- Thread starter HawkI
- Start date

I guess line segments can be multi-dimension as you might need multi-variables to define a particular line segment....This is always 1-dimensional, because line segments are 1-dimensional.

eg. line segment from {1,2,3} to {1,5,3} is of course different from line segment that is from {1,6,4} to {1,9,4} even though they have same length.

Both of your examples are 1-dimensional. Every line segment is 1-dimensional. You are confusing the dimension of the line segment with the dimension of the "ambient" space in which you place it. If you think about this for a second, it's obvious why distinguishing these would be terrible. For example, dimension is something which should be intrinsic to an object/set. If you distinguish the lines in your example then you have the situation that a line can be 1-dimensional unless you rotate it and then its dimension can change. This is actually a simple example of a more general problem which would be that dimension suddenly depends on the arbitrary choice of coordinates used to represent the object. This would effectively make dimension a useless notion. All of this can be made very precise but I'm avoiding that at the moment. I'm happy to go into more detail if necessary.I guess line segments can be multi-dimension as you might need multi-variables to define a particular line segment.

eg. line segment from {1,2,3} to {1,5,3} is of course different from line segment that is from {1,6,4} to {1,9,4} even though they have same length.

tldr: Lines are 1-dimensional. It doesn't matter what ambient space they live in.

I agree lines or open line segments are one dimensional but not closed line segments.Both of your examples are 1-dimensional. Every line segment is 1-dimensional. You are confusing the dimension of the line segment with the dimension of the "ambient" space in which you place it. If you think about this for a second its obvious why distinguishing these would be terrible. For example, dimension is something which should be intrinsic to an object/set. If you distinguish the lines in your example then you have the situation that a line can be 1-dimensional unless you rotate it and then its dimension can change. This is actually a simple example of a more general problem which would be that dimension suddenly depends on the arbitrary choice of coordinates used to represent the object. This would effectively make dimension a useless notion. All of this can be made very precise but I'm avoiding that at the moment. I'm happy to go into more detail if necessary.

tldr: Lines are 1-dimensional. It doesn't matter what ambient space they live in.

Line segment - Wikipedia

"If

Well ok then, to be fair, if the radius is a straight line, length or height, then it's 2D.A diagonal line has both height and length.

A 2D shape has both height and length.

Case closed.

You are misunderstanding what this says. The vector space, $V$ is one dimensional if an open line segment is an open subset. This doesn't say anything about the dimension of the line segment. A line or line segment is 1-dimensional whether its open, closed, or half open.I agree lines or open line segments are one dimensional but not closed line segments.

Line segment - Wikipedia

"IfVis a topological vector space, then a closed line segment is a closed set inV. However, anopen line segmentis an open set inVif and only ifVis one-dimensional "

Why bother asking if you are going to ignore the answer? This is just plain wrong. A line is 1-dimensional. I don't know what you mean by it has length and height, but it's simply wrong.A diagonal line has both height and length.

A 2D shape has both height and length.

Case closed.