• a_g_marut@lemmy.worldEnglish
    2·
    5 days ago

    Transcript for screenreader users:

    Part 3 of 5! Press forward to continue! [orange rolling back to the right]

    Supermarkets often use this packing to stack oranges and other round fruits in their grocery section.

    Grocery store worker: Customers will appreciate this conjectural maximum-efficiency orange-packing! [gesturing towards two pyramids of oranges]

    Kepler computed that the density of this packing was about 74% – that is, about 74% of the available space was occupied by the spheres, and the remaining 26% by the gaps between the spheres.

    Kepler: There is probably not a better way. [standing with Raleigh and Harriot around a pyramid of cannonballs]

    He conjectured that this was the maximum possible density: there was no other way to pack spheres that could achieve a density of, say, 75%.

    [Kepler pointing with a rod at a pie chart with the top and left three quarters labeled “cannonball” and the lower right quarter labeled “not cannonball”]

    Tao: This “Kepler conjecture” attracted the attention of many brilliant mathematicians for centuries. [four people labeled Gauss, Thue, Toth, and Rogers are standing behind Tao]

    It was only solved in 1998, after countless works culminating in a 100-page paper by Thomas Hales and Samuel Ferguson, who also needed extensive computer calculations to complete the proof.

    Ferguson: We have packed the maximum number of theorems into this proof. [Hales, scratching his head, is standing not far from Ferguson]

    This by itself didn’t revolutionize the way cannonballs or oranges were stacked.

    Grocery store customer: Wow, is that PROVEN maximum-efficiency orange-packing? [looking at a pyramid of oranges]

    Worker [next to another orange pyramid]: Kepler-approved.

    Tao: But once mathematicians study one question, they are naturally led to explore other related questions, that often venture quite far from the motivation of the original problem.

    Tao: The Kepler conjecture is about packing spheres in three dimensions. Mathematicians asked: what happens instead in two dimensions? Four? A billion? [outline of a sphere superimposed over x-y-z coordinate axes]

    This is perhaps a problem if you want to IMAGINE the mathematical object…

    Geometer, tauntingly: Fingers getting tired from all those symbols, Thompson? [Algebraist and Geometer writing on a chalkboard, the algebraist having written out the equations x sub 1 squared equals one, x sub one squared plus x sub two squared equals one, and so on for three and four x variables, and the geometer having drawn a pair of points, a circle, and a sphere and labeled them 1D, 2D, and 3D]

    But not if you want to do the math.

    Algebraist: Where’s your precious “visual intuition” now, Jenkins!? HAHAHAHA! [The algebraist has written out the equations for an n-sphere with five and six x-variables, while the geometer has drawn a question mark and labeled it 4D, two question marks labeled 5D, and three question marks labeled 6D, and is crying from one eye.

    Tao: But higher-dimensional spheres aren’t just useful for making geometers cry.

    Bonus panel: same as Parts 1 and 2