People are brainwashed by their prior experience: they have seen so many puzzles with specific assumptions that they assume them to exist here, as well.
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.
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 triangles don't have to be similar to make it a real pyramid. They only have to be similar of you want the truncation to be made parallel to the base.
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.
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.
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.
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.
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).
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.
-----
(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.)
Try approaching the problem from another direction: imagine you're positioning three planes (which will form the side faces of the truncated pyramid, but imagine infinite planes).
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.
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.
> I'd argue this is not in the spirit of the question
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.
> 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:
>entire thread is mostly suffering from excessive pedantry
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.
> I only see the drawing of a solid and a statement that it's impossible for such solid to exist.
> 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.
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."
Here, 'impossible' is defined very loosely, as 'not what it appears to be'. But what does it appear as? Try imagining it, per Necker Cubes, as a concave shape.
I understand the argument some people are making as "my intuition tells me it's impossible because $X" and that makes sense.
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.
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.
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.
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.
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.
It reminds me of a variation of the bridges of konigsberg problem where in this diagram as presented, it has an even number of vertices and edges, and an implied even number of faces, but the vertices have an odd number of edges, which breaks the need for the number of edges per vertex to be even to do a hamiltonian path. (not a mathematician, can't remember this rule specifically)
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.
I can see how it's impossible. If the top and right faces are planar, the front face can't be.
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.
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.
Funny, when I first looked at the sketch, I saw the front left face as curved (and extending the lines would result in a flat vertical surface at the top that spirals upward).
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.
It states that "The drawing appears to represent a polyhedron with two triangular faces and three quadrilateral faces."
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.
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!
--------
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.
-------
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.
It's an object that could exist, except that ABED would not be a plane. It's a twisted plane. I don't know what the name of that shape is but you can take a plane and twist it in a way that every point along a straight line between any two points on the surface is also on the surface, but yet the surface is not an actual plane.
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.
62 comments
[ 47.3 ms ] story [ 230 ms ] threadThe 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.
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.
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
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.
[1] https://news.ycombinator.com/item?id=29879862
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.
-----
(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."
Yes, I can also recommend Futility Closet for HN readers.
https://www.futilitycloset.com
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.
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
https://news.ycombinator.com/item?id=29879788
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.
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.
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.
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.
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.
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.
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.
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