Why are the formulas for the sphere so weird? (major upgrade of Archimedes' greatest discoveries)

In today’s video we’ll make a little bit of mathematical history. I'll tell you about a major upgrade of one of Archimedes' greatest discoveries about the good old sphere that so far only a handful of mathematicians know about.

00:00 Intro to the baggage carousel
01:04 Archimedes baggage carousel
04:26 Inside-out animations
04:59 Inside-out discussion
10:38 Inside-out paraboloid
12:43 Ratio 3:2
13:28 Volume to area
18:40 Archimedes' claw
20:55 Unfolding the Earth
29:43 Lotus animation
30:38 Thanks!

Those fancy conveyor belts are called a crescent pallet conveyors, and sometimes "sushi conveyors" because they were originally designed for carrying sushi plates. https://en.wikipedia.org/wiki/Conveyo... Andrew also dug up an American patent dating back to 1925 https://patents.google.com/patent/US1...

Great wiki page on Archimedes
https://en.wikipedia.org/wiki/Archimedes

In "On the sphere and the cylinder"
https://en.wikipedia.org/wiki/On_the_...
Archimedes derives the volume and area formulas for the sphere. The proofs used in this work are quite complicated and conform to what was acceptable according to Greek mathematics at the time. His original original ingenious proof most likely involved calculus type arguments. Marty and I wrote about this here https://www.qedcat.com/archive_cleane... and here https://www.qedcat.com/archive_cleane... Also check out this page https://en.wikipedia.org/wiki/The_Met...

Why is the formula for the surface area the derivative of the volume formula? Easy:
V'(r) = dV/dr = A(r) dr / dr = A(r).
A nice discussion of the onion proof on this page I'd say check out the discussion of the onion proof on this page https://en.wikipedia.org/wiki/Area_of...
B.t.w. this works in all dimensions the derivative of the nD volume formula is the nD "area" formula. https://en.wikipedia.org/wiki/Volume_...

Wiki page on Cavalieri's principle
https://en.wikipedia.org/wiki/Cavalie...
Includes both hemisphere = cylinder - cone and paraboloid = cylinder - paraboloid
Video on the volume of the paraboloid using Cavalieri by Mathemaniac
   • Without integration, why is the volum...  

Henry Segerman: https://en.wikipedia.org/wiki/Henry_S...
Henry's video about his 3d printed Archimedes claw:    • Slide-glide cyclides  
Henry's 3d printing files: https://www.printables.com/model/651714

Andrew Kepert: https://www.newcastle.edu.au/profile/...
Andrew's playlist of spectacular video clips complementing this Mathologer video:
   • Lunes and Cyclides  
All of Andrew's animations featured in this video plus a few more (actual footage of a fancy baggage carousel in action, alternative proof that we are really dealing with a cylinder minus a cone, paraboloid inside-out action, inside-out circle to prove the relationship between the area and circumference of the circle, etc.)

There is one thing (among quite a few) that I decided to gloss over at the end of the video but which is worth noting here. At the end it’s not straight Cavalieri. Before you apply Cavalieri, you also need to put some extra thought into figuring out why the flat moon that runs along the semicircular meridian can be straightened out into something that has the same area (straighten meridian spine with interval fishbones at right angles). Here I was tempted to include a challenge for people to figure out why the red and blue surfaces in the attached screenshot have the same area: https://www.qedcat.com/ring.jpg

Funniest comment: Historians attempting to reconstruct the Claw of Archimedes have long debated how the weapon actually worked. The sources seem to have trouble describing exactly what it did, and now we know why. Turns out it was a giant disc that slid beneath the waters of a Roman ship, then raised countless eldritch crescents which inexplicably twisted into a sphere, entrapping the vessel before dragging it under the waves, all while NEVER LEAVING ANY GAPS in the entire process. No escape, no survivors, fucking terrifying. No wonder that Roman soldier killed Archimedes in the end, against the Consul's orders. Gods know what other WMDs this man would unleash on the battlefield if he were allowed to draw even one more circle in the sand. The Roman marines probably had enough PTSD from circles.

T-shirt: One of my own ones from a couple of years ago.
Music: Taiyo (Sun) by Ian Post

Enjoy!

Burkard

P.S.: Thanks you Sharyn, Cam, Tilly, and Tom for your last minute field-testing.