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I personally don’t see the image as “impossible”, in terms of seeing it as a projection of a 3D object. I didn’t interpret ABDE as being flat (A, B, D, and E c

by psanan 5y ago

I personally don’t see the image as “impossible”, in terms of seeing it as a projection of a 3D object. I didn’t interpret ABDE as being flat (A, B, D, and E coplanar), and I didn’t expect G, I, and H to be intersections of lines in 3D.

jerf 5y ago
If my "geometric intuition" is working properly, the "problem" is that the figure in the picture wouldn't meet in a point. There would be a line at the top, and it wouldn't be a pyramid. But there's nothing "impossible" about that. The impossibility simply seems to be an assertion of impossibility. It feels like it's a problem similar to spending to much time doing "2 + 5 = _" problems and thinking the equality symbol is directional, in this case, spending too much time looking at figures that do meet at a point and thinking that is obligatory for all figures.
- thaumasiotes 5y ago
  > If my "geometric intuition" is working properly, the "problem" is that the figure in the picture wouldn't meet in a point. There would be a line at the top, and it wouldn't be a pyramid. But there's nothing "impossible" about that. The proof given in the article seems fine. Assuming the figure has three flat faces, the arrangement of those faces is impossible. A figure such as you describe, with a line on the top, would not be ruled out by the proof, but the depicted figure cannot match that description. For a quick summary-style restatement of the proof: 1. Consider the three sides (as opposed to the top and bottom) of the shape to be flat. Each of them will come to a separate point. Those three points are labeled G, H, and I. 2. We can easily show that the point G lies in the same plane as each side of the shape. We can symmetrically show that this is also true of H and of I. 3. When G, H, and I are the same point, this doesn't restrict the sides in any meaningful way - no matter what the "angles" of three planes are, you can always translate them such that they'll all intersect at an arbitrary point. 4. But when G, H, and I are all different points, there is only a single plane that contains them all. ("Three points determine a plane".) This tells us that the three faces of such a shape would all be coplanar, which obviously can't happen. ----- (5. You are positing that, for example, G and H might coincide while I is a different, second point. But the depicted figure doesn't satisfy that description.)
  - thaumasiotes 5y ago
    Try approaching the problem from another direction: imagine you're positioning three planes (which will form the side faces of the truncated pyramid, but imagine infinite planes). How would you position them so that they didn't come to a single point? It can't be done; the first two planes will form an infinitely long "trough" in more or less the shape of a Λ (well, an X, but we're only interested in the part below the intersection), and then, wherever the third plane cuts through, you have the single point that a three-sided pyramid requires.
    - JumpCrisscross 5y ago
      > It can't be done Cheeky: it can, but they must be parallel.
      - thaumasiotes 5y ago
        Good luck positioning parallel planes to form the sides of a pyramid. ;D
- norrius 5y ago
  Sure, if you want to allow non-flat, curved faces, this body is possible. I'd argue this is not in the spirit of the question, similar to the triangle statue mentioned in the article.
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  - marcosdumay 5y ago
    > I'd argue this is not in the spirit of the question What question? I don't see any question. I only see the drawing of a solid and a statement that it's impossible for such solid to exist. And the only thing that is supposed to make the solid impossible is that it's named a "pyramid". What only means that the author uses a definition of that word that is more lenient than the strict usage I see in use, and more strict than the lenient usage. It's an interesting math problem, that exists on the contexts of its definitions (like any other). But given that the definitions aren't stated, it's not reasonable to expected people to come aware of them.
    - Jtsummers 5y ago
      > It's an interesting math problem, that exists on the contexts of its definitions (like any other). But given that the definitions aren't stated, it's not reasonable to expected people to come aware of them. That's why I linked to the the definitions in my own comment: https://news.ycombinator.com/item?id=29875085 https://news.ycombinator.com/item?id=29875085 https://aitopics.org/download/classics:E29CE08E https://aitopics.org/download/classics:E29CE08E This entire thread is mostly suffering from excessive pedantry because the linked blog failed to properly frame the problem.
      - WhitneyLand 5y ago
        >entire thread is mostly suffering from excessive pedantry It is? The comments here don’t seem pretentious and dogmatic to me, I prefer to use pentantry for cases where basically people can tell they’re being a little bit of a dick. It seems here in the comments people are simply saying, it’s hard to see the contradiction, that they can’t see any contradiction, and I think their implication is not to be a dick, it’s to hope someone will reply and say well here’s how it works or to correct a mistake. In other words to simply get to the bottom of understanding.

by psanan 5y ago

jerf 5y ago

If my "geometric intuition" is working properly, the "problem" is that the figure in the picture wouldn't meet in a point. There would be a line at the top, and it wouldn't be a pyramid. But there's nothing "impossible" about that. The impossibility simply seems to be an assertion of impossibility. It feels like it's a problem similar to spending to much time doing "2 + 5 = _" problems and thinking the equality symbol is directional, in this case, spending too much time looking at figures that do meet at a point and thinking that is obligatory for all figures.

thaumasiotes 5y ago
> If my "geometric intuition" is working properly, the "problem" is that the figure in the picture wouldn't meet in a point. There would be a line at the top, and it wouldn't be a pyramid. But there's nothing "impossible" about that. The proof given in the article seems fine. Assuming the figure has three flat faces, the arrangement of those faces is impossible. A figure such as you describe, with a line on the top, would not be ruled out by the proof, but the depicted figure cannot match that description. For a quick summary-style restatement of the proof: 1. Consider the three sides (as opposed to the top and bottom) of the shape to be flat. Each of them will come to a separate point. Those three points are labeled G, H, and I. 2. We can easily show that the point G lies in the same plane as each side of the shape. We can symmetrically show that this is also true of H and of I. 3. When G, H, and I are the same point, this doesn't restrict the sides in any meaningful way - no matter what the "angles" of three planes are, you can always translate them such that they'll all intersect at an arbitrary point. 4. But when G, H, and I are all different points, there is only a single plane that contains them all. ("Three points determine a plane".) This tells us that the three faces of such a shape would all be coplanar, which obviously can't happen. ----- (5. You are positing that, for example, G and H might coincide while I is a different, second point. But the depicted figure doesn't satisfy that description.)
- thaumasiotes 5y ago
  Try approaching the problem from another direction: imagine you're positioning three planes (which will form the side faces of the truncated pyramid, but imagine infinite planes). How would you position them so that they didn't come to a single point? It can't be done; the first two planes will form an infinitely long "trough" in more or less the shape of a Λ (well, an X, but we're only interested in the part below the intersection), and then, wherever the third plane cuts through, you have the single point that a three-sided pyramid requires.
  - JumpCrisscross 5y ago
    > It can't be done Cheeky: it can, but they must be parallel.
    - thaumasiotes 5y ago
      Good luck positioning parallel planes to form the sides of a pyramid. ;D
norrius 5y ago
Sure, if you want to allow non-flat, curved faces, this body is possible. I'd argue this is not in the spirit of the question, similar to the triangle statue mentioned in the article.
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- marcosdumay 5y ago
  > I'd argue this is not in the spirit of the question What question? I don't see any question. I only see the drawing of a solid and a statement that it's impossible for such solid to exist. And the only thing that is supposed to make the solid impossible is that it's named a "pyramid". What only means that the author uses a definition of that word that is more lenient than the strict usage I see in use, and more strict than the lenient usage. It's an interesting math problem, that exists on the contexts of its definitions (like any other). But given that the definitions aren't stated, it's not reasonable to expected people to come aware of them.
  - Jtsummers 5y ago
    > It's an interesting math problem, that exists on the contexts of its definitions (like any other). But given that the definitions aren't stated, it's not reasonable to expected people to come aware of them. That's why I linked to the the definitions in my own comment: https://news.ycombinator.com/item?id=29875085 https://news.ycombinator.com/item?id=29875085 https://aitopics.org/download/classics:E29CE08E https://aitopics.org/download/classics:E29CE08E This entire thread is mostly suffering from excessive pedantry because the linked blog failed to properly frame the problem.
    - WhitneyLand 5y ago
      >entire thread is mostly suffering from excessive pedantry It is? The comments here don’t seem pretentious and dogmatic to me, I prefer to use pentantry for cases where basically people can tell they’re being a little bit of a dick. It seems here in the comments people are simply saying, it’s hard to see the contradiction, that they can’t see any contradiction, and I think their implication is not to be a dick, it’s to hope someone will reply and say well here’s how it works or to correct a mistake. In other words to simply get to the bottom of understanding.