Potttery and Paraboloids

Here's a bowl from Cenote. The picture comes from Arlen Chase's PhD thesis. I've aligned the image, added a grid, and plotted a parabola.

The fit is good out to a radius of 9 centimeters. That doesn't prove much, because pretty much any smoothly curved bowl will fit some parabola for some internal radius.

Working on the assumption of a 9cm radius of pool of mercury, with a y height of 1cm at that radius, the equation of the parabola would be approximately:

y = x*x / 81

a = 1/81

a = 1/(4f)

4f = 81

f is approximately 20 centimeters.

Given focal length of .2 meters, we can derive the required rotational velocity:

f * rpm * rpm = 447

rpm = sqrt(447/.2) = sqrt(2235) = 47

So, the bowl would be rotating at around 47 revolutions per minute.

These numbers are problematic for my theory. The focal length is too short assuming the goal is to reflect the image via a small mirror above the center of the bowl, then into some type of eyepiece that must be 9cm from the center axis (otherwise the viewer's head would obstruct the incoming light.) Secondly, 47 rpm is really fast: even with a forty foot string suspending the bowl, that would be 5 twists per foot of twine to get even five minutes of rotation. At 47 rpm, a forty foot string would be vibrating quite intensely, probably enough to ruin the optical surface of the mercury.

Of course, lower rotational speeds would be reasonable. The mercury would still form a parabola, it would just be deeper in the center of the bowl, and thus a less efficient use of the metal. The more scarce the mercury was, the more we would expect the rotational speed to conform to the parabola of the bowl.

I ran though the math for other similar bowls in the Cenote paper. They all produced numbers in the 40-70 rpm range.