10mo ago

If a 1D figure has length, a 2D figure has area, and a 3D figure has volume, are there names for what's inside 4D, 5D figures and so on?

17 comments

I think you are looking for Lebesque measure, wikipage.
Quote: "For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, hypervolume, or simply volume."
- Wonderful answers all around, but this seems to be the succinct, specific one-word answer: it's a Lebesgué!
You'd just continue saying 'volume', alternatively 'k-dimensional volume' or 'volume of the n-dimensional object'. Like for spheres: https://en.m.wikipedia.org/wiki/Volume_of_an_n-ball
The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:[1]
- I'm going to start calling area "2-dimensional volume"
  
  Only if you also call length "1-dimensional volume".
A popular example of a four-dimensional polytope is the Tesseract, which is just a 4D cube. Four dimensional and beyond polytopes have what is called a hypervolume. This can be calculated by using Lebesgue measure, which is beyond my understanding of mathematics.
Fun fact: four-dimensional analysis is common in the development of modern parallel supercomputing!
4D ounces
- To freedom
  
  is the only chance I have
4D also has duration
- Only if time is your fourth dimension. OP is likely asking about a fourth spatial dimension, since that's much more in keeping with the progession of 1D > 2D > 3D
  
  Ah, I see
- In specific applications where it is useful to consider time as a 4th spacial dimension.
  So if you're not talking about relativity, it's probably not.

17 comments