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Now I'm curious if that holds for 4d shapes. Maybe someone with a better understanding of calculus or 4d geometry can answer Edit: apparently it does for pyram

Now I'm curious if that holds for 4d shapes. Maybe someone with a better understanding of calculus or 4d geometry can answer Edit: apparently it does for pyramids, so it should for cones as well. General formula for volume of n-dimensional pyramid is A*h/n where A is the volume of the base.

sethammons 3y ago
It intuitively holds for 1, 2, and 3 dimensions. I don't know how to visualize the 4th dimension, but if it holds, it should take up a quarter of the "space" (point -> area -> volume -> space time?) If the 4th dimension is time, does that mean a quarter of the "space" is spread out over time? What does that even mean? Anyone familiar with n-dimensional space able to weigh in?
- zoky 3y ago
  It's probably better to think of time as the "zeroth" dimension, in that everything that exists in any other dimension has to exist for some duration of time first. Of course, we can still consider theoretical objects that exist outside of time, just like we can easily consider two-dimensional objects despite being in an existence with three spatial and one temporal dimension, but for an object to physically exist it must at least exist in time. When you look at time as the zeroth dimension rather than the fourth, it should be obvious that the fourth dimension is simply another spatial dimension. Think about how a two-dimensional plane is a cross section of a three-dimensional object, and line would be a cross section of a plane, and a (zero-dimensional) point is a cross section of a line. So too can our three-dimensional universe be looked as a cross section of a theoretical four-dimensional existence. A three-dimensional object that moves in the fourth dimension would simply cease to exist in our universe, and a four-dimensional object that moves in the fourth dimension would change which 3D cross section is visible to us.
- nkrisc 3y ago
  I found this video explaining quaternions to actually be a great way to explain (and visualize) higher spatial dimensions: https://youtu.be/d4EgbgTm0Bg https://youtu.be/d4EgbgTm0Bg When you think about the nth dimension as a projection of the n+1th dimension, it begins to make a lot more sense. That doesn’t mean you can necessarily visualize the 4th spatial dimension directly, but you can at least visualize how the 3rd dimension can be a projection of the 4th, just like how we can easily see that the 2nd dimension as a projection of the third.
  - withinboredom 3y ago
    There was a paper that showed “bee dances” were very similar to some 6th dimensional shapes projected into 3d. IIRC, it was retracted due to people not taking it seriously and making it into a joke (maybe it was, but I wouldn’t know). But anyway, people who can visualize more than 3 dimensions should be considered magical wizards.
- shawnz 3y ago
  Time is just one possible meaning you can assign to a dimension if it's convenient for you to do so in order to model a problem.
- hydrolox 3y ago
  No, not really, and I don't think the other reply about it being a 0th dimension makes much sense either. The key word is `spacial-dimensions`, whenever math talks about just spacial dimensions, there is no time dimension. In other words, a 0-dimensional 'world' would just be a point, 1 dimensional a line, etc. In something like flatland, where there are living beings, of course there would be a time dimension. But if you are just talking about the volume of shapes, all the dimensions are spacial. So when we go from 3D -> 4D in this scenario, we are adding an extra spacial dimension perpendicular to all the other 3. The idea that 4th D is time is more of something from physics, although the fact that we call it the 4th dimension doesn't mean much, it could be called 1st 2nd 3rd dimension and then a spacial one gets moved to a different number.
contravariant 3y ago
All of these are examples of how the integral of x^n is 1/(n+1) x^(n+1). Combined with the fact that rescaling a n dimensional object by a factor x makes it x^n times bigger and that the (n+1)-volume can be expressed as an integral of n-volumes. It will generalize to any shape consisting of stacked rescaled slices of some other shape where the scaling factor increases linearly.

sethammons 3y ago

It intuitively holds for 1, 2, and 3 dimensions. I don't know how to visualize the 4th dimension, but if it holds, it should take up a quarter of the "space" (point -> area -> volume -> space time?) If the 4th dimension is time, does that mean a quarter of the "space" is spread out over time? What does that even mean? Anyone familiar with n-dimensional space able to weigh in?

zoky 3y ago
It's probably better to think of time as the "zeroth" dimension, in that everything that exists in any other dimension has to exist for some duration of time first. Of course, we can still consider theoretical objects that exist outside of time, just like we can easily consider two-dimensional objects despite being in an existence with three spatial and one temporal dimension, but for an object to physically exist it must at least exist in time. When you look at time as the zeroth dimension rather than the fourth, it should be obvious that the fourth dimension is simply another spatial dimension. Think about how a two-dimensional plane is a cross section of a three-dimensional object, and line would be a cross section of a plane, and a (zero-dimensional) point is a cross section of a line. So too can our three-dimensional universe be looked as a cross section of a theoretical four-dimensional existence. A three-dimensional object that moves in the fourth dimension would simply cease to exist in our universe, and a four-dimensional object that moves in the fourth dimension would change which 3D cross section is visible to us.
nkrisc 3y ago
I found this video explaining quaternions to actually be a great way to explain (and visualize) higher spatial dimensions: https://youtu.be/d4EgbgTm0Bg https://youtu.be/d4EgbgTm0Bg When you think about the nth dimension as a projection of the n+1th dimension, it begins to make a lot more sense. That doesn’t mean you can necessarily visualize the 4th spatial dimension directly, but you can at least visualize how the 3rd dimension can be a projection of the 4th, just like how we can easily see that the 2nd dimension as a projection of the third.
- withinboredom 3y ago
  There was a paper that showed “bee dances” were very similar to some 6th dimensional shapes projected into 3d. IIRC, it was retracted due to people not taking it seriously and making it into a joke (maybe it was, but I wouldn’t know). But anyway, people who can visualize more than 3 dimensions should be considered magical wizards.
shawnz 3y ago
Time is just one possible meaning you can assign to a dimension if it's convenient for you to do so in order to model a problem.
hydrolox 3y ago
No, not really, and I don't think the other reply about it being a 0th dimension makes much sense either. The key word is `spacial-dimensions`, whenever math talks about just spacial dimensions, there is no time dimension. In other words, a 0-dimensional 'world' would just be a point, 1 dimensional a line, etc. In something like flatland, where there are living beings, of course there would be a time dimension. But if you are just talking about the volume of shapes, all the dimensions are spacial. So when we go from 3D -> 4D in this scenario, we are adding an extra spacial dimension perpendicular to all the other 3. The idea that 4th D is time is more of something from physics, although the fact that we call it the 4th dimension doesn't mean much, it could be called 1st 2nd 3rd dimension and then a spacial one gets moved to a different number.

contravariant 3y ago

All of these are examples of how the integral of x^n is 1/(n+1) x^(n+1). Combined with the fact that rescaling a n dimensional object by a factor x makes it x^n times bigger and that the (n+1)-volume can be expressed as an integral of n-volumes. It will generalize to any shape consisting of stacked rescaled slices of some other shape where the scaling factor increases linearly.