[HN Gopher] Explaining Huffman's Impossible Pyramid
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Explaining Huffman's Impossible Pyramid
Author : signa11
Score : 57 points
Date : 2022-01-08 05:57 UTC (2 days ago)
(HTM) web link (mathblag.wordpress.com)
(TXT) w3m dump (mathblag.wordpress.com)
| [deleted]
| teddyh wrote:
| 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.
|
| That's what my intuition tells me, in any case.
| SilasX wrote:
| 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.
| aaroninsf wrote:
| I am in agreement with y'all.
|
| 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."
| analog31 wrote:
| 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.
| msandford wrote:
| 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.
| whatshisface wrote:
| The front plane is bent. You can bend stuff in real life, but
| it's implied by the drawing that it's supposed to be a plane.
| [deleted]
| dsr_ wrote:
| 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.
| [deleted]
| drcongo wrote:
| _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.
| smegsicle wrote:
| From linked article:
|
| > 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.
| drcongo wrote:
| Downvoted, but can anyone explain how it's impossible if the
| base is quadrilateral? [1]
|
| [1] https://imgur.com/a/C3tDi50
| SilasX wrote:
| Thank you! That's what I had in my in this comment:
| https://news.ycombinator.com/item?id=29879788
| Jtsummers wrote:
| 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.
| SilasX wrote:
| 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.
|
| https://news.ycombinator.com/item?id=29879788
| Jtsummers wrote:
| 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.).
| SilasX wrote:
| 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 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.
| drcongo wrote:
| Thanks, that definitely wasn't in my mind when I looked at
| the original diagram and instantly pictured the
| quadrilateral base that makes "possible".
| petargyurov wrote:
| 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.
| GuB-42 wrote:
| 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.
| kd0amg wrote:
| 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.
| righttoolforjob wrote:
| The analysis is wrong. There is no reason to believe that AD and
| BE intersect, etc.
| cylon13 wrote:
| 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.
| righttoolforjob wrote:
| 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 -\\_(tsu)_/-
|
| 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.
| [deleted]
| Claudus wrote:
| 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).
| dragontamer wrote:
| 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.
| red_trumpet wrote:
| If the top triangle is not parallel to the base (which you
| picture by your method), then the two triangles don't match,
| even in a real picture.
| dragontamer wrote:
| 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.
| [deleted]
| Jtsummers wrote:
| 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.
| Jtsummers wrote:
| 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.
| JKCalhoun wrote:
| > I read about Huffman's Pyramid from the consistently excellent
| blog Futility Closet.
|
| Yes, I can also recommend Futility Closet for HN readers.
|
| https://www.futilitycloset.com
| fallingfrog wrote:
| 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.
|
| 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.
| pentagonalhole wrote:
| 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.
| mannykannot wrote:
| 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.
| motohagiography wrote:
| 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.
| psanan wrote:
| 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.
| jerf wrote:
| 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.
| thaumasiotes wrote:
| > 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.)
| thaumasiotes wrote:
| 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 L
| (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.
| JumpCrisscross wrote:
| > _It can 't be done_
|
| Cheeky: it can, but they must be parallel.
| thaumasiotes wrote:
| Good luck positioning parallel planes to form the sides
| of a pyramid. ;D
| norrius wrote:
| 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.
| [deleted]
| [deleted]
| marcosdumay wrote:
| > 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.
| thaumasiotes wrote:
| > 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.
| Jtsummers wrote:
| > 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.
| WhitneyLand wrote:
| >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.
| dandanua wrote:
| The "impossibility" proof actually uses 3 assumptions:
|
| 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.
| whatshisface wrote:
| 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.
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