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.
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.
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 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.
> 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, 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.)
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.
The 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.
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.
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'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.
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.
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.
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.
The "impossible" part is trying to imagine a "flat" ADFC face.
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.
I 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.
That thing has bothered me for a while and I think I finally found why this is impossible.
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.
Readit News
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.
https://news.ycombinator.com/item?id=29877311
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.
-----
(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.)
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.
[1] https://imgur.com/a/C3tDi50
> 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!
--------
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.
-------
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