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What does volume mean in higher dimensions? 2-D has no volume. 3D has a volume. What's volume in 4D and higher? Edit: Just to further the thought. Do all di

by ww520 3y ago

What does volume mean in higher dimensions? 2-D has no volume. 3D has a volume. What's volume in 4D and higher? Edit: Just to further the thought. Do all dimensions have the same unit length? In 3D, the spacial x,y,z dimensions have the same unit length. In 4D like 3D+Time, does the t-dimension have the same unit length as the spacial dimensions? What is 1-unit of t meant vs 1-unit of x? What does it mean to say a sphere in 4D with radius r having the same "distance" from the center in all 4 dimensions? What unit length is r in?

deleted 3y ago
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IshKebab 3y ago
2D volume is area.
rocqua 3y ago
A hyper cube with sidelength 1 has a 'volume' of 1. Then you sorta just define 'volume' (or the proper term 'hypervolume' based on how many hyper cubes you can fit. If the fit isn't perfect, half the size of your cubes, and try again. Whatever limit this system approaches is the true volume. (Edge cases are fractals, and they are already complicated in 3 dimensions) Edit: each dimension has the same unit length, by definition of that dimension. At least, in 'flat' space. Distance is just done by Pythagoras. This is actually meaningful because Distance is unique in being preserved by rotations. And rotations are special because they form a subgroup of all linear transformations. Hence the choice to use Pythagoras for Distance is not arbitrary, it is very natural. Because rotations are very natural.
emaro 3y ago
You need to distinguish between physics and mathematics. The explainer is about math. Here, 2D absolutely has volume (i.e. area). Also, higher dimensionality spaces are the same: each dimension uses the same units, therefore r is always the same. Thinking in "time" dimension doesn't help here imho.
sillymath 3y ago
> What does volume mean in higher dimensions? I think you need a scalar product to define volume. But fuzzy thinking an analogy: Think friendship, define three concepts related to friendship and that they are orthogonal, now establish some numeric scale to measure each concept, then you have just defined a volume to measure the degree of friendship. Unfortunately there is no canonical way to establish those orthogonal features, but they could be obtained applying a linear model to big data sets of measures of friendship.
adrian_b 3y ago
> Do all dimensions have the same unit length? The units of length are whichever you choose them. You are free to choose different units of length for each of the dimensions of a multi-dimensional space, but that does not bring any benefit and it introduces many constant factors corresponding to the ratios between the different units of length in all mathematical formulae. Therefore everybody chooses to have the same unit of length for all dimensions, in order to have the simplest formulae. Of course, this is true for the multidimensional spaces in which a scalar product of the vectors is defined. When there is no scalar product, the units of length for the different dimensions are independent. There is no way to compare them and the question whether they are the same or different is meaningless. In such spaces, angles, areas and volumes are also undefined, only the affine geometry is applicable.

by ww520 3y ago

deleted 3y ago

[deleted]

IshKebab 3y ago

2D volume is area.

rocqua 3y ago

A hyper cube with sidelength 1 has a 'volume' of 1. Then you sorta just define 'volume' (or the proper term 'hypervolume' based on how many hyper cubes you can fit. If the fit isn't perfect, half the size of your cubes, and try again. Whatever limit this system approaches is the true volume. (Edge cases are fractals, and they are already complicated in 3 dimensions) Edit: each dimension has the same unit length, by definition of that dimension. At least, in 'flat' space. Distance is just done by Pythagoras. This is actually meaningful because Distance is unique in being preserved by rotations. And rotations are special because they form a subgroup of all linear transformations. Hence the choice to use Pythagoras for Distance is not arbitrary, it is very natural. Because rotations are very natural.

emaro 3y ago

You need to distinguish between physics and mathematics. The explainer is about math. Here, 2D absolutely has volume (i.e. area). Also, higher dimensionality spaces are the same: each dimension uses the same units, therefore r is always the same. Thinking in "time" dimension doesn't help here imho.

sillymath 3y ago

> What does volume mean in higher dimensions? I think you need a scalar product to define volume. But fuzzy thinking an analogy: Think friendship, define three concepts related to friendship and that they are orthogonal, now establish some numeric scale to measure each concept, then you have just defined a volume to measure the degree of friendship. Unfortunately there is no canonical way to establish those orthogonal features, but they could be obtained applying a linear model to big data sets of measures of friendship.

adrian_b 3y ago

> Do all dimensions have the same unit length? The units of length are whichever you choose them. You are free to choose different units of length for each of the dimensions of a multi-dimensional space, but that does not bring any benefit and it introduces many constant factors corresponding to the ratios between the different units of length in all mathematical formulae. Therefore everybody chooses to have the same unit of length for all dimensions, in order to have the simplest formulae. Of course, this is true for the multidimensional spaces in which a scalar product of the vectors is defined. When there is no scalar product, the units of length for the different dimensions are independent. There is no way to compare them and the question whether they are the same or different is meaningless. In such spaces, angles, areas and volumes are also undefined, only the affine geometry is applicable.