https://mathblag.wordpress.com/2015/03/11/explaining-huffmans-impossible-pyramid/ mathblag * About Musings on mathematics and teaching. Explaining Huffman's Impossible Pyramid by David Radcliffe I read about Huffman's Pyramid from the consistently excellent blog Futility Closet. Huffman's Pyramid is a drawing of a figure that cannot exist. 2015-03-10-huffmans-pyramid-1 However, the impossibility of this figure is hardly obvious. Here is the reason: if the slanting lines were extended, then they would have to meet at the apex of a pyramid. However, the lines do not meet. Contradiction! 2015-03-10-huffmans-pyramid-21 Are you convinced yet? If you are, then your geometric intuition is stronger than mine, and you can probably stop reading now. But if you are skeptical, as I was, then please continue reading. First, we should ask this question: What exactly do we mean by an impossible figure? Surely a figure can't be impossible if we can draw it. Sometimes we can even make a physical model of an impossible object, such as this impossible triangle sculpture in Perth. perth_tri_front I think that what we mean by an impossible figure is that the figure appears to have contradictory properties. The sculpture shown above is not actually a triangle, but it appears to be a triangle from a certain angle, and this apparent shape is inconsistent with other things that we can observe about the sculpture. Let's consider Huffman's Pyramid again. We'll label the points to make it easier to talk about. 2015-03-10-huffmans-pyramid-21 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. Here's why. Consider the plane P containing the face ABED. The plane contains G, since G is on the line AD. The plane also contains H and I, since they are on the line BE. Therefore, P contains the triangle GHI. Similarly, the plane containing the face BCFE must also contain the triangle GHI. But there is only one plane that contains the triangle GHI, so the two planes must be the same. Therefore, the entire shape lies in a plane, and it cannot be a polyhedron. This seems to suggest that pentahedra can't be too irregular. It would be interesting to explore this notion further. Edit: Greg Ross suggests that the figure might be possible if there is another hidden edge. Can anyone explain this? Share this: * Print * Email * More * * Tumblr * Pinterest * * LinkedIn * Facebook * * Reddit * Twitter * * Like this: Like Loading... Related Published: March 11, 2015 Filed Under: Uncategorized 5 Comments to "Explaining Huffman's Impossible Pyramid" 1. [5e840] David Radcliffe says: March 12, 2015 at 1:06 am I get it now. If the back face has a hidden edge (AF or CD) then the lines AD and CF are skew, so they do not cross at G. Reply 2. [34f76] Joshua says: March 12, 2015 at 1:14 am 4th edge: How do you know that the points G, H, and I exist? Well, I and H are both the unique points of intersection of two coplanar lines (by assumption, ABED and BCFE are planar faces). However, G only exists if lines AD and FC are also coplanar. if there is more in the back, then AD and FC need not be coplanar and G doesn't exist. Reply 3. [8cb92] Rutha says: April 2, 2017 at 10:55 am Taylor Swift is a racist white supremacist she licked the KOKKK of Richard Spencer, Benoit Hamon and ANDREW ANGLIN and she voted for Donald Trump TWICE. Taylor Swift the hkb RACIST also listens to the necro music NECROPEDOSADOMASO!!! Google "Taylor Swift NecroPedoSadoMaso" and SEE FOR YOURSELF! The songs talk about jizzing in the pussy of 12 year old GIRLS!!! BOYCOTT TAYLOR SWIFT to fight FASCISM and PEDOPHILIA! Reply 4. [cab3d] Mathias Beinling says: October 18, 2017 at 6:17 am The three points E, F and B already define a plane. If you draw C outside that plane (like in the drawing above), it is no wonder that ED and FC do not meet. And no, they do not meet at H. One line is behind the other. Reply 5. [cropp] Explaining Huffman's Impossible Pyramid - pentestswap says: January 10, 2022 at 9:09 am [...] Read More [...] Reply Leave a Reply Cancel reply Enter your comment here... [ ] Fill in your details below or click an icon to log in: * * * * * Gravatar Email (Address never made public) [ ] Name [ ] Website [ ] WordPress.com Logo You are commenting using your WordPress.com account. ( Log Out / Change ) Google photo You are commenting using your Google account. ( Log Out / Change ) Twitter picture You are commenting using your Twitter account. ( Log Out / Change ) Facebook photo You are commenting using your Facebook account. 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