🧵 The surprising geometry of high-dimensional space In 2 or 3 dimensions geometry feels familiar. Volumes fill the interior. Distances between points vary widely. Raise the dimension and the picture changes completely. The volume of the unit ball concentrates in a thin shell near the surface. The ratio of the volume of the unit ball to the volume of the unit cube falls rapidly toward zero. Almost all the mass sits near the boundary. Interior points become negligible. This is the first signal that high-dimensional geometry follows different rules.

Pick ANY positive integer. Apply TWO ridiculously simple rules. It ALWAYS ends up at 1… supposedly. But after 87 years, the greatest mathematicians, supercomputers, AND AI still can’t prove it works for EVERY number. This is the Collatz Conjecture 🧵