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

Video Description

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/Conveyor_belt_sushi Andrew also dug up an American patent dating back to 1925 https://patents.google.com/patent/US1757652A/en 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_Sphere_and_Cylinder 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_cleaned/99.html and here https://www.qedcat.com/archive_cleaned/100.html Also check out this page https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems 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_a_circle 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_of_an_n-ball Wiki page on Cavalieri's principle https://en.wikipedia.org/wiki/Cavalieri%27s_principle Includes both hemisphere = cylinder - cone and paraboloid = cylinder - paraboloid Video on the volume of the paraboloid using Cavalieri by Mathemaniac https://www.youtube.com/watch?v=U2_nvfWIAYE Henry Segerman: https://en.wikipedia.org/wiki/Henry_Segerman Henry's video about his 3d printed Archimedes claw: https://youtu.be/KD_hRn_97RI Henry's 3d printing files: https://www.printables.com/model/651714 Andrew Kepert: https://www.newcastle.edu.au/profile/andrew-kepert Andrew's playlist of spectacular video clips complementing this Mathologer video: https://www.youtube.com/playlist?list=PL9JP5WCX_XJY9GmMO-kotRR5bRYOPOtn9 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.