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

Видео What's the curse of the Schwarz lantern? канала Mathologer