Explaining Huffman’s Impossible Pyramid
mathblag.wordpress.com
mathblag.wordpress.com
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.
> 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.
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.
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.
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).
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.
That’s what my intuition tells me, in any case.
This is IMHO very different from usual "impossible figure" drawings where there is no interpretation that makes it work.
In particular with you observation that this is a different class of "impossible figures."
Though I can also understand the counter-argument, that one can also construct objects which from a privileged perspective also "technically" are possible solutions for such figures.
I believe the (compelling) counter-counter-argument is that such solutions are AFAIK unique to privileged perspectives, but in this case, there is a whole set of such perspectives. You can rotate the thing through quite a range and still assert you're looking at an impossible pyramid.
Adding a single edge to defeat the premise that it's a "pyramid" is I suspect a formalizable distinction which reduces the impossibility (as others have said) to whether you want to hinge all on the word "pyramid."
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.
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.
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.
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.
> 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.
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.
There is no such "ADFC" face that can result in a triangle in the shape of DEF.
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To imagine what "ADFC" could possibly look like, I suggest trying to visualize the ABC-base at first, and then trying to imagine different "ADFC" faces to match the ABC-triangle with the DEF-triangles.
The only way you can get this shape to look like this, is if ADFC is bent or twisted somehow. No "straight/flat" ADFC could possibly connect the bottom with the top.
In order for the shape to be impossible ADEB, BEFC and ACFD have to be coplanar. If you add a hidden AF or CD edge, making ACFD not coplanar, it becomes possible.
To make things simple, let's assume a simple, orthogonal projection (discard the z). We will use degrees of freedom for that. All vertices have their x, y position fixed, that's their 2D coordinates in the drawing, we don't know z yet because it is discarded by the projection. The problem becomes: for each vertex, find the z coordinate. The constraints are: ADEB, BEFC are coplanar, DBFE are not (making the entire thing a flat shape would be cheating).
So, make the z coordinates for D, B, F, E anything we want, it will make a small tetrahedron, nothing wrong with that. Now, because A has to be in the DEB plane, its z coordinate is fixed, but it can be calculated, no problem. Same thing with C, it is in the BEF plane and its z coordinate is fixed and can be calculated. You can do that in every case. So as long as you are not looking behind the "pyramid", you can always find a 3D shape that matches the projection.
Now if we add the constraint that ACFD are coplanar, that's when you have a problem. All points are fixed, and you have to play with your 4 degrees of freedom that are the z coordinates of D, B, F, E to make ACFD coplanar. Playing with the z, you can do translation, scaling on the z axis and shearing on xz and yz, that's 4 independent transforms, all your degrees of freedom are used up. None of them help making your vertices coplanar, except if you flatten everything, which, as said before, is cheating.
Now, maybe we can also make the impossible possible using fancier projections. Perspective projection has a FOV parameter, it may be an extra degree of freedom we can play with, but that enough maths for today.
EDIT: missed a transform, now, it works
I'd wonder if there is some topology theorem that generalizes hamiltonian paths into higher dimensions, where the difference between a graph and a solid is whether it has a hamiltonian path. e.g. this shape is "impossible" as represented because of the lack of a hamiltonian path.
For it to be a real pyramid, you only need the side edges lines to meet at the apex, which always happen when the top and bottom triangles are similar.
Similar triangles are a sufficient, but not necessary condition.
If G isn't on line AD, then BE and CF can't be a side of the same truncated pyramid.
Yes, I can also recommend Futility Closet for HN readers.
It becomes possible again if you cut the front face into two triangles, that being the "hidden edge" the author mentions.
The pyramidal explanation sounds like it's using a theorem that I didn't know before, but without proving it. You could say the explanation deserves its own explanation. It's an interesting fact, that the lines have to intersect, nonetheless.
The quasi-pyramid is impossible if you assume that there are no hidden nodes and edges. That assumption is not an explicit requirement, so you can posit any number of nodes and edges that are hidden from the specific perspective of the drawing -- it could be as crenellated as the interior of an English muffin.
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.
My intuition tells me that there's an extra node not pictured, let's call it Z. There's a triangle ADZ that's out of view and a planar surface DZCF that mirrors DABE.
I'm sure my intuition is wrong too, but without pulling out some kind of CAD I can't really see why.
1. A, D, E, B lie in one plane
2. B, E, F, C lie in one plane
3. A, D, F, C lie in one plane
Any of these assumptions can be wrong. It's easy to perceive that number 3 is not flat, if you think that 1 and 2 are flat.
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.