The helicoid versus the catenoid: Geometrically induced bifurcations

The minimal surfaces bounded by a frame formed of a double helix and two horizontal rods are studied. The vibration equation shows that the helicoid is the stable surface when its winding number is small. The catenoid is locally isometric to the helicoid so that their vibration spectra are strongly related. While the catenoid is known to undergo a discontinuous transition to two disks, the helicoid is shown to become unstable through a continuous transition to a ribbon-shaped surface obtained experimentally, numerically, and analytically in the limit of infinite height. The normal forms of the bifurcations confirm the analysis.