The James Webb Space Telescope sits 1.5 million kilometers from Earth, but it isn't orbiting Earth, and it isn't quite orbiting the Sun either. It circles an empty patch of space, a gravitational parking spot where the pull of two massive bodies cancels out. These are Lagrange points, invisible real estate in the sky, and this episode traces them from an 18th-century mathematical nightmare to the key to humanity's future infrastructure in space.

It explains why adding a third body turns elegant orbital math into unsolvable chaos, how Euler and Lagrange found five points of equilibrium on a chalkboard centuries before computers, and the counterintuitive physics that lets a single fixed sunshield keep Webb in permanent darkness. Then it looks forward: fuel depots in the deep-space freeze, rotating space cities at L4 and L5, and a proposed magnetic shield at Sun-Mars L1 that could let the Red Planet grow its atmosphere back.

• The restricted three-body problem: why a third object makes long-term orbits impossible to solve
• L1 and L2: Earth as a drag chute or a booster, and why Webb only needs one heat shield
• L3 and the counter-Earth myth: why a hidden planet behind the Sun can't actually exist
• The stable points: L4, L5, Trojan asteroids, and the dream of space cities that never drift
• Terraforming from a ghost point: the magnetic shield that could thicken the Martian atmosphere