General Question

Pazza's avatar

How many dimensions does space have?

Asked by Pazza (3114 points ) September 1st, 2012

I’ve been a CADist for many years now and have recently been teaching myself 3d CAD, anyway, to calculate volume you need 3 dimensions X,Y,Z or length, width, height etc. This really only works for what you might call standard geometric solids, once you start adding curves (never mind compound curves) all hell brakes loose, and an approximate volume based on Pi is used.

So it occurred to me that to find the exact volume of a sphere, or portion thereof, you need to add more and more dimensions/lines of dissection to gain a more accurate volume, until you get to the point where you would need an infinite number of dissection lines to gain a perfectly accurate volume.

So the question is, why do we say space is 3 dimensional, and not infinitely dimensional?

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26 Answers

ragingloli's avatar

Because space only has 3 dimensions. The number of crossections you use for your approximation is a part of the third dimension.

Pazza's avatar

@ragingloli – ???
Don’t know if I understand that?
Is that because according to mainstream science a two dimensional object doesn’t exist therefore a third dimension is required to calculate the volume because volume only comes into existence in said third dimension?

Lightlyseared's avatar

I’m not entirely sure I understand what you are asking. The volume of any 3D shape no matter how many weird curvy sides it has, even a sphere, can be desribed and calculated easily using x,y,z planes only. Thats why they are called 3 dimensional. You could add a forth dimension, time, to describe how the shape changes over time but I that would probably complicate this all too far…

ragingloli's avatar

@Pazza
The problem with the infinite crosssections is not that you need more dimensions, but that the dimension on which you line up the crosssections has infinite subdivisions.

gasman's avatar

As I understand the question, you’re trying to approximate the volume of a curved solid using rectilinear slices. Is that what you mean by “dissection lines?” This method was used successfully by Archimedes in ancient times, and is the basis for methods of calculus based on integration, which considers slices that grow thinner and more numerous until the curved figure is exactly attained as a limit.

But it doesn’t require more dimensions than the standard 3 of solid geometry.

wonderingwhy's avatar

The infinite slices are still bounded in 3 dimensions (though they don’t have to be). You’re simply slicing that 3d space up n times. This might help.

Something about your question reminds me of the Zeno Paradox or what’s the smallest distance between a and b when a ≠ b.

Pazza's avatar

(Sorry. I had to go shopping)
I think what I’m asking, is, why is what we call 3 dimensional space defined as having 3 dimensions?
Why do dimensions have to propagate at 90 degrees to each other, and not 89 or 72 or 36, or any other direction?

Also, I was kind of amazed the other day that I suddenly realised that Pi is infinitely recurring.
Amazed in the same way as realising (at about 35!) that Stevie Wonder was blind and played the piano. But then I thought, curves are sort of infinitely variable along their length, and the Pi ratio is a ratio between a straight line and a curve. So mentally I can settle on that, although mentally, curves now fascinate me immensely.

Or, if for arguments sake you have a bubble. This bubble starts off with no volume and zero surface area. Now you start to stretch the bubble in a straight line thus creating 1 dimension, now reverse what you just did, and stretch the bubble out in a different direction creating 1 dimension again. In each case you created 1 dimension. Now again reverse what you just did. Now stretch the bubble out in those 2 directions at the same time, why is that now described as having 2 dimensions? Surely both those dimensions are the same? why does it matter what direction you travel? In my mind, I can’t separate the 2, they are either the same dimension, or they are infinite dimensions?

Now although I conceptualised stretching the bubble in only 1 direction, I actually can’t see how you could do this in actuality, I can only see the bubble stretching out in all directions simultaneously, the same way electromagnetic waves propagate from a source.

hiphiphopflipflapflop's avatar

As a CAD person, you “live” in the space that mathematicians call R^3, E^3 , or ‘3-dimensional Euclidean space’. Every point in that space has a unique identifier which is a real number triplet (x, y, z). But this identifier is valid only with respect to the origin and axes you pick in the beginning. The origin (0,0,0) can be anywhere. Normally you would choose the axes orthogonal so as to make them fully independent, but if you wanted to, you could skew them around so long as any two pair weren’t made co-linear with each other (basically, you turn it into a linear algebra problem in going back and forth between an orthogonal basis and a non-orthogonal basis).

Once you pick the origin, the length and the orientation of the three basis vectors, the identifiers for all points in the space (relative to that system) are now fixed. We can now talk about objects embedded in this space, and this is an important distinction which I’m not sure you are aware of (kind of like the imaginary fish that doesn’t realize the importance of the water in which it is swimming because it is always there).

dabbler's avatar

@hiphiphopflipflapflop basically points to the fundamental misunderstanding behind all those other dimensions you’re asking about. “Orthogonal” is the key word.

Any object in physical space can be described fully using three orthogonal dimensions. Orthogonal dimensions are completely independent of each other.
All the multiple dimensions you’re imagining can be reduced to one or more of three orthogonal dimensions.

LostInParadise's avatar

Just to make things interesting, do you know that mathematicians work with what they call fractal dimensions? The Sierpinski gasket is a good example.

Basically, it works like this. First consider a square of length 1, which has area 1. If we double the length, the area is 2^2. One way of defining dimension is ratio of log of area (for objects in the plane) divided by log of scaling factor.

For the square, that gives log (2^2)/log(2) = 2.

For the Sierpnski gasket, doubling the length gives 3 copies, so the area increases by a factor of 3. The dimension is log(3)/log(2) = about 1.58.

dabbler's avatar

P.S. The original question describes the process of integration in calculus. (“find the exact volume of a sphere”)
Those increasingly infinitesimal slices are all in the direction of the same dimension, they are not separate dimensions.
Grab some intro to calculus, sounds like you’re ready for some fun with that. Some of the most elegant mathematics you’ll ever run across.

Pazza's avatar

@dabbler – (I just had to google orthogonal! lol), I mentioned CAD because I’ve used it for many years, and it caused me to form an opinion of what space is from the point of view that, the whole on screen CAD environment is imaginary, yet as far as the user (observer) is concerned it has volume, and very specific laws, and tools to manifest objects, and its base measurement (metric version) is 1mm dissected up to the 16th decimal place. This 16th decimal place could be described as the plank length, now as far as the computer and user are concerned, there is an actual distance between 2 points at this smallest CAD scale, yet in reality, there is no space between these 2 points, as the computer would have to go to the 17th decimal place to measure a distance between them.

But expanding further, none of the points have any space between them other than that which the computer generates with the graphics card, so devoid of objects the CAD environment becomes a sort of (I’m going to use the term but I don’t like it!) 3D grid of infinitely small node points with no space between them and ultimately no volume?

This is how, rightly, or wrongly, I have come to view the cosmos.

Also, my current understanding is that, existence is defined as having 3 dimensions, that is to say that something with only 1 or 2 dimensions cannot exist, if that is the case, how did the universe arise from a singularity?

I have read somewhere that the singularity still had a plank length, or was packed into a diameter with the plank length, but it is my understanding, that to measure any object, you would need something to measure it against, since size is relative, and since nothing else existed at that time, how could the singularity have volume, if it had no volume, it couldn’t be described as being 3 dimensional and therefore could not exist?

There are people far cleverer than I here, and I’m not sure that even made any sense so please be gentle…...

@LostInParadise – no offense, but the wind from that just messed up my hair as it flew over my head. sorry.

Pazza's avatar

@dabbler – Yes my original question revolved around finding the volume of a sphere or a portion thereof, a sphere manifest in space has volume, it makes no apology for the volume it has, yet we cannot measure it precisely, that perplexes me immensely.

Yet if we have 2 spheres, the first with volume 1n, and the second with volume twice that of the first, we can now measure the volume of the second precisely using the first as our base measurement, but if we now said that, the first sphere had a volume that of a fundamental particle with a diameter of 1 plank, then how do we assign volume to the first sphere if we have nothing to measure it by other than another fundamental particle, to my mind, this particle with diameter of plank length has no volume, therefore, anything else made from a collection of these particles will also have no volume, but strangely, we’ll still be able to measure it? ouch…...

hiphiphopflipflapflop's avatar

OK, some distinctions to make… there’s space as far as your CAD program is concerned, spaces as mathematicians think of them, and ‘real’ space as an object of physical inquiry.

In regards to the latter, we have no real conception of what spacetime is really like close to the scale of the Planck length because we lack the ability to probe it directly. There are a number of things like the isotropy of light propagation that tell us that it can’t be as simple as a 4D cubic lattice of points spaced a Planck’s length (and Planck’s time) apart.

In R^3, there can always be found points between any two given non-coincident points. (Indeed, there are always infinitely many… and this is a higher-order infinity than the infinity of the integers.) This isn’t the case in your CAD program. This dodges the problem of having to specify points with infinite accuracy. Point sets in continuous spaces like R, R^2 and R^3 can have really odd properties like @LostInParadise brings up.

A sphere in R^3 has the exact, closed-form solution to the volume as (4/3)*pi*r^3.

Pazza's avatar

@hiphiphopflipflapflop
“In R^3, there can always be found points between any two given non-coincident points. (Indeed, there are always infinitely many”

Is that describing space (cosmos space) as being fractal in nature?

dabbler's avatar

“yet we cannot measure it precisely” ... maybe not in the CAD program.
But yes we can calculate lots of volumes precisely with calculus methods.
The formulae that we have for volumes of spheres and other shapes are the same as the results of a calculus/limits/integration approach to finding the volume given the fundamental characteristics of those shapes.

Pazza's avatar

@dabbler – So are you saying in real world we can measure spheres precisely without Pi?

hiphiphopflipflapflop's avatar

@Pazza this “always infinitely more points” is what makes mathematical spaces ‘continuous’. When someone speaks of a fractal, I think of it describing an object embedded in a space rather than it referring to the space (getting back to the first distinction I was trying to make in my first post).

There are different theories for what Planck-scale real space is like. We don’t have enough empirical data to tell which ones (or all of them!) are wrong right now.

Whenever someone talks about a real object being a fractal, it is generally to be assumed that this only fits over a certain range of scale, with the fractal nature breaking down at some lower limit. Famous “real world fractals” are mountains and coastlines. They cease to follow the mathematical properties of “true fractals” once you get down to the atomic scale.

Pazza's avatar

@hiphiphopflipflapflop – It is my understanding that there are no objects in space, only objects manifest from space, and that size is relative to the point of observation, as in CAD world if there were infinite decimal places and not restricted to 16, the observer would be able to zoom in or out ad-infinitum as though observing an object (or space) fractal in nature.

Question, is there a difference between a mathematical space and a real-world space?

dabbler's avatar

Why can’t we use Pi? Did I miss something?
In any case, in the real world, there is Archimedes old method of displacement. Fill a tub full to overflowing. Set a container to catch any further overflow and submerge your sphere or whatever shape you want into the tub. The volume of overflowed water equals that of the submerged object.

Pazza's avatar

@dabbler
“But yes we can calculate lots of volumes precisely with calculus methods.”
Sorry being a layman, and trying to say I didn’t really understand what you had posted.

The way I read it was that you can calculate a spheres volume using calculus.
My lack of understanding of mathematics as opposed to you missing something. I can conceptualise things in my head, but you give me some numbers to crunch and its lights out in my noggin.

As for the dunk a sphere in a bucket measuring method, I get that, however, when you break that down your still measuring newtonian billiard ball quantities, and I suppose from that perspective you can measure volumes precisely, but to measure the distance around a curve using Pi?

I suppose, my problem really is that I have a conceptually perfect curve, circle, sphere in my head, and I’m trying to get a handle on how whatever measurement you come up with for the distance around it, circumference of it, or volume of it, will have a figure which is infinitely recurring, and it makes my brain hurt.

But then I suppose there’s probably no such thing in the real world as a conceptually perfect curve, circle, or sphere since they would all be made from a bundle of fundamental particles. But then that still leaves me pondering on what size a fundamental particle is?

(my apologies for ducking out, but I’m up early in the morow, and its 12.30 in UK land many thanks for everyones efforts in trying to get a non-mathematical brain to understand how mathematics can describe the universe.)

dabbler's avatar

@Pazza, I think you would enjoy calculus. Don’t worry about the numbers, try to grock the qualitative concepts.
Keep in mind that while mathematics deals with “ideal” concepts (perfect curve, circle, or sphere) that are not as such found in our everyday reality, mathematical solutions are very powerful. When you get around to working out the difference between the ideal solution and reality, congratulations, you’re an engineer!

Also, You have a lot of curiousity about the world and I applaud that. I’ll suggest that you try to keep your science, what you really can show and prove, distinct from philosophical questions that science has no grip on at this time, like what’s the size of a fundamental particle.
Both the science and the philosophical conundrums are worth considering and discussing but it’s worth using different tools and liberties on each of those categories of thinking.

—-ref your earlier post:
“existence is defined as having 3 dimensions, that is to say that something with only 1 or 2 dimensions cannot exist, if that is the case, how did the universe arise from a singularity?”
Let’s take that apart a bit at a time, I think you are laboring under a boatload of misconceptions…

“existence is defined as having 3 dimensions” Not so, “space” is defined as having three dimensions. Physical existence (never mind what can happen on the astral plane for a moment) is defined by the best theoretical thinkers these days as having either ten or eleven dimensions. Space is three of those, time is another. And there are other collapsed dimensions necessary to explain some partical physics and quantum interactions.

“that is to say that something with only 1 or 2 dimensions cannot exist” I don’t think that is a fair conclusion at all. Depends on what you mean by “something”. A line has one dimension. A plane has two.

“How did the universe arise from a singularity?” Nothing about our descriptions of space says anything about whether the universe arose from a singularity. That section of science is pretty theoretical and certainly worth exploring for whatever answers are out there than can fit into our human awareness.

Pazza's avatar

@hiphiphopflipflapflop
” Famous “real world fractals” are mountains and coastlines. They cease to follow the mathematical properties of “true fractals” once you get down to the atomic scale.”

Is that because once you get down to the fundamental particle, you can’t break the object down any further?

In any case, from a ‘real object’ point of view, it seem to me that mountains and coastlines don’t really exist, since its just the particles grouped together, in the same way a tornado doesn’t really exist, its just the movement of the atoms moving through themselves which manifest into a mountain, coastline or tornado.

This posed the question in my head a few years back, that if solid objects are manifest from atoms, atoms, manifest from fundamental particles, then, what are fundamental particles manifest from? Which led me to the conclusion that they are manifest from the void, or as I’ve heard it recently called, the source field.

It would also seem to me that, from our point of reference, the boundary zoom scales are at the planck length, and the edge of the cosmos. It would be interesting to know, if on zooming in down to the planck length would be the same relative scale as zooming out to the edge of the cosmos? Maybe those boundaries are the edges of other universes? That is to say, the edge of the cosmos would be a planck length in the universe above, and the planck length in this universe is the boundary to the universe below, an on and on ad-infinitum, hence the ancient term “as above, so below”

Tht would also make mind the centre of the universe.

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dabbler's avatar

Also note there is some difference between measuring the volume of something (practical, real-world, get out the tape measure), and calculating the volume (given a model/ideal description, theoretically determine its volume, use Pi if you need it). Intention and methods distinguish the two.

That may seem semantic and pedantic but there is a world of practical difference.

SaveTheRhinos's avatar

4, X,Y,Z and Time, just like earth.

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