-17, 06:49 PM
Is there a term for 4d volume?

like
1d = length
2d = area
3d = volume
4d ?

NEOWatcher
2008-Sep-17, 06:57 PM
Is there a term for 4d volume?

like
1d = length
2d = area
3d = volume
4d ?
Wouldn't a context for the 4th dimension need to be there before a meaningful name is given?

I've heard it referred to as spacetime, but that is a lay perspective. Although that does demonstrate the need for a context.

Other than that, I've only heard 4d space.
Since the ones you gave are all in the common language, then maybe we don't have one because there is no common language expression of something not experienced.

tdvance
2008-Sep-17, 06:57 PM
I've only heard it called:

"volume"
"4d volume"
"hyperdimensional volume"

myself.

Disinfo Agent
2008-Sep-17, 07:04 PM
Also "hypervolume", or "4-D hypervolume".

NEOWatcher
2008-Sep-17, 07:07 PM
But, those are all still based on the word "volume" (3d).
I don't think that's what he had in mind.

tdvance
2008-Sep-17, 07:12 PM
The Wikipedia article on "hypersphere" lists a formula for the "volume of a hypersphere" in any number of dimensions. That's consistent with the few times I've heard mathematicians speak of measuring sizes in many dimensions. I'd thus think "volume" is the generic term for any number of dimensions (defaulting to "three" unless context says otherwise), and if the number of dimensions needs to be specified, "4d volume" or something similar.

"measure" is also a generic term, but maybe too generic, as there are other kinds of measures besides the "Lesbegue Measure" that give length, volume, etc. that we're used to.

Disinfo Agent
2008-Sep-17, 07:39 PM
Right, it's the 4-dimensional Lebesgue measure. After 3, mathematicians stopped giving them distinct names.

tommac
2008-Sep-17, 09:05 PM
But, those are all still based on the word "volume" (3d).
I don't think that's what he had in mind.

It will work I guess ... basically ... I am trying to explain a vision I had about binary gravity.

there would need to be a small thing ( maybe 1 pl x 1 pl x 1pl x 1pt ) that would either exist or not exist. But I dont know how to word that space-time bit.

mugaliens
2008-Sep-17, 09:30 PM
Is there a term for 4d volume?

like
1d = length
2d = area
3d = volume
4d ?

4d = time?

also

5d = content
6d = pressure
7d = mass
8d = acidity

We could really go on, as there are many ways of describing it, but I take it you're just talking about linear measurement (length), and not such things as pressure.

DrRocket
2008-Sep-18, 02:39 AM
Is there a term for 4d volume?

like
1d = length
2d = area
3d = volume
4d ?

The general term used in mathematics is "measure" which refers to a fairly general notion of the size of a (measurable) set. In most cases the measure used is either stated explicitly or is implicitly taken to be Lebesque measure. Lebesque measure coincides with your intuitive notion of length, area or volume in cases in which your intuition applies. In geometric discussions in higher dimensions, the term "volume" is also used. If you use the term "volume" you will not be misunderstood.

thorkil2
2008-Sep-18, 03:58 AM
4d = temporal interval

tommac
2008-Sep-18, 02:02 PM
4d = time?

also

5d = content
6d = pressure
7d = mass
8d = acidity

We could really go on, as there are many ways of describing it, but I take it you're just talking about linear measurement (length), and not such things as pressure.

? content? pressure? How is pressure a dimension? Isnt pressure just a state?

tdvance
2008-Sep-18, 03:48 PM
Pressure is force per area, and force is energy over distance, so pressure is thus energy over volume. Energy is mass, so 7d fails for that reason :)

energy--not normally considered a dimension, but in the 4-momentum vector, a kind of dual of a 4-dimensional spacetime point, the coordinate that corresponds to time (in a given frame of reference) in the latter corresponds to energy in the former. Since time is a single dimension, it's measured by length. So, pressure is thus:

dual of length divided by volume.

This is partly tongue-in-cheek, but a real analysis of the situation along these lines might not be too far from reality.

mugaliens
2008-Sep-18, 07:17 PM
? content? pressure? How is pressure a dimension? Isnt pressure just a state?

dimension: pounds per square inch

state: yes, we have pressure vs no, we do not

astromark
2008-Sep-18, 07:20 PM
4d = volume... I do not except that.

In my 3 dimensional world. Volume is the sum of area by depth... length, breadth, depth. = 3 dimensions.

It can be sensibly argued that time could be the 4th dimension.

How many times do we have this conversation?

mugaliens
2008-Sep-18, 07:26 PM
Pressure is force per area, and force is energy over distance, so pressure is thus energy over volume.

Yes, pressure = force / area

But...

:wall:

No, force is not energy over a distance. Rather, energy is force over a distance:

Work = Energy = Heat = kW-hr = force x distance (J, or N-m)

This is force applied to something as it travels a distance, not force per unit area.

Watt = work done per unit of time (J/s, or V-A)

tommac
2008-Sep-19, 12:19 AM
4d = volume... I do not except that.

In my 3 dimensional world. Volume is the sum of area by depth... length, breadth, depth. = 3 dimensions.

It can be sensibly argued that time could be the 4th dimension.

How many times do we have this conversation?

Yes so say I had a wanted to calculate a volume of space over a period of time. How would I refer to that?

What I am really looking to name is a cubic planck length over a planck time. To create a 4d bit of space-time.

undidly
2008-Sep-19, 02:25 AM
A point has no dimension.P =0.
A row of points is a line and has the dimension called length.L = (a number of units) in LINEAR units.
A row (side by side) of lines is an area.A = (a number * another number)of SQUARE units.
A row or pile of areas is a volume.V = (a number * another number * another number) of CUBIC units.
A row , pile or assembly of volumes (into a single object)is a quolume (Q from quad).
Q = (a number * another number * another number * another number) of
QUADRIC units.

Objects or even empty space in 4 physical dimensions may not exist in the real universe but the math can be useful.

tdvance
2008-Sep-19, 02:58 AM
Yes, pressure = force / area

But...

:wall:

No, force is not energy over a distance. Rather, energy is force over a distance:

Work = Energy = Heat = kW-hr = force x distance (J, or N-m)


If energy is force x distance, as you just said, then force is energy over distance.

thorkil2
2008-Sep-19, 07:03 AM
Yes so say I had a wanted to calculate a volume of space over a period of time. How would I refer to that?

What I am really looking to name is a cubic planck length over a planck time. To create a 4d bit of space-time.

See my second post under Motion, Duration, and Time in ATM for a speculative thought about this (I've also just posted a refinement of the first post). I suspect that the temporal width of the Universe is equal to a single quantum of time (if such exists) or 0 if there is no quantum T. But no more here, as that subject probably belongs exclusively in ATM.

astromark
2008-Sep-19, 10:43 AM
No, I am not rushing of to ATM. I see what 'Tomac' is asking as mainstream answers to a question. How we perceive this is fundamental to understanding space time and the relationship of them to us. Them, space and time as a dimension. No, sorry I see that as a result of movement in either. Space or time. There is nothing that does not take some time... and the movement is relative only. I do not see an answer here.

NEOWatcher
2008-Sep-19, 12:24 PM
Yes so say I had a wanted to calculate a volume of space over a period of time. How would I refer to that?

What I am really looking to name is a cubic planck length over a planck time. To create a 4d bit of space-time.
Well; cubic planck length already sounds a bit off, but I know what you mean. A cube with sides of a planck length.

That requires a planck length defined in each of the 3 dimensions.

With that in mind, can't you just use planck length for the 4th dimension, and use a transformation that defines it in terms of (planck) time?

I think without the proper application and/or formulas, the variables cannot be defined.

John Mendenhall
2008-Sep-19, 04:58 PM
4d = volume... I do not except that.

In my 3 dimensional world. Volume is the sum of area by depth... length, breadth, depth. = 3 dimensions.

It can be sensibly argued that time could be the 4th dimension.

How many times do we have this conversation?

Groan. Wish for a 4th dimension in one hand, and . . .

OP, start with the Wiki article on 'Dimension'. The most important point is that there is that there is zero, zilch, none, nada evidence of physical dimensions beyond the three we know. Time is considered to be a fourth, and unidirectional, dimension in GR and SR.

Hyper-dimensional spaces are fascinating mathematical curiosities, and so far that's it. Maybe, maybe the string theorists will get 10, or 11, or 21d spaces to work, but nothing testable has shown up yet.

tommac
2008-Sep-19, 07:15 PM
Groan. Wish for a 4th dimension in one hand, and . . .

OP, start with the Wiki article on 'Dimension'. The most important point is that there is that there is zero, zilch, none, nada evidence of physical dimensions beyond the three we know. Time is considered to be a fourth, and unidirectional, dimension in GR and SR.


What particuar part of the article? You just stated that there are 4 dimensions. Thus there must be a 4d volume right? I guess it would be a 3 dimensional volume through two points in time.

NEOWatcher
2008-Sep-19, 07:23 PM
What particuar part of the article? You just stated that there are 4 dimensions. Thus there must be a 4d volume right? I guess it would be a 3 dimensional volume through two points in time.
Dimensions encompass more than spacial concepts.
I can call a 4th dimension "color" if I want, it doesn't have to even be a physical concept, only that there is a common set of variability to the other dimensions.

For example, a sheet of paper is 3 dimensional it has width, length and color.

tdvance
2008-Sep-19, 07:31 PM
Quantum mechanics partial differential equations are written in terms of "phase spaces" which have an enormous number of dimensions. The "volume" in this case would be a measure of the size of the space of possibilities (not a space in the 3d-sense we're used to--and it's not very Euclidean either).

Newtonian mechanics can be written that way too (Lagrange's or Hamilton's equations). In fact, that was discovered first, and the quantum mechanical equations made from that.

loglo
2008-Sep-19, 08:12 PM
What's wrong with the regular old spacetime interval?

tommac
2008-Sep-19, 08:17 PM
Dimensions encompass more than spacial concepts.
I can call a 4th dimension "color" if I want, it doesn't have to even be a physical concept, only that there is a common set of variability to the other dimensions.

For example, a sheet of paper is 3 dimensional it has width, length and color.

Ok then how do I define a 3d object with color as the 4th dimension. What would I call that?

tommac
2008-Sep-19, 08:20 PM
What's wrong with the regular old spacetime interval?

Space-time interval? Wouldnt that only be a 1d vector?

Space * time + a direction?

tdvance
2008-Sep-20, 12:01 AM
I guess space-time interval could be defined as a vector, but I've mainly seen it used as a scalar (i.e. as "proper time"--the arc length between endpoints of the geodesic, a geodesic strangely being the "longest" distance between two points in the funny General Relativity geometry since a clock runs at its fastest in a free-fall frame of reference, i.e. one following a geodesic).

mugaliens
2008-Sep-20, 09:28 AM
If energy is force x distance, as you just said, then force is energy over distance.

Provided your use of the term "over" is synonomous with "divided by," yes.

Typically, work is spoken of as a force applied over a distance, and that use of the term over does not imply division, but rather, a force applied to something between one point and another.

Semantics.

sirius0
2008-Sep-20, 01:16 PM
Well, on the one hand I work with pumps. On the other I have been tortuously self studying GR.

In pumps we have volumetric flow rate. Which in my preffered units is (m^3)/hr. This probably is not what you want but it is a time dependant volume. If it takes a long time to get a 3d volume through then I have a low rate. If it is quick then I have a high rate. Most Newtonian style physics has time as a 1/t or (t^-1) and GR has the time-like (ct) on the opposite sign to the space-like dimensions (-,+,+,+) or (+,-,-,-). The two concepts aren't that different in the sense that the amount of time elapsed is inversely proportional to the amount of "volume" achieved as demonstrated above with the pump. But also in space-time as the axiom there is that a 'straight line is that which takes the longest time between two points'. So the longer an object exists from the moment of it's manufacture to it's destruction the smaller it's space-time volume. So indeed as thorkil2 suggests a plank length of time might be all time.

tommac
2008-Sep-21, 01:34 AM
Well, on the one hand I work with pumps. On the other I have been tortuously self studying GR.

In pumps we have volumetric flow rate. Which in my preffered units is (m^3)/hr. This probably is not what you want but it is a time dependant volume. If it takes a long time to get a 3d volume through then I have a low rate. If it is quick then I have a high rate. Most Newtonian style physics has time as a 1/t or (t^-1) and GR has the time-like (ct) on the opposite sign to the space-like dimensions (-,+,+,+) or (+,-,-,-). The two concepts aren't that different in the sense that the amount of time elapsed is inversely proportional to the amount of "volume" achieved as demonstrated above with the pump. But also in space-time as the axiom there is that a 'straight line is that which takes the longest time between two points'. So the longer an object exists from the moment of it's manufacture to it's destruction the smaller it's space-time volume. So indeed as thorkil2 suggests a plank length of time might be all time.

I dont think it exactly flow rate. Flow rate is really 3d I think.

Basically the area of a circle over time. I want 4d.

undidly
2008-Sep-21, 04:43 AM
I dont think it exactly flow rate. Flow rate is really 3d I think.

Basically the area of a circle over time. I want 4d.

You want pumps?.

A water molecule is about 10^-9 long and does not weigh much.

A 1 cm long line of water (single row)contains 10^9 molecules and does not weigh much.10^9 long.

A 1 cm square of water (single layer)contains 10^18 molecules and does not weigh much.10^9 long,10^9 wide.

A 1cm cube of water contains 10^27 molecules and weighs 1gm.10^9 long,10^9 wide,10^9 high.

A 1cm hypercube of water contains 10^36 molecules and weighs 10^9 as much as the 3D cube,10^9 long,10^9 wide,10^9 high,10^9 in the 4D direction.

The weight would be 10^9 gm.,or 10^6 kgm ,or 10^3 tonne.
Since I000 liters of water weighs 1 tonne we deduce that a 1cm hypercube of water contains 1,000,000 litres.

Pump that.

mugaliens
2008-Sep-21, 04:45 PM
What version of "hyper"cube are you referring to? Tesseract? Penteract? Hexeract?

A 10-cube contains 15,360 times the volume of a cube.

NEOWatcher
2008-Sep-22, 12:07 PM
Ok then how do I define a 3d object with color as the 4th dimension. What would I call that?
Nothing... Or a definition*.
What I'm trying to demonstrate is that the word "dimension" does not infer spacial concepts. It demonstrates independent variable states. There's no implied relationship between these states.

*Mine would be defined as sheet-color.

tommac
2008-Sep-22, 02:48 PM
Nothing... Or a definition*.
What I'm trying to demonstrate is that the word "dimension" does not infer spacial concepts. It demonstrates independent variable states. There's no implied relationship between these states.

*Mine would be defined as sheet-color.


Again, what i am trying to define is a 4d volume which I would like to define as a space-time bit. the 4ds are 3x space + 1x time ... It does seem like a hypercube model would work ( I think ) but really the details are less important ... basically I am looking for a term to define a smallest piece of space-time that can exist. ( This is to be used in my ATM that these space-time bits can be turned on or off by energy )

NEOWatcher
2008-Sep-22, 05:29 PM
It does seem like a hypercube model would work ( I think ) but really the details are less important ...
Until you start working on formula translations. Then the difference between a time unit and one of the 3 spacial units makes the interpretation important.

basically I am looking for a term to define a smallest piece of space-time that can exist.
What's the smallest piece of 3D space? Its the same kind of extension.

Just like a 2d square space can be seperated into 2d squares, and a 3d cubic space can be seperated into 3d cubes, then I would assume a hypercubed space can be seperated into hypercubes.

tdvance
2008-Sep-22, 05:37 PM
"What's the smallest piece of 3D space? Its the same kind of extension."

that would be plank volume. I'm not sure if its "shape" is well-defined though. Time can be quantized as well--and a plank unit of time is the same "length" as a plank length in space, with the usual convention of using the speed of light to map space units to time units. So, it's reasonable to have a "plank 4-volume" for which, if any two events happen within the same plank 4-volume, it is not well defined which happened first or which is closer to the observer.

tommac
2008-Sep-22, 08:02 PM
for my math I would like to replace space units with # of bits and time units with number of bits .... where 1 plank length = 1 plank time.

so have 4d spacetime units ( or 1 space time unit that spands a 3d volume and a 1d time )




Until you start working on formula translations. Then the difference between a time unit and one of the 3 spacial units makes the interpretation important.

What's the smallest piece of 3D space? Its the same kind of extension.

Just like a 2d square space can be seperated into 2d squares, and a 3d cubic space can be seperated into 3d cubes, then I would assume a hypercubed space can be seperated into hypercubes.

tommac
2008-Sep-22, 08:05 PM
I would think that if it happened in the same plank-4 volume then they would be the same event.


"What's the smallest piece of 3D space? Its the same kind of extension."

that would be plank volume. I'm not sure if its "shape" is well-defined though. Time can be quantized as well--and a plank unit of time is the same "length" as a plank length in space, with the usual convention of using the speed of light to map space units to time units. So, it's reasonable to have a "plank 4-volume" for which, if any two events happen within the same plank 4-volume, it is not well defined which happened first or which is closer to the observer.

NEOWatcher
2008-Sep-22, 08:07 PM
for my math I would like to replace space units with # of bits and time units with number of bits .... where 1 plank length = 1 plank time.
so have 4d spacetime units ( or 1 space time unit that spands a 3d volume and a 1d time )
Well, it appears to me that tdvance has more insight into that part of it than I have. But; if "a plank" is sufficient, then why not?

tommac
2008-Sep-22, 08:15 PM
Well, it appears to me that tdvance has more insight into that part of it than I have. But; if "a plank" is sufficient, then why not?

It may or may not be. We may find that we could interpolate the bit size based on scientific evidence. I am using a plank because it is pretty small.

Has anyone proposed the size of a gravitron?

mugaliens
2008-Sep-22, 09:58 PM
for my math I would like to replace space units with # of bits and time units with number of bits .... where 1 plank length = 1 plank time.

so have 4d spacetime units ( or 1 space time unit that spands a 3d volume and a 1d time )

Planck length: 1.616252 x 10-35 meters

Planck time: 5.39124 x 10-44 seconds

c: 299,792,458 meters/second

Planck length is the distance light travels in a vacuum during one interval of Planck time.

ie, PL = c * PT

sirius0
2008-Sep-22, 11:50 PM
It may or may not be. We may find that we could interpolate the bit size based on scientific evidence. I am using a plank because it is pretty small.

Has anyone proposed the size of a gravitron?
I propose one of about 5m radius. That's about the size of the last gravitron I had a ride in.:)

Frozen thought is a flaw in some of the logic in this thread. this is where a preset of 3d in the mind actually tricks us into considering a stationary state in a constantly changing one, i.e. the steady state doesn't very often exist, it is meant to be a mental convenience but can have us multiplying what has already been divided by time; in the classic physics sense.

If indeed we live in a 4d universe then volumes we already calculate are already 4d This weirds me out because I only use 3 axis, but again I am only thinking in terms of a frozen moment of that volumes existence. Remember that the time-like dimension is (relative to the space-like) negative r= -(ct)+(x1)+(x2)+(x3) so the smallest delta(ct) will have the largest time elapse. Yuk I don't like it!

dodecahedron
2008-Sep-23, 12:36 AM
Is there a term for 4d volume?

like
1d = length
2d = area
3d = volume
4d ?

Bulk. Cite: http://tetraspace.alkaline.org/glossary.htm
Also a search of fourth dimension volume bulk and 4d bulk supports the use of 'bulk' as the fourspace analogue of volume. Plus I remember this being mentioned back when The Discovery Channel and The Learning Channel were actually good.

mugaliens
2008-Sep-23, 04:30 PM
If indeed we live in a 4d universe then volumes we already calculate are already 4d This weirds me out because I only use 3 axis, but again I am only thinking in terms of a frozen moment of that volumes existence. Remember that the time-like dimension is (relative to the space-like) negative r= -(ct)+(x1)+(x2)+(x3) so the smallest delta(ct) will have the largest time elapse. Yuk I don't like it!

Think of it in terms of triple integrals, integrating curves from a to b for each of x, y, and z.

That gives us volume where time=t.

To give us volume over time, simply apply a fourth integral, such that you're integrating curves from a to b for each of x, y, z, and t.

What might we measure over time? How the mean temperature contained within the volume changes over time; proportions of elements engaged in radioactive decay; the pressure within the volume, and how it rises as the volume shrinks over time.

That looks like 4 dimensions to me...

astromark
2008-Sep-23, 07:13 PM
As every thing we do envolves a period of time is a fact. As minute and difficult to define as a planked length or billions of years. Time would seem to be the measure of a interval in the progression of this universe. Separate from the much more obvious definition of the three dimensional space we move through. Until you can offer proof of some thing different, and you haven't yet... I like this. and quote; Mugaliens...

"To give us volume over time, simply apply a fourth integral, such that you're integrating curves from a to b for each of x, y, z, and t.

What might we measure over time? How the mean temperature contained within the volume changes over time; proportions of elements engaged in radioactive decay; the pressure within the volume, and how it rises as the volume shrinks over time.

That looks like 4 dimensions to me..."

and me.