The painter's paradox, a.k.a. Gabriel's horn paradox a.k.a. Torricelli's horn paradox has been done to death on YouTube. So why do it again? Well, being all about some remarkable features of 1/x, this topic nicely complements the previous two videos that were also dedicated to 1/x. Now the first Mathologer trilogy is complete! Also, I thought of a couple of nice twists to make this treatment of the painter's paradox really stand out from the crowd (I hope :)

00:00 Intro
02:59 What's that horn?
07:09 A 700 year old trick
12:26 What's the volume exactly?
16:33 What paradox?
18:17 Fun fact.
20:12 Thanks

Must check out.
Wiki page on the horn: https://en.wikipedia.org/wiki/Gabriel...
(also has a nice discussion of the converse of the paradox at the end)

Wiki page on Torricelli: https://en.wikipedia.org/wiki/Evangel...
(the anagram that featured in the video also gets a mention on this page)

Very nice write-up of the history of the horn by Paolo Mancosu and Ezio Vailati https://www.jstor.org/stable/233514 https://tinyurl.com/3rff7ubb

Wiki page on Oresme: https://en.wikipedia.org/wiki/Nicole_...
(has a page from one of his books with some graphs. Somehow overlooked this one. Would have been nice to flash this in the video :(

Stanford Encyclopedia of Philosophy entry on Nicole Oresme (in particular, see the section on math) https://plato.stanford.edu/entries/ni...

Wiki page on the Method of shells: https://en.wikipedia.org/wiki/Shell_i...

Fun/intersting: The opposite of this is a vuvuzela, which is a horn with finite surface but infinite volume (as in loud)

One interesting observation is that both area and volume being finite or infinite is independent of what unit we use. Therefore it does make sense to compare finite/infinite volumes and surface areas of shapes. And one interesting observation in this respect is that solids of infinite volume and finite surface area don't exist. On the other hand, that also means that areas and volumes scaling differently is not part of the resolution of our paradox which only depends on the volume being finite and the surface area being infinite.

Music at the end: Young rich pixies - Year of life
T-shirt: TONEWAVE Music Tee Geometric Overtone Frequency Musician Sacred Geometry Tshirt https://www.etsy.com/ca/listing/21276...

Enjoy!

Burkard