jeshaw2

I like the idea of staying within the dimensions of discourse, but I think you make my point with the classic example. When you draw a triangle (basically a 2-dimensional object) upon a sphere (a 3-dimensional object) you distort the triangle in the 3rd dimension. One may then measure the distance between any two corners of the triangle in one of two ways: The distance as measured along the surface of the sphere (the geodesic), and the straight line distance (on a Euclidean plane) through the sphere. By extension then, if I "draw" a 3-dimensional pyramid on a 4-dimensional surface, would I then have three ways to measure distances between the points? Further, if we use the method of determining distances as the determinant of being flat or curved, then what about the measurements about the pyramid? Would it be flat, curved, or something else? So, how do I deduce that the triangle is flat or curved from your information? And how do I extend that logic to a 4-dimensional object?

Just as an aside; In his latest book A Briefer History of Time, Stephen Hawking makes the statement "...in relativity, there is no real distinction between the space and time coordinates, just as there is no real difference between any two space coordinates." I take it then, that all four dimensions have a combined spatial and temporal essence. In fact, in space time, I'm not sure there is something distinctly spatial as differentiated from something that is temporal. Any who, back to my question at hand then... Is it consistent with current theory to say that a 4-dimensional object in space time becomes curved, or warped, by its distortion in the 5th dimension?

I'm trying to get a layman's understanding by what is meant by "Space Time is curved or flat" From my readings, space time has 10 dimensions; from my perceptions, it has 4 dimensions. I'm gravitating toward conceptualizing space time curvature and flatness in the following way: A 2-dimensional object such as a sheet of paper (ignoring its thickness) becomes curved if it is distorted in the 3rd dimension. A 3-dimensional object becomes curved if distorted in the 4th dimension (leading to things like a Klein bottle). Thus, an n-dimensional object becomes curved by its distortion in the (n+1)-dimension. Similarly, an n-dimensional object is said to be flat if it is not distorted in the (n+1) dimension. Does this make sense or is there something else I should read? If it makes sense, then if Space Time (a 10-dimensional object) is curved (or flat but distorted by gravity), is there an 11th dimension?