Spheres Part 2 (2026-04-09)

Transcript for screenreader users:

Part 2 of 5! Press forward to continue! [an orange]

Tao: There are admittedly some grains of truth to all of these caricatures.

[behind Tao are those caricatures, and the words "Also, extremely normal-looking people!" with arrows pointing to one more person and some empty space]

Tao: And while we ARE unusually fond of pursuing abstract and technical questions primarily for reasons of intellectual curiosity, the remarkable thing is that such pursuits can have unexpected practical benefits many years after they were first investigated. [In the background, Shannon continues to blow fire out of a trombone.]

Eugene Wigner famously called this the "unreasonable effectiveness of mathematics in the natural sciences".

Wigner: The fundamental structure of the universe runs on math we developed for GAMBLING?!

Tao: One example is the story of "sphere packing". [Tao holds up three or four oranges]

In the early 1600s, Sir Walter Raleigh asked the English mathematician Thomas Harriot for the most efficient way to stack cannonballs together.

Raleigh: There's GOT to be a better way. [gesturing towards a pyramid of cannonballs, five to a side, on a ship, with water in the background stretching all the way to the horizon]

Harriot studied the question in detail but could not definitively answer it, and wrote about it to an eminent German colleague, Johannes Kepler.

Kepler: Finally, something I'll be remembered for.

Kepler viewed this as an abstract mathematical problem about packings of infinite three-dimensional space by spheres of unit radius.

Kepler: There's less to keep track of if you just make it infinitely large. [holding a stack of small cannonballs]

One such packing is known as the hexagonal close packing; it is layer upon layer of spheres arranged in a hexagonal lattice pattern, with any sphere on one layer lying balanced on three spheres on the previous layers.

[hexagons arranged in an 8-by-8 honeycomb pattern] [a circle tangent to six other circles of the same size, with red line segments connecting the centers of the six circles, forming a hexagon] [spheres arranged the same way, but tilted to indicate their 3-dimensionality, with the same red hexagon, and also two new spheres at opposite ends, making the hexagonal arrangement of spheres into more of a rhombus]

Bonus panel: same as Part 1