Saturday, October 14, 2017

234: Le Grand K

Before we start, let me apologize for the delay in getting this episode out.  My old ISP imploded recently, not even giving its users the courtesy of domain or email forwarding, so I had to spend some time straightening out my online life.   Note that this also means the Math Mutation rss feed URL has changed— if you are using iTunes, I think this will be transparent, but if using another podcatcher, you will need to go to to grab the new address.  

Anyway, on to today’s topic.   Recently reading about the silly lawsuit against Subway for selling foot-long sandwiches that were technically less than a foot long, I had a great idea for a startup business.   I would sell measuring tapes and rulers where every unit is 10% smaller than normal, a great boon to businesses such as Subway that make money by the meter.    Sadly, I soon realized that most weights and measures are standardized by international bodies, and such a business would violate various laws.   But that got me a little curious about how these international measurements are settled upon.   After all, how do I know that a meter measured on a ruler I buy today in Oregon will be exactly the same as a meter stick held by a random ice miner in Siberia?    Do companies just copy each other when they manufacture these things?  How do we keep these things consistent?

In most cases, the answer is simple:  objective definitions are created in terms of fundamental physical constants.   For example, a meter is the distance travelled by light in a vacuum in one 299,792,458th of a second.    With a second being defined in terms of the decay of a cesium-133 atom.   OK, these may sound like somewhat exotic definitions, but they are in principle measurable in a well-equipped physics laboratory, and most importantly, will give the same measurements any time the appropriate experiment is repeated.   But I was surprised to discover there is one odd man out:  the kilogram.   Rather than being defined in terms of something fundamental to the universe, a kilogram is literally defined as the mass of one particular hunk of metal, a platinum-iridium sphere in France known as the International Prototype Kilogram, or IPK, nicknamed Le Grand K.

It is strange that in this modern day and age, we would define mass in terms of some reference like this instead of fundamental constants.   But if you think about how you measure mass, it can be a bit tricky.   Usually we measure the mass of an object by checking its weight, a simple proxy that works great as an approximate measure, when you happen to live on a planet with noticeable gravity.   Once you care about tiny differences like millionths and billionths, however, you realize there is a lot of uncertainty as to the exact relationship between weight and mass at any point on earth—  you need to know the exact force of gravity, which can depend on the altitude, local composition of the earth’s crust, position of the moon, etc.   However, if you compare to other objects of known mass, all these issues are normalized away:  both masses are affected equally, so you can just use the counterbalancing masses to measure and compare.   Thus, using a prototype kilogram, and making copies of it for calibrating other prototypes, is a very practical solution.

Scientists did an amazing job defining the initial prototype:  they wanted it to be equal to the mass of a cubic decimeter of ocean water at 4 degrees Kelvin under one atmosphere of pressure, and the IPK apparently meets that ideal with an error roughly comparable to the mass of a grain of rice.    Unfortunately, recent measurements have shown that the IPK has lost about 50 micrograms over the last half-century relative to copies of it in other countries.   This is despite an amazing level of caution in its maintenance:  climate control, filtered air, and special cleaning processes.    There are various theories about the root cause:  perhaps minuscule quantities of trapped gases are slowly escaping, maybe the replicas are gaining dirt due to not-quite-careful-enough handling, or maybe even mercury vapor from nearby thermometers is playing a role.   But whatever the cause, this is a real problem:  now that high-tech manufacturing is almost at the point of building certain devices atom-by-atom, even tiny levels of uncertainty in the actual value of a kilogram are very bad.

Thus, there is a new push to redefine the kilogram in terms of fundamental constants.   One idea is to define it based on the number of atoms in a carefully-prepared sphere of pure silicon.   Another is to use the amount of voltage required to levitate a certain weight under controlled conditions.     A more direct method would be to define the kilogram in terms of an exact number of atoms of carbon-12.   All these share the problem that they depend on fundamental constants which are themselves only measurable experimentally, to some finite degree of precision, which adds potential error factors greater than the uncertainty in comparing to a copy of the IPK.  However, the precision of most of these constants has been steadily increasing with the advances of science, and there seems to be a general feeling that by the close of this decade, Le Grand K will finally be able to be retired.

And this has been your math mutation for today.


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