24 October 2010

A new (and wrong) value for pi

For the past several weeks I have been trying to develop a method to measure the volumes of snail shells. The method I am working on involves the immersion of an object with an unknown volume in water and several sets of weight measurements. The method is not yet applicable to snail shells, because I can't quite figure out how to seal their apertures in a practical and reversible way so that water won't leak inside. Nevertheless, I have carried out a test using 6 small balls, including 3 balls from old computer mice, 1 metal ball bearing and a glass marble.

After I measured the volumes and the diameters of the balls, it dawned on me that I could use the data to calculate a value for pi. Here is the plot of measured volumes against measured radii.

This plot is, of course, based on the equation for the volume of a sphere: V=(π4/3)r3. Therefore, pi can be calculated from the slope. The value I got was 3.16, which is 0.6% higher than the true value of 3.14.

There are several sources of error that could have resulted in my erroneous result. For example, if the calipers I used to measure the diameters were off by even 0.1 mm, that could have created an error of about 0.4%. Also, the balls I used are probably not perfect spheres. Another likely source of error is the value I used for the density of water, which depends on the temperature and the purity of the water used.

If and when I figure out how to apply this method to snail shells, these errors will not be important as long as they are equally applicable to every specimen; I am mainly interested in the relative volumes of a sample of shells rather than the absolute values of their volumes.

Tests are continuing.


leptotila@gmail said...

What about filling the shell with water, use a swab to dry off the outside of the shell, and dry off your fingers or forceps, dump the water from the snail shell into a container. Repeat until you've done it 10 or 20 times. Measure volume of collected water, divide your measured volume by the number of replicates (e.g., 10 or 20), to get a measurement average. This way doesn't require sealing the aperture, but does require drying off the outside of the shell each time you fill the shell with water.

leptotila@gmail said...

or would you have to do this 100's of times to get a measurable volume of water...? (depending on size of snail shell, I guess)


Theoretically possible & you wouldn't have to empty out the water to measure its volume; you could simply weigh the shell before & after, which would be more accurate. But there are several practical problems:

1. Because air gets trapped in the apex, it is very difficult to fill a shell with water completely. That's why empty snail shells float in water.
2. Many field-collected shells have debris trapped in them, which prevents them from getting filled completely.
3. Because most adult shells are opaque, one can't see if the shell is clean or filled completely.