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Comment by Jtsummers | Hacker News Reader

https://aitopics.org/download/classics:E29CE08E - Source of the image

The image being discussed is on page 16 of the PDF (p. 310 by the document's page numbers).

EDIT: From my very quick, pre-coffee and woken-up-early-by-the-cats reading:

To everyone saying, "Well, if we choose a different set of assumptions it becomes possible." Yes. The discussion in the paper I link above goes into the assumptions used and rationale for why figures would be impossible in the context of his discussion. Basically, if you start with the image and treat it as an accurate (as accurate as it can be, a necessarily lossy process in most cases) 2d representation of a 3d object/scene, is there a valid 3d interpretation? In the case of the pyramid, with the assumption that it is an accurate drawing of a 3d pyramid, it's an impossible 3d pyramid. You'd have to add more information for it to become possible.

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Too late to edit again, but here is one of the critical assumptions from the original paper:

> One assumption we shall make throughout this paper is that all pictures are taken from a 'general position'; that is, that a slight change of the position from which the picture is taken would not change the number of lines in the picture or the configurations in which they come together. In the case of pictures of polyhedra this eliminates the possibility of pictures in which two vertices of the objects in the scene are, by coincidence, represented at the same point in the picture, or two edges in the scene are seen as a single line in the picture, or a vertex is seen exactly in line with an unrelated edge. [p. 298]

This is important, since, again, it addresses a lot of the comments here on how to make the image represent a possible object/scene. With this assumption, the "pyramid" is impossible. In the next paragraph (same page) Huffman goes on to address this:

> Furthermore, if this assumption leads us to judge as impossible an object or set of objects which we know to exist (and therefore by definition 'possible') we can conclude that the camera was probably not in a general position (or that some other assumption was unjustified). In that case we can either move the camera slightly and retake the picture, or go to an augmented list of local configurations which are possible and reanalyze the picture accordingly.

But there's a solution where the camera is in a general position. See this thread:

https://news.ycombinator.com/item?id=29877311

Which I was also in, just look at Huffman's paper, it's not that long, which lays out his various assumptions which lead to why the figure would be an impossible pyramid. That it has a valid interpretation does not make it suddenly a possible pyramid just a possible polyhedron, if you introduce information that isn't evident in the image.

There's no sense in discussing Huffman's Impossible Pyramid without also considering Huffman's definition for impossible. If we discard his definition, what we're talking about is something else entirely.

Sure, and I addressed your point there[1], that this still requires a non-standard, new definition of "impossible figure".

Furthermore, I don't see how it introduces information that "isn't evident in the image" when several people automatically assumed a 4 sided base (me among them) without trying to be clever.

[1] https://news.ycombinator.com/item?id=29879862

"assumed", because it wasn't presented. It wouldn't need to be assumed if it were present. That's pretty much the definition of assuming.

It's "assumed" only in the sense that the form of any occluded part of an image is "assumed". By that standard, "ACFD is a face/plane" isn't presented either, and, relative to the image alone, is "assumed" in the same sense.

Again, I'm not saying that this blog post presented the problem statement effectively. That is literally why I linked the original. So that we don't have to assume anything about what Huffman meant in his formulation because he stated his assumptions and definitions. I'm only saying that given Huffman's formulation it is an impossible pyramid. All the other possible "solutions" are not solutions to how this can be a pyramid, instead they are solutions to how it can be something else. Which is covered in Huffman's paper, which you apparently cannot be bothered to read.

To summarize: If this were a real image of a real object, then it is, by definition, a possible object, but it is not, per the image, a possible pyramid. The camera would have to be shifted (or lighting altered or something changed) in order to discern what it actually was (which could be a pyramid if some distortion were discovered, but could also be any of a number of other possible objects, including, as one person suggested, the corner of a room through a peculiarly shaped viewport).

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