What's the curse of the Schwarz lantern?

Second coop with Andrew. This time it's about the Schwarz lantern a very famous counterexample to something that mathematicians believed to be obviously true. A 3D cousin of the famous pi = 4 paradox. 00:00 Intro 00:39 Troll math: the pi=4 meme 02:25 Archimedes chops off corners 05:51 Archimedes boxing of pi 07:40 Schwarz lantern 16:59 Area formula 17:12 Schwarz pi = 4 memes 20:17 Folding flat 21:12 Andrew's VR experiment 22:24 Soda can --- lantern 23:26 Thanks 25:00 Andrew's Christmas tree The pi = 4 troll math meme on reddit: https://www.reddit.com/r/math/comments/e6q4r/troll_math_pi_4_crosspost/ Good discussion by GoldPlatedGoof https://static.nsta.org/pdfs/QuantumV1N4.pdf The Vi Hart video: https://youtu.be/D2xYjiL8yyE Closely related, the Staircase paradox: https://en.wikipedia.org/wiki/Staircase_paradox Good discussion of the history of this sort of paradox http://tinyurl.com/3pe2eaav Great wiki page on the Schwarz lantern: https://en.wikipedia.org/wiki/Schwarz_lantern Schwarz's original paper: https://archive.org/stream/gesammeltemathem02schwuoft#page/310/mode/2up Diamond-Cut Can https://www.toyo-seikan.co.jp/e/technology/can/decorationshape/diamondcut/ Lots of nice ideas (coffee cans, soda can creasing, collapsing cylinders, patent for effectively collapsing cans) https://beachpackagingdesign.com/boxvox/pseudo-cylindrical-concave-polyhedral-packaging The IKEA looking lantern features in the paper: Application of paper folding technique to three-dimensional space sound absorber with permeable membrane: Case studies of trial productions http://tinyurl.com/223948tz Great article in Quantum magazine (there is also an article about Gabriel's horn in the same issue): https://static.nsta.org/pdfs/QuantumV1N4.pdf Andrews Geogebra app in action: Schwarz Lantern in Geogebra 3D & AR: https://youtu.be/YyKX00VukFw Schwarz Lantern christmas tree: https://youtu.be/Ay1s4hizg1o Geogebra 3D AR mode on an iPad how-to: https://youtu.be/yQAiBxTGbAg Schwarz lantern origami with Deepali Karanjavkar: https://youtu.be/ar7dMtsA-dY Crushmetric water bottle (really impressive can crushing inspired toys): https://www.crushmetric.com/ Hydraulic press vs. metal pipe (square and circular) https://www.youtube.com/watch?v=TM5dyY8zfxs Mel Brooks: "May the Schwarz be with you" https://www.youtube.com/watch?v=-6cho-gsaZM Music today: Flat Rock by Bennett Sullivan T-shirt: Google hohoho cubed t-shirts for lots of different versions ((ho)³ A case where (ab)³ ≠ a³b³ :) Enjoy! Burkard