The analysis is wrong. There is no reason to believe that AD and BE intersect, etc.
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Explaining Huffman’s Impossible Pyramid
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Re: Explaining Huffman’s Impossible Pyramid
#32The drawing appears to represent a polyhedron with two triangular faces and three quadrilateral faces. It appears to me to be 1 triangular face and 4 quadrilateral ones. The one they list as ABC should be ABCX where X is a hidden fourth corner on the base. Which would render the shape possible.
From linked article:
> Edit: Greg Ross suggests that the figure might be possible if there is another hidden edge. Can anyone explain this?
This would appear to be the simple example of such.
Re: Explaining Huffman’s Impossible Pyramid
#33The analysis is wrong. There is no reason to believe that AD and BE intersect, etc.
If AD and BE don't intersect then the quadrilateral ADBE doesn't lie in a plane, and instead you have a curved surface there, which is not what you'd expect from the diagram. The idea is that if you assume all of these surfaces are flat, nothing lines up and the shape can't exist. Certainly you can make something that looks like this, but it won't have perfectly flat surfaces everywhere.
It states that "The drawing appears to represent a polyhedron with two triangular faces and three quadrilateral faces."
There is no reason to believe it actually is a quadrilateral. The initial question just gives you a drawing...
It is not a pyramid either, since all the lines would intersect at the apex.
So what you have is a pyramid-like drawing and some wrongful analysis of it ¯\_(ツ)_/¯
The only other interpretation is that this is just a rough sketch and actually G=H=I and the analysis again is incorrect, since GHI isn't a triangle, it's the same point, the apex, which thus can exist in both planes.
Re: Explaining Huffman’s Impossible Pyramid
#34I don't think it contradicts the discussion, but there might be a rotation of the figure where all the lines appear to intersect. For this reason, while pyramid is demonstrably impossible, the converse is not the case: seeing the lines intersect doesn't necessarily mean it's possible.
Re: Explaining Huffman’s Impossible Pyramid
#35Earlier quoted context omitted.
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 symbo…
> 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,…
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.
Re: Explaining Huffman’s Impossible Pyramid
#36Earlier quoted context omitted.
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 symbo…
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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Re: Explaining Huffman’s Impossible Pyramid
#37It's an object that could exist, except that ABED would not be a plane. It's a twisted plane. I don't know what the name of that shape is but you can take a plane and twist it in a way that every point along a straight line between any two points on the surface is also on the surface, but yet the surface is not an actual plane.
Edit: Go to wolfram alpha and graph
graph y=(z-zx)/2 from x=-1 to x=3 and z=-1 to z=1
and you'll see an example.
Re: Explaining Huffman’s Impossible Pyramid
#38The implied fact, which makes it an impossible figure, is the assumption that ACFD is a plane. If, on the other hand, there is an edge CD or an edge AF, the figure becomes possible. That’s what my intuition tells me, in any case.
Yes! Thank you! I was wondering what's so impossible about this and I think that's what I was missing -- yes, if you go with the assumption that the back side is a plane, then it's not possible, but that was never my initial assumption. I interpreted it as like a ... 3D trapezoid (?), with a rear support for D and fourth point on its base (though maybe that has its own issue).
This is IMHO very different from usual "impossible figure" drawings where there is no interpretation that makes it work.
Re: Explaining Huffman’s Impossible Pyramid
#39The drawing appears to represent a polyhedron with two triangular faces and three quadrilateral faces. It appears to me to be 1 triangular face and 4 quadrilateral ones. The one they list as ABC should be ABCX where X is a hidden fourth corner on the base. Which would render the shape possible.
Downvoted, but can anyone explain how it's impossible if the base is quadrilateral? [1] [1] https://imgur.com/a/C3tDi50
Thank you! That's what I had in my in this comment: https://news.ycombinator.com/item?id=29879788
Re: Explaining Huffman’s Impossible Pyramid
#40Earlier quoted context omitted.
Downvoted, but can anyone explain how it's impossible if the base is quadrilateral? [1] [1] https://imgur.com/a/C3tDi50
I upvoted your original comment because it shouldn't have been downvoted. But I'd suggest reading the paper I linked elsewhere in this discussion which helps to explain why Huffman called it impossible. The image is "impossible" under the assumptions of that paper. If it were known to be a real object, then it implies that something about the image is wrong (there is hidden information, like the edge you add in your…
Okay, it's fair to call it impossible if you restrict the domain like that. But, per my other comment, this feels like a non-standard use of "impossible figure", as the conventional meaning is such that, just from the image, there is no reasonable figure that results in that image. That's not the case here.