Smale's inside out paradox

This week’s video is about the beautiful mathematics you encounter when you try to turn ghostlike closed surfaces inside out. Learn about the mighty double Klein bottle trick, be one of the first to find out about a fantastic new way to turn a sphere inside out and have another go at earning the Mathologer seal of approval by accepting the Mathologer inside out challenge.

Latest news (November 7, 2016): Arnaud Chéritat just finished an absolutely stunning animation of the deformation of the outer dome that I talk about in this video. Check it out! https://www.math.univ-toulouse.fr/~cheritat/eversion/Hacon/
Make sure you explore what the sliders can do and rotate the model around with your mouse.

More latest news (November 10, 2016): Arnaud just rerendered his torus eversion in HD using a colourblind friendly colour scheme. Here are links to the versions showing the full torus and the half torus https://youtu.be/INdOWVFb8fk
https://youtu.be/Cw4aTVi8ndQ

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Mathologer inside out challenge
1. Marco Souza de Joode, 2. Max Brain, 3. Cory Williams, 4. Stefan Linden, 5. Saelben Noa, 6. Mehmed Adzemonic, 7. Lachie Miles, 8. Rory McAllister, 9. Alejandro Robles, 10. Marcin Szyniszewski, 11. Sam Jones, 12. Jack Leightcap, 13. Christian Callau, 14. Richard Schank, 15. Daniel Feuerstein, 16. Irene Meunier, 17. Sinom, 18.Denny Eggroll, 19. Joshua Pirie, 20. Grillet Lucien, 21. Lea Werle, 22. Dominic Birkwood, 23. Andrei Maria, 24. Marco Rozendaal, 25. Khalis Totorkulov, 26. Kevin Tsang, 27. Thiasam, 28. Batonkal, 29. Grillet Lucien, 30. Arnaud Cheritat, 31. Cichy Wodór, 32. Manex Vallejo, 33. Matthew Giallourakis, 34. Eric K., 35. Kai Wolder, 36. Mei Li, 37. Mad Cuber, 38. Nelly Lin, 39. Sam Amber, 40. Devansh Sehta, 41. Samuraiwarm Tsunayoshi, 42. Joris van Duijneveldt, 43. Craig Montgomery, 44. Warren Brodsky, 45. Jonathan Fowler, 46. Nathan Petrangelo, 47. 정재윤 , 48. George Milis, 49. TrianguloY, 50. Potii92 (Daniel), 51. Ha Quang Trung, 52. Jerry Stoops, 53. Conall Kavenagh, 54. Dean Reichel, 55. Pavel Klimov, 56. Griffin Keeter, 57. Tian Chen, 58. frobeniusfg (Andrew), 59. Nathaniel Gofourth, 60. Benjamin Seidel, 61. Miloš Stojanović
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Check out the following animations of different ways to turn a sphere and a torus inside out

https://youtu.be/gs_eUoQPjHc Arnaud Chéritat’s sphere eversion (bottom right among the four eversions I show). The animation is joint work between Arnaud Chéritat and Jos Leys (make sure to also check out Jos Leys' channel and website/in my list of recommended channels).

https://youtu.be/kQcy5DvpvlM Arnaud Chéritat’s torus eversion (the half torus version https://youtu.be/jA86M6fdm_Q). Also check out Arnaud’s website for other mathematicial treasures http://www.math.univ-toulouse.fr/~cheritat/

https://youtu.be/sKqt6e7EcCs
https://youtu.be/x7d13SgqUXg the video “Outside in” split into two parts (Thurston’s eversion, top right among the four eversions I show). An absolute must-see !! I think Outside in and what I talk about in this video complement each other very nicely. The clip at 3:00 is also part of Outside in.

https://youtu.be/cdMLLmlS4Dc the automatic “Optiverse” eversion (bottom left among the four eversions I show). Also check out this really nice write-up by John Sullivan http://torus.math.uiuc.edu/jms/Papers/isama/color/

https://youtu.be/876a_0WAoCU the “Holiverse” eversion by Iain Aitchison another (just like me) mathematician from Melbourne, Australia. Read about his eversion here http://www.ms.unimelb.edu.au/~iain/tohoku/Aitchison2010-ss-A*-TerseEversion-arch.pdf

https://youtu.be/wn-qmgOt-Js
https://youtu.be/bGiVPj2P19s “Morin’s eversion” (top left among the four eversions I show). This first animation of an eversion was produced by Nelson Max.

https://youtu.be/FL4JoWlVj98 “deNeve/Hills eversion” Also check out these pages for more details about this eversion: http://www.usefuldreams.org/sphereev.htm and http://www.chrishills.org.uk/ChrisHills/sphereeversion,

For a very nice history of sphere eversions visit this page http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm

Here is a link to Derek Hacon’s notes on his eversion hosted on his son’s Christopher Hacon’s website http://www.math.utah.edu/~hacon/sphereeversion.pdf (Christopher Hacon is also a mathematician). Here is my adaptation using level curves: http://www.qedcat.com/misc/deformation.jpg
(and, of course, now there also Arnaud's animation that I mention earlier on.)

And here is another writeup of Derek Hacon’s eversion by his PhD supervisor E. Christopher Zeemann http://zakuski.utsa.edu/~gokhman/ecz/hacon.pdf

Thank you very much to Arnaud Chéritat, Christopher Hacon and Cliff Stoll for their help with this video.

Enjoy,

Burkard

One more video credit: The nice clip of the punctured torus turning inside out at is based on a video by Greg Mcshane: https://youtu.be/S4ddRPvwcZI

Видео Smale's inside out paradox канала Mathologer