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.
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.
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.
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(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.)
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.
Cheeky: it can, but they must be parallel.
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.
That's why I linked to the the definitions in my own comment:
https://news.ycombinator.com/item?id=29875085
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.
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.
> 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.
You know, we aren't supposed to accuse people of not reading the article.
But this is what the article says:
> The drawing appears to represent a polyhedron with two triangular faces and three quadrilateral faces. The triangular faces are ABC and DEF. The quadrilateral faces are ABED, BCFE, and CADF. It also appears that AD and BE intersect at I, AD and CF intersect at G, and BE and CF intersect at H. If we accept this interpretation of the drawing, then the shape that it represents is impossible.
It would be difficult to be more explicit about the definitions.