Five puzzles for thinking outside the box

Geometry puzzles that benefit from shifting the dimension. Bonus video with extra puzzles: https://www.patreon.com/posts/115570453 The artwork at the end is by Kurt Bruns Thanks to Daniel Kim for sharing the first two puzzles with me. He mentioned the earliest reference he knows for the tile puzzles is David and Tomei's AMM article titled "The problem of Calissons." The idea to include the tetrahedron volume example was based on a conversation with Po Shen Lo about these puzzles, during which he mentioned the case of one dimension lower. I received the cone correction to the proof of Monge's theorem from Akos Zahorsky via email. Also, the Bulgarian team leader Velian Velikov brought up the same argument, and just shot me a message saying "I came across it in a book I found online titled 'Mathematical Puzzles' by Peter Winkler. There, it is attributed to Nathan Bowler" I referenced quaternions at the end, and if you're curious to learn more, here are a few options. This is a nice talk targetted at game developers: https://youtu.be/en2QcehKJd8 This video walks through concretely what the computation is for using quaternions to compute 3d rotations: https://youtu.be/-zsnHbQyRnc My own video on the topic is mainly focused on understanding what they do up in four dimensions, which is not strictly necessary for using them, but for math nerds like me may be satisfying: https://youtu.be/d4EgbgTm0Bg Also, one of the coolest projects I've ever done was a collaboration with Ben Eater to make interactive videos based on that topic: https://eater.net/quaternions Timestamps - 0:00 - Intro - 0:32 - Twirling tiles - 6:45 - Tarski Plank Problem - 10:24 - Monge’s Theorem - 17:26 - 3D Volume, 4D answer - 18:51 - The hypercube stack - 25:52 - The sadness of higher dimensions SEV#3: https://youtu.be/vXBtyYvMx24 ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim https://github.co