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Comment by drcongo | Hacker News Reader

The 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.

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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 diagram, or the viewport is unusual as described in another comment). Which would mean the camera or lighting needs to be adjust to reveal this hidden information to make the "pyramid" (potentially no longer a pyramid) possible.

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.

https://news.ycombinator.com/item?id=29879788

The context of the creation of this "impossible pyramid" is where the definition for "impossible" comes from here. Under Huffman's assumptions (the image is not deliberately misleading, there is no hidden information, each side lies entirely within a plane, and the image is meant to represent a polyhedron), the pyramid is, indeed, impossible. Relaxing his assumptions (which he discusses) would make it possible, but then, well, you've relaxed the assumptions (there's a hidden edge not seen, the image is meant to mislead, each side does not, in fact, lie entirely within a plane, etc.).

I don't know how you can call it misleading if my immediate, natural, non-clever interpretation of the image matches a valid possibility. If anything, it's the insistence that "no no it's got a be a [truncated] pyramid with three sides at the base" that feels bolted on and arbitrary.

Also, the solution I responded to adheres to "each side entirely within a plane"[1] and "is a polyhedron". To make this impossible, you have to add information -- the assumption that this must be a triangular pyramid -- that is not present in the image itself.

Furthermore, the solution here, does not require that you be looking at it at a very precise angle that gets everything to line up perfectly, as you'd need for e.g. the Penrose triangle: https://en.wikipedia.org/wiki/Penrose_triangle

Bottom line, it is a different kind of thing than usual impossible figures.

[1] Assuming I'm interpreting "side" correctly? But any definition where the solution violates it, also feels arbitrary.

I meant "face", not "side".

However, almost all the solutions to make this a possible polyhedron make it not a pyramid and operate under the assumption that there is hidden information (that is, that the image is misleading, deliberate or not). It still remains an impossible pyramid in Huffman's context.

>I meant "face", not "side".

Okay but if you're going to ground your position in "hey, this is just what you get when you're super rigorous about terms and assumptions and all", it's incumbent on you to be a little more careful about that.

>However, almost all the solutions to make this a possible polyhedron make it not a pyramid and operate under the assumption that there is hidden information (that is, that the image is misleading, deliberate or not). It still remains an impossible pyramid in Huffman's context.

And again, that's adding information that is not present in the image itself, the exact problem you claimed would invalidate the offered solution (in this subthread), and it's still a different definition of "impossible figure" than the conventional one.

Thanks, that definitely wasn't in my mind when I looked at the original diagram and instantly pictured the quadrilateral base that makes "possible".

Thank you! That's what I had in my in this comment: https://news.ycombinator.com/item?id=29879788

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.

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