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:
  / troll_math_pi_4_crosspost  

Good discussion by GoldPlatedGoof
https://static.nsta.org/pdfs/QuantumV...
The Vi Hart video:
   • Rhapsody on the Proof of Pi = 4  

Closely related, the Staircase paradox:
https://en.wikipedia.org/wiki/Stairca...

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...

Schwarz's original paper:
https://archive.org/stream/gesammelte...

Diamond-Cut Can
https://www.toyo-seikan.co.jp/e/techn...

Lots of nice ideas (coffee cans, soda can creasing, collapsing cylinders, patent for effectively collapsing cans)
https://beachpackagingdesign.com/boxv...

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/QuantumV...

Andrews Geogebra app in action:
Schwarz Lantern in Geogebra 3D & AR:    • Schwarz Lantern in Geogebra 3D & AR  
Schwarz Lantern christmas tree:    • Schwarz Lantern christmas tree  
Geogebra 3D AR mode on an iPad how-to:    • GeoGebra 3D Mobile App – AR howto  

Schwarz lantern origami with Deepali Karanjavkar:
   • How To Fold Yoshimura Polyhedral Patt...  

Crushmetric water bottle (really impressive can crushing inspired toys):
https://www.crushmetric.com/

Hydraulic press vs. metal pipe (square and circular)
   • Crushing Long Steel Pipes with Hydrau...  

Mel Brooks: "May the Schwarz be with you"
   • "May The Schwartz be with you !"  


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