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Explaining Huffman’s Impossible Pyramid

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Re: Explaining Huffman’s Impossible Pyramid

#21

Post by kd0amg »

petargyurov wrote: post #4

I didn't really get it at first, in fact, I might still not be getting it, but if you draw a line from A to C, then you can see that the triangle ABC will not be similar to DEF... when it should be? I don't know.

The issue is about which edges are in the same plane as which other edges. If this is a polyhedron, AD and BE must be coplanar, and BE and CF must be coplanar. Extending the edges into lines, they don't form a 3-way intersection. G gets projected to the same screen point as some point on line AD, but the only way G can be on line AD is if AD and FC are coplanar. If all three of AD, BE, and FC are coplanar, the entire figure is a flat pentagon (and DE, BE, and FE are not edges).

If G isn't on line AD, then BE and CF can't be a side of the same truncated pyramid.

Re: Explaining Huffman’s Impossible Pyramid

#22

Post by norrius »

jerf wrote: post #15
psanan wrote: post #6

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.

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.

Re: Explaining Huffman’s Impossible Pyramid

#23

Post by pentagonalhole »

It's a corner of a room, as seen through a pentagonal hole. The pentagonal hole is the planar pentagon ABCFD. The corner of the room is at E, where 3 planes meet. ED, EF and EB are the visible parts of the edges where two of the planes meet. The only constraint on the shape of the pentagonal hole is to be possible to be positioned such that, from the viewer point of view, the vertices B, F, D to be seen as if they are on the edges of the room corner.

Re: Explaining Huffman’s Impossible Pyramid

#24

Post by drcongo »

drcongo wrote: post #14

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.

Downvoted, but can anyone explain how it's impossible if the base is quadrilateral? [1]

[1] https://imgur.com/a/C3tDi50

Re: Explaining Huffman’s Impossible Pyramid

#25

Post by Jtsummers »

Jtsummers wrote: post #5

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 rationa…

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.

Re: Explaining Huffman’s Impossible Pyramid

#26

Post by Jtsummers »

drcongo wrote: post #24
drcongo wrote: post #14

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.

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.

Re: Explaining Huffman’s Impossible Pyramid

#29

Post by dragontamer »

If you draw a "dotted line" from "A to C", it becomes blatantly obvious that the backside must be curved or warped in some degree (where the "dotted line" represents the bottom base). Assuming that the front-faces are flat, the triangle on the top of the figure is too warped to match the readily assumed triangular base.

In fact, my brain is bouncing between many different interpretations of the base when I saw this figure (first one is a quadrilateral base, as discussed in the blogpost) I don't consider myself to be that good with geometry, but I did play with CAD for a bit back in college, which probably built my instincts up. I imagine that machinists and mechanical engineers would instinctively see the "impossibility" of flat-faces and a 3-sided bottom base.

Us computer engineers really won't work with real-world geometry enough to really get the instincts that those mechanical engineers get though. That's fine. They can taunt us with their superior geometry skills while I'll taunt them with my superior C++ skills!

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I did require the additional guidelines to see the problem. But an additional set of guidelines (ex: dotted-triangle base of ABC) would really make it more obvious.

Re: Explaining Huffman’s Impossible Pyramid

#30

Post by cylon13 »

righttoolforjob wrote: post #28

The 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.
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