Richard Howard <rich@richardehoward.com> wrote:

Large telescope mirrors start as extremely accurate sections of a sphere made by carefully randomized grinding of two surfaces against each other. The only two surfaces that fit together in all orientations are spheres.

Nitpick: Flat surfaces also fit together in all orientations. I suppose a flat surface can be considered part of a sphere of infinite radius. Indeed, the method you describe is also used to make extremely flat mirrors, except that three surfaces are used, two at a time of which are ground against each other. I recently mentioned this in another forum in which we were discussing whether heliographs could be used for long distance communication on the Ringworld (an immense fictional solid ring around a sun-like star) without high technology. I argued that it should be possible to send something like Morse code over distances up to about 100 million kilometers if the recipient has good binoculars, even if the heliograph mirror is no larger than the Great Pyramid, a size that pre-technological cultures could obviously build things.

Starting with a rough glass sphere and a rough hemispherical hole in a plate, continued random grinding produces a perfect sphere (and a perfect hemispherical hole).

How was the spherical gyro in Gravity Probe B, supposedly the roundest object ever made, constructed? My understanding is that it wasn't round enough, and the measurement of the frame dragging effect was below the noise floor, though some claim the noise could be compensated for. It's undisputed that it did measure the much larger space-time curvature effect. (When Magellan's crew returned to Europe after sailing consistently west, they thought that was because the world was round. They were mostly right, but if they had kept very close track of the curvature, they'd have noticed that they were short by about an inch. The missing inch was because of the space-time curvature around the Earth.) (Copernicus, Galileo, Kepler et al believed that the celestial sphere stood still and Earth rotated. They were mostly right, but if they had kept close track, they would have noticed that the celestial sphere rotates by about 0.04 seconds of arc per year due to frame dragging.) As for Rupert's drops, you can find high-speed videos on YouTube. Some show that the drops aren't harmed by being shot by a rifle on their thick part. Others show that the slightest tap on the narrow part makes them explode, and that the shock travels up the drop at several kilometers per second. Never watch this in person without safety goggles!