Jump into an evacuated hole drilled straight through a uniform, static Earth-like sphere. Accelerate to 7.9 km/s (or 18 000 m.p.h.) at the center, then decelerate back to zero at the **antipodes** 42 minutes later! Step out of the hole upside down — or return 84 minutes after you left.

Last fall, as part of his senior thesis, Yuchen Gan ’21 and I used computer simulations to generalize this famous result to uniform spinning planets, where **Coriolis** and **centrifugal** effects force the tunnels into arcs curving away from the center and intersecting the surface in multiple places. We discovered many families of **periodic** tunnel networks that connect multiple surface locations even at non-equatorial latitudes, as in the animation. Such tunnels could ideally provide energy-free communication and transportation for the planets’ inhabitants.

But in January, in a wonderful **aha!** moment, we were surprised and delighted by a dramatic perspective change: the motion of an object or passenger (a “terranaut”) freely falling through the tunnel system is both spiky concave arcs with respect to the planet *and* a smooth convex **ellipse** with respect to inertial space! We subsequently proved mathematically that the inertial motion is that of a two-dimensional harmonic oscillator, and the ellipses are **centered** (not **focused**) on the planet.

Download a higher-resolution QuickTime MOV version of the animation with or without the red elliptic trace.

Thanks, Mark! I enjoy reading your posts as well.