Guide to N Dimensions

 Wikipedia Commons 2 22 Spacetime Curvature
Thinking about dimensions other than the three we're used to can rattle one's mind. That's why it's usually left to stoned conversationalists and theoretical physicists. To help the rest of us navigate flatland, fractal landscapes, and hyperspace, New Scientist put together a concise and fun tour titled "Beyond Space and Time." From New Scientist (spacetime curvature illustration from Wikimedia Commons):
What is a dimension?

The most intuitive description is the oldest one: the number of dimensions a system possesses is the number of independent directions you or anything else can move in. Up and down count as only one dimension because up-ness and down-ness are two sides of the same coin: the further up you go, the less down you are. The same connection exists between left and right, and forwards and backwards, but not between up and right, down and backwards, and so on. Thus, the geometers of Ancient Greece recognised, we live in a three-dimensional world.

So far, so simple, but then things start to unravel. Our place in the cosmos is defined as much by time as it is by space. As long ago as the late 18th century, the Frenchmen Jean le Rond d'Alembert and Joseph-Louis Lagrange recognised that the mathematical language needed to address time was very similar to that which described space. Time, the mathematicians of the day rapidly came to agree, was a fourth dimension.

That opened the floodgates. Once untethered from its origins in physical space, the concept of a dimension began to lose its focus. It came to be used as a general term to describe the number of independent coordinates or variables needed to determine the state of any object.
"Beyond space and time: Fractals, hyperspace and more"


  1. I always say 7 dimensional space takes 7 coordinates to locate a point.

    Like an outline:


    If you are 5 levels deep into an outline, then your item is located by 5 coordinates, which makes it a kind of model for 5 dimensional space.

  2. I think software that makes taste recommendations for you represents you as a point in, say, 50-dimensional space. You are a dot along the 50 or more mutually orthogonal axes of “likes grapes” “wears brown belts” “skis regularly”. It then looks for your neighbors in 50-D space and makes recommendations to you based on their purchase histories.

    I think that’s how it works, anyway.

  3. I thought this little video about imagining the first 10 Dimensions was pretty cool. it has two parts, here is a link to the 1st part.

  4. #4 – already posted on BB.

    And as I was about to say: the articles on New Scientist seem just like a slightly more accurate and explanatory version of that video. So then I expect to see as much vitriol against these articles as we did against the video? Yes?

  5. “Thinking about dimensions other than the three we’re used to can rattle one’s mind. That’s why it’s usually left to stoned conversationalists and theoretical physicists.”

    – So what does a stoned theoretical physicist think about?

  6. I use the following example for my Grandma: we want to study the physics of shuffleboard. Now, we require two dimensions in order to figure out where the rock is on the board. However, if we want to figure out where the rock is going to go, we also need to know its velocity, which we can record using an arrow (direction and magnitude). Thus, we’re up to four dimensions.

  7. @9: It’s not meant to show “down-ness” around the Earth. It’s a visual metaphor, not a literal depiction.

    It abstracts 4(or whatever) dimensions of space into a 2D plane, and uses the 3dimensional object of a rubber sheet as a metaphor.

    Adding more grid would clutter it up and disguise the message.

    The “downness” deformation of the grid isn’t suggesting that all objects near earth in space are attracted to the south pole and that it’s harder to walk north than it is to walk south. That’s ridiculous. If you need to stretch the metaphor somewhat, the “Down” that the “grid” is being deformed towards is the center of the planet. Ish.

  8. Time is a physical dimension, just like the three we perceive more completely.

    The limitations of the human sensorium do not delimit reality, despite Bishop Berkeley. Many things exist that we cannot know with our unassisted senses, and many things can only be partially glimpsed.

  9. #7 ANON

    I’ve long explained it similarly using an arrow in flight as an example. Sure, a picture of an arrow in flight may seem obvious to determine, because the point points towards the direction it travels, but who is to say that it wasn’t an arrow held and dropped, with no forward motion? A candid image of an arrow in air needs something else to describe it. The vectors acting on it help show that there are other ways that things are “real” besides length width and height… other dimensions.

    I’m fighting through a sweaty hangover, so maybe my descriptive capacity is suffering, but I’m sure you get the point.

  10. Per the rubber sheet thing, I’ve also seen visual depictions with a 3D grid that is “scrunched up” towards masses. That’s a bit more accurate (it is indeed the “bending” of what we would normally think of as “straight lines”), but harder to see and, perhaps, harder to understand the significance of. It IS an “up” and “down,” just in more dimensions than we’re used to. You have to “climb out” of the “gravity well” of the Earth to get anywhere else, for example, the deepest part of which is at the center. And the force of gravity would be represented by the “steepness” of the sides. These are concepts we’re used to thinking about in a deformed plane, but less so in a deformed 3space.

    And, um, sorry about all the scare quotes…

  11. #4–

    My main problem with that video is the narrator invokes a lot of hand-waving when it comes to discussing the underlying mathematical concepts. By not discussing the reason why scientists believe there is a fifth dimension (unification of gravity and electromagnetism) and possibly ten or more dimensions (if you subscribe to string theory), the entire video comes off as preachy and improbable; like a philosopher of old describing the heavens as a series of giant, shifting glass spheres.

    As for the linked article, it does a much better job describing the underlying logic for why scientists believe that there are multiple dimensions. I, however, am inclined to believe that since our understanding of quantum physics is still so limited, all this theoretical talk of n dimensions is just hypothetical at best– frameworks built with the sole intent of satisfying the data we have, lacking the elegance and lucidity of a more comprehensive theory like general relativity. Of course, that doesn’t mean it’s necessarily wrong, just that I feel like we only have a small portion of the larger picture figured out.

  12. #6 posted by Anonymous:

    So what does a stoned theoretical physicist think about?

    Me, apparently, and it’s bloody annoying. Every time it happens, I have to change my state.

  13. Is the New Scientist quoting from non-refereed arxiv articles (here and
    here) because they were never published, or because the published versions are copyrighted?

    In any event, if you’re going to refer to a published article you need to give the source. Phys Rev is a totally different source than JIR.

  14. Please. A dimension is simply a vector of possible values. The notion that something has a PLACE, in SPACE, may be as intuitive as orangutans and bananas, but it’s not special, just a local case of an abstract class. Wittgenstein would say that only the model is knowable, while the reality may not even exist, throwing the whole question into a cocked hat.

  15. Not only is the concept of ‘dimension’ not related to physical space, the concept of ‘space’ hasn’t been related for quite some time as well.

    Anything that shares some of the mathematical properties of a physical space can be termed a ‘space’. Most notably Hilbert spaces, which is any space that shares the property of orthogonality with ordinary spaces. (i.e. the ‘dimensions’ of the space are orthogonal. X has no Y component.) Only Hilbert spaces can have any number of dimensions, they’re often infinite-dimensional.

    Usually, the valid solutions to a Schrödinger equation are viewed as a Hilbert space (since they’re orthogonal), and a quantum-mechanical wave function is a vector in that space.

    This isn’t particular to quantum mechanics though. It can also be used to describe the solutions to a classical wave equation, for instance. E.g in the case of a simple vibrating string, each ‘axis’ or ‘dimension’ of the Hilbert space would correspond to a harmonic. (out of which there are an infinite number).

    Since any tone that that string would be capable of producing would be a set of harmonics and their respective amplitudes, which can then be viewed as a vector in that Hilbert space.

  16. @22:

    If you’re going to mention infinite-dimensional spaces, you may as well point out that there are different sizes of infinity, too. Most wavefunctions live in spaces with uncountably many dimensions.

    This isn’t all that scary- it often just means the waves have a (complex) value at every point in space, only mathematically we treat each position as orthogonal direction.

  17. Grikdog, I agree. And furthermore I will hold Wittgenstein for you while you gut-punch him into silence.

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