#### HawkI

Well length is ___________________________
And height is
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When they come together they make a diagonal, so yeah, I think a diagonal line is a 2-D shape inside a 3-D sphere.

#### skipjack

Forum Staff
There is a standard definition of 2D (and 3D), but you're not using it. Also, note that the word "radius" has more than one meaning.

• HawkI

#### HawkI

Ok then, what are the definitions of radius and what is the standard definition of 2D?

This all started with such a seemingly simple question.

#### re-x

...This is always 1-dimensional, because line segments are 1-dimensional.
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.

• HawkI

#### SDK

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.
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.

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

• Maschke and HawkI

#### re-x

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.
I agree lines or open line segments are one dimensional but not closed line segments.
Line segment - Wikipedia

"If V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional "

• idontknow

#### HawkI

A diagonal line has both height and length.
A 2D shape has both height and length.
Case closed.

#### HawkI

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

#### SDK

I agree lines or open line segments are one dimensional but not closed line segments.
Line segment - Wikipedia

"If V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional "
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.

A diagonal line has both height and length.
A 2D shape has both height and length.
Case closed.
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.

#### bamboum

Confusion between description of an object and its space position. A point is a 0D object but one needs three numbers for position. A segment is a 1D object but one can have 1 number in its one dimensionnal space or 2 numbers if the segment is in a plane (length, angle) 3 in the space (length, two angles)