How to build and solve a 4D Rubik's cubes in physical 3D (no simulator!)

I’ve been meaning to make this video about building and solving physical 4D Rubik’s cubes ever since Melinda Green sent me one of her brilliant physical 2x2x2x2s back in 2017! Why did it take so long? Well, this one was especially tricky to get right, and I think it’s probably the video that took me the longest to put together. I really hope you like this one :) 00:00 Intro 03:53 Warm-up 08:59 4d 14:13 Fancy moves 20:24 Solve 26:20 Melinda 31:27 Coding challenge 31:46 What's next? 33:54 Gallery of animations 36:10 Thank you ! Ed Collen's amazing hedgehog simulator (now with full macro support, avoid Chrome browser, also see my macros for this simulator at the bottom of this blurb) https://2x2x2x2.vercel.app/cube Mitchell Manning's similarly amazing but quite different simulator (avoid Chrome browser) https://rebelkeithy.github.io/TheHedgehog/ Burkard's hedgehog simulator (just permutations, avoid Chrome browser) https://www.qedcat.com/2x2x2x2%20hedgehog Melinda's 2x2x2x2 home page (go there straightaway !) https://superliminal.com/cube/2x2x2x2/ Melinda's YouTube page. Check out her 4D Twisty puzzle playlist. https://www.youtube.com/c/melindagreen Zasharan2 Melinda's 2x2x2x2 simulator (now also with some macros for twisting corners of the 2x2x2 sides, swapping corners of the 2x2x2 sides, resolving the half-turn parity, and performing a monoflip, just in case you need them/would like to experiment) https://zasharan2.github.io/2x2x2x2 A playlist of different ways to perform a gyro in Melinda's puzzle https://www.youtube.com/watch?v=d2Fh_1m0UVY&list=PLx1mIVtz33hJbAiFSsfsQ_IlAB1_fuAVc Joel Karlsson animation showing the correspondence between a different unfolded version of the real 2x2x2x2 and Melinda's puzzle. https://www.youtube.com/watch?v=QvhkGRUeGco Everything about higher-dimensional twisty puzzles https://hypercubing.xyz MagicCube4D is a fully functional four-dimensional analog of Rubik's cube plus dozens of other beautiful 4D puzzles. https://superliminal.com/cube/ Relevant Mathologer videos: Cracking the 4D Rubik's Cube with simple 3D tricks (solving the 3x3x3x3 in MagicCube4D) https://www.youtube.com/watch?v=yhPH1369OWc Can you solve THE Klein Bottle Rubik's cube? https://youtu.be/DvZnh7-nslo A simple trick to design your own solutions for Rubik's cubes https://youtu.be/-NL76uQOpI0 One of my Mathematica files---an animation of the hedgehog giro (.nb and .pdf) https://www.qedcat.com/4dRubik Hyperspeedcube, the name says it all https://hypercubing.xyz/software/hyperspeedcube/ Piles of other simulators https://hypercubing.xyz/software/ Grant Staten solves Melinda's 2x2x2x2 in under a minute https://www.youtube.com/watch?v=eIEu38wsjtM The Unpopular Cuber solves the 3x3x3x3 using a simulator (the crazy action clip at the end of this video are a couple of seconds from this solve) https://www.youtube.com/watch?v=2GPtYwmSeIU Some mathematical articles worth checking out: The Rubik tesseract by H. J. Kamack and T. R. Keane https://udel.edu/~tomkeane/RubikTesseract.pdf Rubik's Tesseract by Dan Velleman in Mathematics Magazine (1992), 65:1, 27-36. https://www.jstor.org/stable/2691357 n-dimensional sequential move puzzle https://en.wikipedia.org/wiki/N-dimensional_sequential_move_puzzle Thank you very much to Melinda for all her help with this video. Also thank you to Vivian and Cristian from Monash's FutureLab for their help with 3d printing the connectors that hold the cubies in my hedgehog together. Nice insight: Two swaps of the same adjacent cubies in one of the hedgehog sides results in a half-turn of the other side. Try in one of the simulators. Music: Morning Mandolin by Chris Haugen Enjoy! Burkard P.S.: Here are my macros if you want to use them in Ed's hedgehog simulator (save as a .json file and import). { "groups": [ { "name": "Ungrouped", "macros": [ { "name": "corner twist", "steps": "Ly' Ix' Lz' Ix' Lz Ix Lz' Ix Lz Ix' Lz' Ix' Lz Ix Lz' Ix Lz Lz Ix Lz' Lz' Ix' Lz Ix' Lz' Ix Lz Ix Lz' Ix' Lz Ix' Lz' Ix Lz Ix Lz Ix' Lz' Ly" }, { "name": "corner swap", "steps": "Lz Ly Lz Lz Ix' Lz' Lz' Ly' Ix' Ly Lz Lz Ix Lz' Lz' Ly' Ix Ly Lz Lz Ix Lz Ix' Lz' Ix Ix Lz' Lz' Ly' Ix Ly Lz Lz Ix Lz' Lz' Ly' Ix Ly Lz Lz Ix' Lz' Lz' Ly' Ix' Ly Lz Lz Ix Lz Ix Lz Ly' Lz'" }, { "name": "halfturn parity", "steps": "Ix Lz Ix Lz' Ix' Lz Ix Lz' Ix Lz Ix Lz' Ix' Lz Ix Lz' Ix Lz Ix Lz' Ix' Lz Ix Lz' Ix Lz Ix Lz' Ix' Lz Ix Lz' Ix Lz Ix Lz' Ix' Lz Ix Lz'" }, { "name": "monoflip", "steps": "Ly' Ix' Lz' Ix' Lz Ix Lz' Ix Lz Ix' Lz' Ix' Lz Ix Lz' Ix Lz Lz Ix Lz' Lz' Ix' Lz Ix' Lz' Ix Lz Ix Lz' Ix' Lz Ix' Lz' Ix Lz Ix Lz Ix' Lz' Ly xy Lz' yx Ix Ly' Lz Ix Lz' Ix' Lz' Ix' Lz Ix Lz' Ix Lz Ix' Lz' Ix' Lz Ix Lz' Ix Lz Lz Ix' Lz' Lz' Ix' Lz Ix' Lz' Ix Lz Ix Lz' Ix' Lz Ix' Lz' Ix Lz Ix Ly Ix' xy Lz yx" } ] } ] }