space curvature

Higher-dimensional geometry (previously "Polyshapes").

space curvature

Postby Adam Lore » Mon May 04, 2009 2:30 am

I was reading somewhere that all space is curved, including Euclidean 3-space. (I think. I believe by Wendy?)

Can anyone expand on that?
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Re: space curvature

Postby wendy » Mon May 04, 2009 12:17 pm

All space is indeed curved, where 'flat' simply means 'curved at the same measure of all-space'.

There are some things to be noted here. Firstly, space is not curved in anything: it is a property of space itself. It has nothing to do with space-time, or the device where they show black holes in deep pits, but rather a geometric thing.

The thing where one has balls sitting in pits on a surface, is a kind of representation of space, where downwards gravity effects real curvature. It has time, (ie time is not one of the coordinates), since it's intended to be a billiard table, where the slope and speed traces out a trajectory of a thing.

In geometry, curvature has to do with the perimeter of a circle, when compared with its length. For homogeneous isotropic spaces, the sort of thing that we talk about with hyperbolic, euclidean and spherical geometry, this measure depends the measure of the radius against a universal radius. For real space, it varies in direction and place.

Were ye to take a circle, whose circumference is say, 2.pi, the diameter gives a chord of 2. However, the chord from the centre of the circle to its perimeter can be greater or lesser than one. When it's less than 1, the curvature is positive, and greater than 1, it's negative. Zero curvatrure gives euclidean space, but this is zero-curvature, is not necessarily flat.

A curve can be enbedded in a curve of lesser curvature (or greater radius). Ye can draw a circle of diam 20 inches on a sphere of diameter 30 inches, or on the plane, or on a sphere of 20 inches. You can't do it on a 10-inch sphere. On the 20-inch sphere, the circle is straight (that is, it bisects the length of the circuferences at each point). You can't draw a euclidean line on any sphere, but ye can draw one on a hyperbolic plane. It's horridly a crooked line. The horizon of the hyperbolic space is not a straight line, but is very bent.

None the same, a Euclidan plane, of curvature zero, can be represented by a polytope like {3,6} or {4,4}. In euclidean space, this is flat. since all-space is zero curvature. So we see for example, a polytope {4,4} is a tiling of squares, with half-space as interior. In hyperbolic space, the thing has angles less than 180°, can for example have angles like 120° or even 90°. It's still zero-curvature, but it's not flat.

When it is said that one can do away with gravity, by making space curved at any point, the thing is that one supposes at a point P, that different degrees of arc have different lengths, and that space is in tension according to the length. Where space is negatively curved, the length of each degree nearer the mass has a greater length, and a body at rest is pulled towards this direction more than others: it falls in the direction where the curvature is least: ie the degrees are longer. Likewise, a straight line bisects the circumference at each point, will tend to vere towards a mass, because points near the mass are longer than those opposite, and the half-circuference is less than 180°.
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Re: space curvature

Postby Adam Lore » Wed May 06, 2009 5:46 am

So..

Euclidean space has zero curvature, and it's flat, right? Or the flat objects (such as a {4,4} tiling), are flat in it, and the space is curved?
But if it's zero curvature, doesn't that mean it's not curved?

Or do you mean that something that we consider flat is only said to be flat because it lines up with our particular curvature? That in a different space there is a different curvature, so neither should be considered standard? Or something like that?

Also, what in the world is half-space? And how does a tiling have an interior?

Also, when you say that space is not curved into anything, is this fact agreed on? Do geometers and scientists have a different understanding of space curvature?

(thanks for the help!)
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Re: space curvature

Postby wendy » Wed May 06, 2009 7:34 am

Curvature is a term that corresponds to C ~ 1/R² of space. SWhen C <> 0, changing the linear dimensions of something like a square makes the angles change too. When C=0, you are effectively multiplying a/R by k, to get ak/R, which is still 0, so Euclidean geometry can be resized without distortion.

Flat is a term relative to all-space, in as far as the curvature of a given subspace (eg a plane), is the same as all-space. This means, that regardless of which sides of the plane we make solid, two copies will join together and move freely over eachother with out making a space inside it.

The two are different in non-euclidean geometries.

It is possible to study geometry leaving curvature to the later bits: Euclidean and non-euclidean geometries have many common elements. Letting R go to very large values makes the geometry look euclidean anyway. For example, the model of cosmology assumes that space is essentially euclidean, with local deformities in accordance with Einstein's relativity, and a gross curvature that is somewhere near zero (but of unknown sign). In the long run, this means that we don't know if space is hyperbolic or spheric.

A planotope has half-space (eg all space where z<0), bounded by a tiling in the space where z=0. In effect, it's a very large polytope. In euclidean space, the nature of zero-curvature means that the face planes must fall in the same plane regardless of size. In non-euclidean geometry, like spherical, one can suppose that the size of a polytope (eg a polygon), is increased so that it makes a great circle (with a hemisphere as interior). The effect here is half-space bounded by a 'tiling'. Making the size different (smaller for spheric, larger for hyperbolic), makes the thing depart from the equator to an equidistant.

So for example, you could have {5,3} as a tiling in E2 (surface of a sphere), or as a planothedron in E3 (surface of a glome), or by reducing the size, a polytope with a lesser margin-angle (eg the cell of a tiling {5,3,3}, the {5,3} has a margin-angle of 120°. The distinction is for those who fail to get this point.
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Re: space curvature

Postby Adam Lore » Sun May 10, 2009 3:17 am

Thanks, you're always a great help.
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