Combining Spatial Dimensions

[TaiLiu]

So this is probably a naïve question.

For two spatial dimensions, to calculate the area of a rectangle, you multiple the length and height. For three spatial dimensions, to calculate the volume of a rectangular prism, you multiple the length, width, and height.

What about four spatial dimensions or more? Do you multiple all four spatial dimensions together to obtain its “volume”? Is this even a coherent thing to ask?

[jayem]

So this is probably a naïve question.

For two spatial dimensions, to calculate the area of a rectangle, you multiple the length and height. For three spatial dimensions, to calculate the volume of a rectangular prism, you multiple the length, width, and height.

What about four spatial dimensions or more? Do you multiple all four spatial dimensions together to obtain its “volume”? Is this even a coherent thing to ask?

Yes but...

In the same way you have the length of each side of a rectangle (and total perimeter), the area of each side of a cube (as well as an overall surface area, side lengths and a total perimeter), you have a '3 dimensional volume' associated with each side of the hypercube (as well as the 'surface volume', surface area, individual areas,

So you need to maintain the term Volume for the 3-dimensional aspects, and need a new word for the 4th dimensional anologue, hyper-volume is normally used (and bounding volume rather than surface volume).
I think after 4 dimensions, the terms used are picked for easy expansion onwards.

https://en.wikipedia.org/wiki/Hypervolume


Further addition, when dealing with time as the '4th dimension', it doesn't play as nicely (where you'd expect to add, you often are taking it off)

[TaiLiu]

Thank you! That helps a lot.

(No worries about time. Just interested in spatial dimensions for now!)