About how many dimensions there really are

Cecil's Columns/Staff Reports
1

http://www.straightdope.com/mailbag/mdimensions.html

I tried to do a search for this topic but drew a blank, so here it goes:

I felt that the reply was a little unsatisfying. OK so we use 3 dimensions, no surprises there, but are there, or could there be, more dimensions?

I often toy with the idea by comparing the difference between the 3 dimensions. For instance, we 3 dimensional guys can get an overview of any 2 dimensional field if viewed far enough away, so would it be possible for a 4th dimension guy to see everything on earth, from it’s core up to every building, cave, creature etc. just by looking at it from a distance?

Could a 4th dimension guy pick up a book and read every word of it without even having to open it? We read in 2 dimensions (and a 2 dimensional guy would have to read in 1 dimension), so wouldn’t they read in 3?

Would a 4 dimensional object cast a 3 dimensional shadow?

Anyway, I think it’s an interesting idea. If anybody knows any websites or books dealing with this directly, I’m all ears.

2

The answer to all your questions is “Yes”.

Not everything is symmetrical, though. For example…

There are an infinite number of regular polygons in 2-space, but only five regular polyhedrons in 3-space, and only two regular poly-whatsits in all higher spaces.

In 4-space, anything analogous to Newton’s law of gravity wouldn’t produce stable orbits.

Only very simple networks can exist in 2-space, but infinite complexity is possible in 3-space.

3

As far as how many dimensions there are, most people agree that they experience three spatial dimensions (with relative freedom of movement within those dimensions) and one dimension of time (which only allows ‘forward’ movement). Theories of relativity claim that the dimension of time is bound to the three dimensions of space such that they can be thought of as a four dimensional space-time construct. Experiments have shown that these theories work out to be true (at least in the macro world, they don’t fully explain workings at the quantum level).

Which brings us to the quantum level where theories such as string theory or supersymmetry claim that quantum level happenings can be explained if we posit that there are more spatial dimensions (6, 11, and 25 of them even!). However, no experiments have shown that these dimensions do in fact exist. The theorists say that perhaps that the reason that we haven’t directly experienced these extra dimensions is because they have wrapped themselves up into themselves into tiny dimensional ‘bubbles’ too small for us to detect, but still large enough to have a quantum effect.

Peace.

The fifth dimension is the age of aquarius.

4

Depends on your definition of “regular”, and it also depends on the curvature of the space. In a positively-curved 4-dimensional space, for instance, you can have a regular hypersolid with each “face” being a pentagonal dodecahedron. But you can always have analogues to the tetrahedron and the cube.

And moriah, Special Relativity, which links space to time, works just fine in quantum mechanics. It’s not until you get to General Relativity, which also brings in mass and gravity, that you have problems. And we’ve never observed anything which can be explained by string theory but which can’t be explained by the theories we do have. Where string theory comes in is there are some hypothetical situations where our current theories can’t make any predictions at all, much less correct ones, while string theory can make predictions there which we don’t know if they’re correct.

5

Look, the whole notion of “dimension” is a construct and model. We can use three dimensions conveniently to model a lot of the world around us, but it’s still just a mathematical model. There’s no “reality” to three dimensions.

We are NOT “three-dimensional” creatures, except if that’s the way your mind wants to envision us. To a painter or sculptor, for instance, we are creatures of color, space, shape, and textures. To a poet, we are creatures of mind and emotion. We like the number three, it’s got lots of nice rhythmic wossname to it, so we think in terms of threes*, but it’s a purely artificial construct.

  • Example: we think of three primary colors; we put three lights on a traffic light; we think of time in terms of past, present, future; we think of things as animal, vegetable, or mineral; and on and on.
6

It seems to me that some particular mathematical models are so correlative to reality that it might as well be reality (I know, I know, Kantian epistemological categoricals apply).

E.g., if I have eight marbles, then ‘8’ is my mathematical construct to denote the quantity. If I add four, the equation ‘8+4=12’ has such an invariable similitude to reality, that the mathematical construct seems to be a valid expression of that reality.

And so, when we deal with geometrical perpendiculars and can only find three in our realm of experience (x,y,z, Cartesianally) which can then become a model of spatial reality as we experience it with such an invariably ‘correctness’ in modeling and predicting reality, well then, I say that the three Cartesian spatial dimensions is reality and not a construct.

Peace.

Let me prove that your ‘image of me’ is a true reality and not a construct of your mind… here, place your finger in my mouth between my teeth…

7

This is classic apples and oranges stuff, Dex. No matter how metaphorical you want to get, spatial dimensions are a different class of fruit than colors and emotions.

As JWK already indicated, we must live in three spatial dimensions because no other number will explain the physicality of our world. There is absolutely nothing artificial about it. In fact, all you are doing is saying that we can construct artificial categories to classify how we see the world. Your own argument negates itself.

8

In a recent issue of Science (Oct 10, 2003), there was an article reviewing an upcoming paper to appear in a physics journal (Phys. Rev. Letters, I think, but I don’t have the article with me now) where surveys of the cosmic background radiation were analyzed and it was concluded that the universe fits something similar to this dodecahedral model that Chronos mentions. As I understood it (and cosmology is not my strong suit) in this model, opposite faces of this dodecahedron are similar and linked so that an object travelling out one face would enter the opposite face. OK cool, I thought, but this is just a mathematical model, right? Well, later in the article, it was claimed that the verasity of this model could be tested by looking for duplicate physical structures in the universe; that we should be able to see our duplicate somewhere. Is anyone familiar with this work and can shed some more light on the meaning of it? Is it a model or reality?

9

<< As JWK already indicated, we must live in three spatial dimensions because no other number will explain the physicality of our world. >>

Bosh. It depends on what “physicality” you want to describe. Your “three perpendicular axes” are useless to describe temperature, for instance. Or color. Or radio-wave length. Yet those are all part of “physicality.”

And the three dimensional system does not work well at a macro or micro level – you don’t try to describe the motion of electrons using a three-dimensional model, nor the motion of the galaxies.

I don’t disagree that there are lots of times when a three-dimension system is a useful model. Especially when dealing with small, local areas, like your backyard or city, or perhaps the hemisphere. But it’s only a model, and there are occasions when it is not at all useful. The mathematical model of dimensions is not limited to three. If I find it useful to add temperature or color as additional dimensions, I can do so; I can still do math in finding distances etc. in multiple dimensions.

10

Multiple dimensions in math? Sure. No problem. Hey, if you want the math to handle infinite dimensions, you can do so. The math is there and anyone crazy enough to want to play with it can do so.

But time is not a “spatial” dimension. Neither is temperature.

And there’s nothing local about it either. If you want to earth to go around the sun, then you want three spatial dimensions. More won’t do. We can only be here to argue about this if there are three spatial dimensions. A mathematical model said so.

The distinction seems to me to be quite clear. Obviously you can extend the math any way you want. But if you do, you shift the definition of what is included in your system away from “spatial.” I’m not understanding why you are arguing against this.

11

I perceive that we may be using somewhat different definitions of dimension here.

In Dex-land, a dimension can be thought of as simply a piece of information used to describe a point (or, alternatively, a slot within a vector used to describe a point).

To wit, we could imagine that a point could be characterized by lenghth, width, height, temperature (Celsius), pressure and time as follows:

(l,w,h,C,P,t)

But length and width are surely more similar than length and Pressure, right? Sure. For one thing, if we wanted to calculate the distance between two points, we would use the first 3 slots and ignore the last 3. To calculate velocity, on the other hand, we would have to look at the last slot -time- as well but we could still ignore slots 4 (temperature) and 5 (Pressure).

In other words, the common definition of dimension (of which l, w and h are the only members) refers to those components that can have a certain set of equations applied to them, among them distance (d = sqrt(l^2 + w^2 + h^2)) and, oh, rotation say.

(If we allow calculus, then we could add equations that would include time in our framework.)

To distinguish the 2 definitions, we might label them as “dimension” and “spatial dimension” or perhaps “spatial-temporal dimensions” if you want to work with (l,w,h,t).

12

Sigh. I’m not arguing philosophy, I’m arguing that there is a difference between reality and any mathematical model of reality.

You cannot calculate the motion of galaxies by just using the three “spatial” dimensions.

My office is on the 23rd floor of a building. My friend has an office on the 23rd floor of an adjacent building. I can see him through the window, across the street. The distance between our two offices from the formula for distance in a vector space (the one given by flowbark) is (let’s say) 50 feet. But the actual path-distance to get from my office to his office involves going down 23 stories and across the street and up 23 stories.

The three spatial dimensions of length, width, and height are almost never used in describing the rotation of the earth around the sun, one uses polar coordinates.

Three spatial dimensions are not sufficient to describe the location of subatomic particles.

My point is we have a mathematical model. It does not always and entirely coincide with the real world. Look, I’m not saying it’s not a useful model. I’m saying that people get confused about reality and mathematical models. There’s no such thing as a “point” in the real world, nor a “line.” The mathematical model of dimensions involves coordinates – when was the last time you gave someone driving directions using coordinates? In fact, you didn’t mention the “height” direction at all, although someone driving along the surface of the earth surely changes “height.”

Mathematics is mathematics, and reality is reality. The one can be used to model the other – one apple plus one pear gives two fruit, just like in mathematics. But adding one lit match to one teaspoon of gunpowder doesn’t yield “two” of anything. The mathematical model of addition (for instance) is very useful for dealing with money but not very useful in dealing with love.

Similarly, the mathematical model of a three dimensional world is useful in many situations. However, if you set your Cartesian coordinates (let’s say) centered on the sun, you can’t even draw a straight line, the spatial dimensions get “bent” by gravity.

13

I’ve been hoping that someone would ask this. The dodecahedral-Universe model is sound science, but it happens to be wrong. As you point out, it can be tested by looking for similar structures in different parts of the Universe. The best tests come from looking for those structures in the cosmic microwave background, using the same data that the group you mentioned used (but using it in a completely different way). One of my professors is on the team that’s doing just that. They haven’t yet completely finished their search for non-trivial topologies (they still have a few weeks of supercomputer time ahead of them, for that), but they have completed enough of their search that they can conclusively rule out the dodecahedral model, as well as all of the other topologies which could explain the same results that the dodecahedral model attempted to explain. Too bad; it would have been nice to know the full and exact extent of the Universe, but the Universe is notorious for not always giving us what we want.

And Dex, I’m going to have to weigh in on the other side of the argument, on how “real” the dimensions are. Of course just the count of dimensions isn’t enough, all by itself, to predict anything. But you don’t need any more spatial dimensions than three to do anything we know how to do in physics. There are times when we need to take one temporal dimension into account, but that still leaves exactly three spatial dimensions. And there are some things which we don’t yet really know how to do, which seem to require more than three spatial dimensions. But if and when we ever do get string theory (or its equivalent) truly figured out, and it requires n dimensions (whatever n is), then I think it’ll perfectly fair to say that there “really are” n dimensions.

And I hate to break it to you, but any discussion about the relationship of reality to the models which describe it is necessarily a philosophical discussion.

14

Chronos and others- FYI, there’s a good discussion of the dodecahedral universe model and it’s refutation at the BadAstronomer’s board:

http://www.badastronomy.com/phpBB/viewtopic.php?t=8620
Yes, too bad. A Far Side cartoon with God playing soccer with the universe could’ve come out of this if the hypothesis had stood up.

15

<< any discussion about the relationship of reality to the models which describe it is necessarily a philosophical discussion.>?>

Perhaps. But until you show me a line that has no thickness, I’m going to say that the three-dimensional mathematical model is NOT the same as the real world universe.

BTW, as a credential, I have Ph.D. in mathematics (Northwestern University, 1973).