The body, computed from its own formula
Sloan's second shape with one extra harmonic in its phase, at the exact parameters of results E1 and E2. The surface below is evaluated in your browser, not drawn by hand.
Blue marks the stable equilibrium at colatitude pi/2, longitude pi; red the unstable one at pi/2, 0. The real body departs from a sphere by about 2 percent, so the slider magnifies that departure for display. At 1x you see the body itself.
Flat in one direction, steep in the other
Near the stable equilibrium the landscape of the squared centroid distance is a long shallow trough. That flat direction is why the body is so sensitive: a tiny shape error can move the minimum a long way.
Axes are proportional to 1 over the square root of each eigenvalue, so the ratio of the two axes is 16.2 to 1 (the square root of the condition number, about 264).
| Stable eigenvalues | 0.0066 to 1.741 |
| Unstable eigenvalues | -1.596 to -0.0068 |
| Condition number | about 260 |
| Definite on the box for | r0 <= 0.005 |
Row F2. Computer-assisted, no second method. The drawing scales the same eigenvalue enclosures that the table reports.
Count the equilibria of a harmonic body
Pick a spherical harmonic. The page counts the extrema of the field on a fixed lattice and prints the closed form m(l-m+1) beside it. Two routes, not wired together.
An illustration, not a result. The lattice is fixed (121 by 240, odd by even, because an even row count misses the equator and silently reports fewer). It counts extrema of the harmonic field; it is not an interval computation and it is not in the ledger as evidence.