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I'm going to say something that sounds silly to most people, but isn't it the same deeper reason why a triangle is half of the base times the height? I don't g

I'm going to say something that sounds silly to most people, but isn't it the same deeper reason why a triangle is half of the base times the height? I don't give a rat's ass if there's a proof or not connecting the two. I'm not a mathematician, but it makes sense from a calculus perspective that the number of dimensions ends up as a divisor.

ozyschmozy 3y ago
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
The downvotes are ignorant. See Lockhart's Mathematician's Lament, where he laments the state of math education comparing it to learning music through sheet music and not pickings up an actual instrument. He asks how to visualize the area of a triangle by bounding it in a rectangle, where it becomes obvious that the triangle takes up half the area. He then poses the question about the volume of a pyramid where you can use a similar technique. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....
pseudostem 3y ago
>I don't give a rat's ass if there's a proof or not connecting the two. Life is too short for proofs. - Gilbert Strang (in one of his lectures)
- tinus_hn 3y ago
  In the same vein: Beware of bugs in the above code; I have only proved it correct, not tried it. Donald Knuth
  - sdkgames 3y ago
    The only effective way to raise the confidence level of a program significantly is to give a convincing proof of its correctness. [0] Edsger W. Dijkstra CONTEXT: Argument three is based on the constructive approach to the problem of program correctness. Today a usual technique is to make a program and then to test it. But: program testing can be a very effective way to show the presence of bugs, but is hopelessly inadequate for showing their absence. The only effective way to raise the confidence level of a program significantly is to give a convincing proof of its correctness. But one should not first make the program and then prove its correctness, because then the requirement of providing the proof would only increase the poor programmer’s burden. On the contrary: the programmer should let correctness proof and program grow hand in hand. Argument three is essentially based on the following observation. If one first asks oneself what the structure of a convincing proof would be and, having found this, then constructs a program satisfying this proof’s requirements, then these correctness concerns turn out to be a very effective heuristic guidance. By definition this approach is only applicable when we restrict ourselves to intellectually manageable programs, but it provides us with effective means for finding a satisfactory one among these. [0] Edsger Dijkstra - Turing Award Lecture - The Humble Programmer - 1972 https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD340.html https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD340...
Solvency 3y ago
While I agree about proofs it's not intuitive to me. A triangle can clearly be visualized as half a square. Look at any videogame square made of 2 polygons. A 2D projection of a cone isn't intuitively 1/3 of a cylinder with the same dimensions.
gniv 3y ago
Here's an attempt at making sense of that intuition. Let's compute the volume of a pyramid that's inside a cube. Imagine that you're inside a cube room (this will make it easier to identify the faces). You are sitting in a corner. From your corner draw lines to the four ceiling corners -- a cube diagonal, two face diagonals and a cube edge. This is the pyramid. Now draw lines to the corners of one of the walls opposite you -- it's the same pyramid, but on its side. And lastly draw lines to the other wall opposite you -- again the same pyramid. You have now covered the entire cube with three identical pyramids, so the volume of the pyramid is a third of the volume of the cube. The interesting part is that the bases of these pyramids are on the three dimensions, giving the intuition that the /3 is due to the dimension. This proof might even generalize in higher dimensions. Edit: I think this is also the gist of the top answer in TFA, but it's weirdly formulated imo.
- justinpombrio 3y ago
  That's a beautiful proof! And yeah, it generalizes to higher dimensions. If you're in the corner of an N-dimensional hypercube, then the hypercube has 2n faces of dimension n-1. You're touching half of these faces, leaving n faces to form pyramids out of. And just like in the 3D case, they're arranged symmetrically so they must all have the same volume. EDIT: got around to reading TFA and it's also the top answer, so there's a nice visualization. Diving deeper into the proof, say the hypercube is a unit hypercube, spanning from (0, ..n, 0) to (1, ..n, 1) -- where "..n" means "a sequence of length n". Then: Each corner is a point (b_1, ..n, b_n) where each b_i is 0 or 1. You're located at (0, ..n, 0). The hypercube has 2n faces, which can be represented as a pair (i, b) where 0<=i<n and b is 0 or 1. The face (i, b) touches the 2(n-1) corners whose i'th coordinate is equal to b. You touch half of those faces: specifically the faces (i, 0) for each 0<=i<n. You can draw a pyramid to the other half of the faces, since you don't touch them. These pyramids must all have the same volume, by the symmetry of coordinate permutations, which is an operation that preserves volume.
- dadoomer 3y ago
  Very cool! Thanks for sharing.
- dp-hackernews 3y ago
diffeomorphism 3y ago
It is exactly that and you even can use the same proof for a triangle, a cone or higher dimensional analogies. You always get 1/dimension. Proof: Take height 1 for simplicity of notation. Then we get the volume as the integral over crosssections (Fubini). Each cross section is rescaled by a factor (1-z), where z is the height. Rescaling changes the cross sections measure by (1-z)^(n-1). Integrate that to get 1/n. Done.
JadeNB 3y ago
It makes sense that the number of dimensions ends up somewhere, but I think there's no particularly obvious a priori reason why it should be a denominator. For example, the length of the long diagonals of a unit cube in n dimensions depends on the dimension, but it's \sqrt n, not anything divided by n (well, it's n^{3/2} divided by n, but that's not much use). Of course one can fill in why in this case it happens to be the dimension in the denominator—several sibling posts, particularly gniv's (https://news.ycombinator.com/item?id=36562093 https://news.ycombinator.com/item?id=36562093), have done so nicely—but I was responding specifically to the disclaimer: > I don't give a rat's ass if there's a proof or not connecting the two. I'm not a mathematician, but it makes sense from a calculus perspective that the number of dimensions ends up as a divisor. which seemed to suggest that it should be obvious without mathematical reasoning.

ozyschmozy 3y ago

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

The downvotes are ignorant. See Lockhart's Mathematician's Lament, where he laments the state of math education comparing it to learning music through sheet music and not pickings up an actual instrument. He asks how to visualize the area of a triangle by bounding it in a rectangle, where it becomes obvious that the triangle takes up half the area. He then poses the question about the volume of a pyramid where you can use a similar technique. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....

pseudostem 3y ago

>I don't give a rat's ass if there's a proof or not connecting the two. Life is too short for proofs. - Gilbert Strang (in one of his lectures)

tinus_hn 3y ago
In the same vein: Beware of bugs in the above code; I have only proved it correct, not tried it. Donald Knuth
- sdkgames 3y ago
  The only effective way to raise the confidence level of a program significantly is to give a convincing proof of its correctness. [0] Edsger W. Dijkstra CONTEXT: Argument three is based on the constructive approach to the problem of program correctness. Today a usual technique is to make a program and then to test it. But: program testing can be a very effective way to show the presence of bugs, but is hopelessly inadequate for showing their absence. The only effective way to raise the confidence level of a program significantly is to give a convincing proof of its correctness. But one should not first make the program and then prove its correctness, because then the requirement of providing the proof would only increase the poor programmer’s burden. On the contrary: the programmer should let correctness proof and program grow hand in hand. Argument three is essentially based on the following observation. If one first asks oneself what the structure of a convincing proof would be and, having found this, then constructs a program satisfying this proof’s requirements, then these correctness concerns turn out to be a very effective heuristic guidance. By definition this approach is only applicable when we restrict ourselves to intellectually manageable programs, but it provides us with effective means for finding a satisfactory one among these. [0] Edsger Dijkstra - Turing Award Lecture - The Humble Programmer - 1972 https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD340.html https://www.cs.utexas.edu/~EWD/transcriptions/EWD03xx/EWD340...

Solvency 3y ago

While I agree about proofs it's not intuitive to me. A triangle can clearly be visualized as half a square. Look at any videogame square made of 2 polygons. A 2D projection of a cone isn't intuitively 1/3 of a cylinder with the same dimensions.

gniv 3y ago

Here's an attempt at making sense of that intuition. Let's compute the volume of a pyramid that's inside a cube. Imagine that you're inside a cube room (this will make it easier to identify the faces). You are sitting in a corner. From your corner draw lines to the four ceiling corners -- a cube diagonal, two face diagonals and a cube edge. This is the pyramid. Now draw lines to the corners of one of the walls opposite you -- it's the same pyramid, but on its side. And lastly draw lines to the other wall opposite you -- again the same pyramid. You have now covered the entire cube with three identical pyramids, so the volume of the pyramid is a third of the volume of the cube. The interesting part is that the bases of these pyramids are on the three dimensions, giving the intuition that the /3 is due to the dimension. This proof might even generalize in higher dimensions. Edit: I think this is also the gist of the top answer in TFA, but it's weirdly formulated imo.

justinpombrio 3y ago
That's a beautiful proof! And yeah, it generalizes to higher dimensions. If you're in the corner of an N-dimensional hypercube, then the hypercube has 2n faces of dimension n-1. You're touching half of these faces, leaving n faces to form pyramids out of. And just like in the 3D case, they're arranged symmetrically so they must all have the same volume. EDIT: got around to reading TFA and it's also the top answer, so there's a nice visualization. Diving deeper into the proof, say the hypercube is a unit hypercube, spanning from (0, ..n, 0) to (1, ..n, 1) -- where "..n" means "a sequence of length n". Then: Each corner is a point (b_1, ..n, b_n) where each b_i is 0 or 1. You're located at (0, ..n, 0). The hypercube has 2n faces, which can be represented as a pair (i, b) where 0<=i<n and b is 0 or 1. The face (i, b) touches the 2(n-1) corners whose i'th coordinate is equal to b. You touch half of those faces: specifically the faces (i, 0) for each 0<=i<n. You can draw a pyramid to the other half of the faces, since you don't touch them. These pyramids must all have the same volume, by the symmetry of coordinate permutations, which is an operation that preserves volume.
dadoomer 3y ago
Very cool! Thanks for sharing.
dp-hackernews 3y ago

diffeomorphism 3y ago

It is exactly that and you even can use the same proof for a triangle, a cone or higher dimensional analogies. You always get 1/dimension. Proof: Take height 1 for simplicity of notation. Then we get the volume as the integral over crosssections (Fubini). Each cross section is rescaled by a factor (1-z), where z is the height. Rescaling changes the cross sections measure by (1-z)^(n-1). Integrate that to get 1/n. Done.

JadeNB 3y ago

It makes sense that the number of dimensions ends up somewhere, but I think there's no particularly obvious a priori reason why it should be a denominator. For example, the length of the long diagonals of a unit cube in n dimensions depends on the dimension, but it's \sqrt n, not anything divided by n (well, it's n^{3/2} divided by n, but that's not much use). Of course one can fill in why in this case it happens to be the dimension in the denominator—several sibling posts, particularly gniv's (https://news.ycombinator.com/item?id=36562093 https://news.ycombinator.com/item?id=36562093), have done so nicely—but I was responding specifically to the disclaimer: > I don't give a rat's ass if there's a proof or not connecting the two. I'm not a mathematician, but it makes sense from a calculus perspective that the number of dimensions ends up as a divisor. which seemed to suggest that it should be obvious without mathematical reasoning.