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A plane is defined by being 2-dimensional. But if it is curved (like a hyperbolic plane), it requires an extra dimension, that is, it is curved in a 3-dimensional space. So it seems, 2-D coordinates wouldn't be enough to describe this plane. What is it's dimension? Should it be considered as 2 < D < 3 (some sort of fractal dimension)?

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I'm not sure whether this question is suitable for Philosophy... please comment.

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    This question would probably fit right in with the content at Math.SE, but as for here, it only looks to have a tinge of philosophy in it. I would recommend this be migrated to Math, where it can be answered much more technically.
    – commando
    Jul 7, 2012 at 15:41
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    @Cerberus A plane can in fact be curved, as is the case for the surface of a sphere. You need only two coordinates to describe any point on the surface (distance in two directions from an arbitrary point on the surface), yet the plane is curved, enough so to actually loop on to itself.
    – commando
    Jul 7, 2012 at 17:40
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    @cerberus: the plane with a point at infinity is topologically equivalent to the surface of a sphere. You can have many, many different metric structures on the plane, and transport them to the sphere. In the category of euclidean planes, the complex line, the usual plane, and the punctured sphere are equivalent. The algebraic description that you gave for the sphere is one that relies on it being embedded in 3-d space, which is why you have 3 different variables in it. In fact it can be described with only 2 variables, which is why its a 2-d surface.
    – Mozibur Ullah
    Jul 8, 2012 at 1:00
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    @cerberus: its not as complicated as it looks. Think of a circle after removing a point, you can unroll it to get a straight line. That point is normally called the point at infinity. So, doing everything in reverse, a line with a point at infinity added to it gives you a circle. (The reason why its called a point at infinity is to do with stereographic projection, and is best illustrated with a diagram.) The category of planes encompasses all the different descriptions of the plane, in the same way say, we can have many different words in many different languages describing a table.
    – Mozibur Ullah
    Jul 8, 2012 at 2:10
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    This above conversation and the tone of the answers below seem to support my recommendation that this question should really be on Math not Philosophy. I don't see much in the way of philosophical explanations as opposed to technical geometry.
    – commando
    Jul 8, 2012 at 17:15

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I agree that this question is more about mathematics than about philosophy. I'll try though to give a short introduction.

In mathematics, there is not the dimension of something. Instead, it's just a name that pops up in the most different settings for more or less different concepts. So first of all, we've got to say which setting we are talking about. Here are some

  • Vector spaces. This is the most intuitive part, as vector spaces carry very much structure on them, and we got all the standard examples: the real number line, 2D, 3D euclidean space, ... In this case, the dimension is defined as the maximum number of linearly independent vectors, i.e. "independent directions" that you can't already express as a combination of each others. Clearly, in 3D space, we have 3 of these. However, a curved plane is far from being a vector space!

  • Submanifolds: That's probably the correct setting here. A submanifold is something that locally, in a very tiny environment, "looks like our euclidean space". We all know examples of these: The earth's surface is of course very different from a 2D plane -- at global scale. But locally, for me as a human looking around, it can be seen as a plain surface. Mathematically that means that for any point on your surface, there is little environment around that point that, with some bending and stretching, can be turned into the euclidean space.

    That's exactly the case for your hyperbolic plane. At any point, you can bend and stretch the plane and get a little piece of the ordinary 2D-plane. You can't bend it to get 1D or 3D-space etc., so we can rightfully call the hyperbolic plane a 2-dimensional submanifold of 3D-space!

  • Hausdorff-dimension: Now we can think of surfaces that aren't even submanifolds. A surface with cusps or, even worse, a fractal surface is such a case. No matter how you'll bend it, it'll never look like plain euclidean space. In these cases, we'd for example need another concept called Hausdorff-dimension.

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  • is Hausdorff-dimension a synonym for fractal dimension? if not, what is the difference?
    – Tames
    Jul 8, 2012 at 0:44
  • @tames: hausdorff dimension is one of many different fractal dimensions. Its the most popular though, so yes, you can probably regard it as a synonym for it.
    – Mozibur Ullah
    Jul 8, 2012 at 2:08
  • It's not trivial that a curved plane is not necessarily a vector space. For example R^2 is a curved plane and a vector space.
    – user2953
    Mar 10, 2015 at 23:43
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The quick answer is that the hyperbolic plane is two-dimensional. In fact any surface is 2-dimensional no matter how it bends or curves.

That a curved surface requires a third dimension is (mathematically speaking) an artifact of the 3-d ness of the space we live in. After all, when I imagine a straight line, I can imagine it in air, or more fundamentally surrounded by nothing at all. That is not only have I eliminated the air surrounding the line, but also the space occupied the air. Similar reasoning applies to a curved line, or to a curved surface.

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  • You can make a space-filling continuous surface that fills all of 3 dimensional space. Only if you have some extra structure like differentiability can you know that the surface is two dimensional.
    – Ron Maimon
    Jul 15, 2012 at 9:00
  • true, but in the context of this question, and what its asking for this would be a pathological example :).
    – Mozibur Ullah
    Jul 17, 2012 at 21:57
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A plane is two-dimensional.

You can define your two dimensions any way you please. Suppose you had a beach ball with a seam on it, and you drew a line perpendicular to the seam (all the way around so it connected to itself). Choose one of the bisections as your origin. The first dimension can be "distance along the seam" and the second dimension can be "distance along the line". Now you have a plane.

It only appears to not be two-dimensional if you assess the "curved" plane relative to a different dimensioning scheme. If you suddenly use the "X-Y-Z" convention relative to a fixed origin, then your "beach ball" plane looks 3-D.

I suppose you could even put your "X-Y-Z" origin on another planet, and then your "beach ball" plane would "look" 4-D.

To say it more generically:

  • Any two-dimensional plane will look three-dimensional if you apply the proper coordinate system.

  • Any "curved plane" will look two-dimensional if you apply the proper coordinate system.

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  • @Jas-3-1 What would the proper coordinate system be, supposing it is a 2d plane that looks 3d? How many variables are needed? if someone has in mind such a plane as the surface of a sphere, how many coordinates would be necessary to give to a 2nd person, so that he could reconstruct the same plane? (what is necessary to describe the plane in mind as a closed surface and not an open one, for example)
    – Tames
    Jul 9, 2012 at 16:22
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    @Tames The beach ball example did add some complexity since it is a "closed surface", but let's table that for now - it was just a simple illustration so my readers didn't have to envision an infinite plane. To describe 2-D as 3-D, you just need an equation for each dimension in 3-D (X, Y, and Z). Each equation would have two variables ("seam" and "line" in this case.) Now, describing 3-D as 2-D is an entirely different (and fascinating) question! Statically, you can't (must drop a 'D'.) But there are ways to describe a 3-D object to a 2-D observer. (Shift in location or time!)
    – Jas 3.1
    Jul 9, 2012 at 17:02

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