On Beyond Yotta

By The Metric Maven

Bulldog Edition

When I was a boy, I read a number of books by Dr. Seuss. One that immediately captured my interest was On Beyond Zebra. I don’t recall  much about the book at this point in my life other than the fact that it involved additional letters of the alphabet. Each new letter was introduced and illustrated by the author. The idea there might be unknown letters piqued my youthful interest. Here are the new letters that appeared in that book:

The first book I ever read by Isaac Asimov (1920-1992) was: The Universe, From Flat Earth to Quasar, which I recently re-read. The two books may seem far apart, but I made a connection between them when I came across a section on how the sun generates its energy. The Sun uses nuclear fusion to convert mass to energy. This process is understood using the famous equation E = mc2 developed by Albert Einstein (1879-1955).
The pressures at the center of the sun cause four hydrogen atoms to fuse into a single helium atom. After this process occurs there is a mass imbalance, the four hydrogen atoms have more combined mass than the resulting single helium atom, and the extra mass is converted into energy.

Dr. Asimov states that about 4.2 Tg (Teragrams) of mass is converted to energy every second inside of the sun. He uses pre-metric terms to describe this value as “4 600 000 tons of mass per second.” Unfortunately so does Wikipedia: “the Sun fuses about 620 million metric tons of hydrogen each second.” As I understand it 1 million is  106  and a “metric ton” is a Megagram or 106 grams for 4.2 x 1012 grams per second or 4.2 Tg per second. That’s a lot of grams. Dr. Asimov inquires: “Is it possible  for the Sun to support this steady drain of mass at the rate of millions of tons per second? Yes, it certainly is, for the loss is infinitesimally small compared with the total vast mass of the sun.”  The currently accepted mass of the sun is, approximately 2 x 1030 kg. This means it’s 2 x 1033  grams, and the proper metric prefix would be?—oh, well, there isn’t exactly a metric prefix for this value. The last magnifying metric prefix is Yotta, which allows the mass to be written as 2 000 000 000 Yg (Yottagrams). Which by current convention it appears there are about three extra metric prefixes needed to express the mass of the sun with a 2, and a minimum of two extra prefixes to use 2000 as a magnitude.

So what does “infinitesimally small” mean? Well the mass lost each second, divided by the total mass of the sun, is 4.2 x 1012 grams/2 x 1033  grams. This value is one divided by 476.19 x 1018 or 0.000 000 000 000 000 000 002 which is quite a tiny ratio. I believe this is indeed a small enough ratio to be “infinitesimally small.” Recall we are talking about 4.2 Tg per second of mass loss. Each gram has 90 TJ (Terajoules) of energy contained within it’s mass. If my computation is correct, then 378 x 1024 joules are released each second. This would be 378 YJ (Yottajoules) per second. We are approaching the limits of the metric prefix Yotta, and in only 1000 seconds we would have  378 000 YJ and see that a new prefix might be useful to describe the power released.

What is notable is that the mass of the sun is not readily expressed with a metric prefix, and it’s not all that massive for a star. It appears that the masses of stars are indeed astronomical. The most massive star is suspected to be R136a1 which is approximately 256 solar masses (a solar mass is the mass of the Sun). This means it has a mass of 512 x 1033 grams or 512 000 000 000 Yg. Clearly we are on beyond Yotta at this point. While I’ve made it clear in the past that astronomical distances are readily expressed with metric prefixes, this is not the case for stellar masses. One can see why R136a1 is described in terms of an equivalent number of solar masses and the metric system is not employed.

Asimov also makes this surprising statement:

Release of energy is always at the expense of disappearance of mass, but in ordinary chemical reactions, energy is released in such low quantities that the mass-loss is insignificant. As I have just said, 670,000 gallons of gasoline must be burned to bring about the loss of 1 gram (1/27 of an ounce). Nuclear reactions produce energies of much greater quantities, and here the loss of mass becomes large enough to be significant.

What I’ve been able find in my research on this subject is both minimal and contentious. It is mostly stated that the amount of mass lost in chemical reactions is “unmeasurable.” The few who venture to put numbers to paper (including a textbook example) end up with magnitudes on the order of 10-33 grams. One example computation has 70 x 10-33 grams as the amount of mass lost in the given chemical reaction. This would be 0.000 000 070 yg (yoctograms) and would indicate a possible need for at least two more metric prefixes. It appears that, at least in theoretical discussions, it might be useful to have two more metric prefixes on the dividing side of the prefixes.

Currently there are 20 metric prefixes from yocto to Yotta. Adding two more prefixes on the magnification side would be useful for some of this astronomical work. It would probably make sense to add a pair to the reducing prefixes also. This would increase the total number to 24 metric prefixes. This is a lot of prefixes, but is far less than the number of magnitudes scientific notation would allow, which would be 60. What I would propose is to consider adding the new prefixes, but at the same time remove the prefix cluster around unity: deca, hecto, deci and centi. They could be separated  and relegated into a set of atavistic prefixes which are no longer considered proper modern usage. They would be included as an appendix to the modern prefixes for historical reference, but discouraged for modern use. This simplification would reduce the number of prefixes back to 20 and also provide a larger dynamic range for scientific description.

In early grades it makes sense to me that only the prefixes micro, milli, Kilo and Mega would be taught as the Common Set of Prefixes. These would be the prefixes that students would generally encounter in everyday life (if the US was metric and fully engaged). In Junior High and High School the new set of prefixes I’ve proposed could be taught as the Complete Set of Prefixes. I would argue that all students (and their teachers) should have to memorize and use all these metric prefixes (without the prefix cluster around unity) in their instruction. Textbook authors should not shy away from using Megameters for planetary dimensions, Gigameters for the solar system, and all the other appropriate uses of metric prefixes.

People have objected to my proposal that we teach all students to use all the metric prefixes. They employ the argument that the Common Set of Prefixes is all that is needed for an ordinary person, and the Complete Set of Prefixes is for engineers, scientists and technical people. I reject this view entirely. It produces a scientific apartheid that keeps the public from understanding the important issues of the day, which involve engineering and science more and more everyday. What I have discovered when working with large questions, such as how much the salinity of the ocean would change if we dumped all our fresh water into it, or how much carbon is being belched into our atmosphere over a given period of time, is that these problems are tamed using appropriate metric prefixes. They allow an ordinary citizen to comfortably work with the magnitudes involved. If one talks about hundreds of billions of tons, that is a metaphor, and is not information. If the goal of education in the US is to create the most numerate population on the planet, then a good command of the magnitudes of all the metric prefixes is essential.

I would like to see a song which fixes the order of the metric prefixes in a person’s mind from the smallest to the largest, something similar to Tom Lehrer’s Element Song. Some manner of meaningless acrostic or other method of recalling the order of the 1000 based prefixes should also be developed. With the prefix cluster around unity eliminated, all the magnitudes will be of 1000 and any parsed base unit can be determined.  This would allow anyone to look at 1 000 000 000 000 000 grams and immediately relate it to the acrostic or song and “sound out” the size of the number as 1 Pg (Petagram), or conversely be able to take the 1 Pg and work out how many sets of three zeros one would need to express it. This would also be the case for 0.000 000 000 000 001 grams. It  could be “sounded out” as 1 fg (femtogram).

When all the metric prefixes no longer apply, that’s when a modern student should viscerally realize they are discussing dimensions that are so large or so small they are mind blowing, and on beyond yocto and Yotta. These values truly exist in an amazing far distant realm.

The Metric System and Coincidence

By The Metric Maven

On July 4th, many, many years ago in Montana, I went out with a group of people to shoot at prairie  dogs. This was a common activity in rural Montana. It was common enough that the prairie dogs understood what was happening and would not surface above their burrow unless they were certain the hunters were far enough away so a .22 rifle was ineffective. It was very hot, and the prairie dogs seemed to have out-smarted us. We were about to leave, when one of us pulled up their rifle and aimed it at a far-off prairie dog. The target was but a tiny shadow on top of its mound, but stood up high, and defiant in the certainty that it was safe. The entire group began laughing and deriding the fellow with the rifle. “You might as well shoot popcorn at that gopher. You’re just wasting a round” they said with ridicule in their voices.  (Montanans sometimes called prairie dogs gophers) This only strengthened  the resolve of the varmint hunter who said “Oh, yeah, well at least I’ll scare him good.”  It must have been 150 meters to the tiny silhouette. He squeezed the trigger and the Prairie dog flopped over instantly. The entire crowd was stunned. One person said “That’s the most incredible shot I’ve ever seen!”  Three people began running over to inspect the scene. They were about three meters from the mound when they all put on the brakes and started screaming: “rattlesnake!  rattlesnake!” and retreated faster than they had arrived. The prairie dog had not been hit by a bullet, but instead, at exactly the same time the shot was fired, a rattlesnake had struck the unfortunate animal. This incident has always stood out in my mind as one of the strangest coincidences I’ve witnessed.

This coincidence however seems tame in comparison to some metric coincidences I’ve come across. One of the strange arguments that is offered by those against metric has been “metric lengths are unnatural” and the sizes of metric products aren’t a product of “natural” dimensions like those of imperial sizes. In his very first newsletter Pat Naughtin states in his Hidden Metric section:

As vinyl records developed, in the 1920s, they were designed and made 250 millimetres and 300 millimetres in diameter. In English speaking countries, they have been called 10 inch and 12 inch records ever since.

This seemed incredibly implausible. I took my trusty Australian mm ruler and measured a 33 1/3 LP. Wow, it was almost exactly 300 mm!–but not quite. It was actually 301 mm. I kept looking at the discrepancy and wondered if the extra mm was from the pressing process. It was hard to imagine if the specification was 300mm that any company would allow 1 mm of waste on each pressing. I did a web search and to my surprise found RIAA specifications for records. In the tradition of American Medieval Unit standards, it is written in the strange and tangled world of fractions:

The Diameter of a 12″ record is: 11 7/8″ + 1/32″ Now assuming the 1/32″ is a one sided tolerance we have in mm:

12″ => 301.625 mm + 0.79375 mm = 302.42 mm

Average = 302.02 mm

This is very close to 300 mm but LP records almost certainly are not based on metric. In the case of a Big Ten Inch Record, the specification is 9 7/8″ + 1/32″ which in mm is:

10″ => 250.825 mm + 0.79375 mm = 251.6187 mm

Average = 251.42 mm

Once again this is really close to 250 mm, but not quite, again it looks like an Olde English specification, that could be confused for a metric one. The other standard size (45 RPM) is 7″ which is 6 7/8″ + 1/32″ according to the specification.

7″ => 174.625 mm + 0.79375 mm = 175.418 mm

One can easily see that Pat might have mistakenly been told that records were 300 mm, 250 mm and 175 mm in diameter. A quick measurement would seem to verify this fact—except if you are a reasonably exacting Engineer like the Metric Maven, the small discrepancy would begin to bother you. Although it was a German-American inventor, Emile Berliner,  who is credited with developing disc records, there is no mention of metric sizes in Wikipedia.  It is quite interesting that these “metric sizes” appear quite natural and have seemed so to American citizens over the decades. Much to my surprise, vinyl is not dead, and seems to be expanding. Every time I ask for an album at my local independent music store the first question is: “vinyl or CD.”

This metric question of vinyl diameter has been superseded by the introduction in the 1980s of the metric defined CD which is 120 mm in diameter, and is indeed specified as metric, independent of the size of the standard case housings.

While LPs might be interesting, there are much bigger metric coincidences. The first has to do with the fact that John Wilkins, the British man who invented the system part of the Metric System, used the length of a seconds pendulum to define what later has become known as the meter. Early on it was suggested that a seconds pendulum be used to define the meter in order to tie its length to a scientific phenomenon. Unfortunately the length of  a seconds pendulum depends on its latitude, which became a point of contention. One can see in the figure below that a seconds pendulum from a 19th century clock is very, very close to the length of a modern meter. The alternating colored sections are 100 mm in length.

Pendulum from 19th Century Clock with Meter Stick

The next idea was to define a meter as one ten-millionth of the distance from the north pole to the equator. James Clerk Maxwell found the idea of using this distance across the Earth to be frivolous, and made sport of it in his understated way, in his famous Treatise on Electromagnetism. Clearly the distance of a meter, as defined by a seconds pendulum, would be quite different than one ten-millionth of the distance from the north pole to the equator—right? Well, no, amazingly enough the circumference of the Earth through the poles is slightly more than forty million meters (40,007,863 m) and the two suggested values of the meter are remarkably close. I find this a very surprising coincidence.

James Clerk Maxwell proposed that light is an electromagnetic wave and used his theory to predict the expected speed of these waves. His answer was 193,308 miles per second. Later in the 20th century the value would be accepted as 186,282.3959 miles/second. It was Albert Einstein who put the speed of light at the center stage of physics and directly related energy and mass using the speed of light in his famous equation E = mc2.

When the meter was finally defined with a scientific phenomenon, it was in terms of counting a number of wavelengths of light of a given color. The next metric coincidence is that the speed of light, when expressed in meters per second, is 299,792,458 meters/second. Don’t see the coincidence? Well this is only 0.07% from 300,000,000 meters per second. I use this approximate value almost daily in my Engineering work. Myself and my peers all use 3.0 x 108 meters per second for hand calculations. It is a nice round number and easy to remember. For instance, let’s compute the wavelength of an electromagnetic wave of 3 GHz (3.0 x 109 Hz) in free space. It’s 99.93 mm if one  uses the exact value for the speed of light. When the 3.0 x 108 m/s approximation is used it is exactly 100.00 mm. The error is 69.17 μm! Yes, micrometers!

When used for everyday engineering computations, there is no need to remember the exact value of the speed of light, as the approximate one is so close, there is no reason to bother. This is an amazing coincidence.

Another coincidence that I find quite interesting (and will discuss in a future blog) is that in Boulder, Colorado one cubic meter of air has a mass of almost exactly 1 kilogram. On the coasts it is about 1.2 Kg.

Here are some other metric coincidences:

The width of a human male hand is about 100 mm.

The length of a stretched human pace is about one meter (1 m)

The distance from the Earth to the Sun is almost exactly 150 Gigameters (150 Gm)*

The volume of the Earth is very close to one Yottaliter (1 YL).

The distance across the Milky Way Galaxy is about one Zettameter (1 Zm).

The diameter of the local group of galaxies is about one hundred Zettameters (100 Zm)

And one engineered non-concidence is that the circumference of the Earth is almost exactly 40 Megameters (40 Mm)

I find these metric coincidences far more interesting than the Fourth of July rattlesnake coincidence. The meter and its divisions seem to me much more attuned to the natural world than the contrived, inconsistent and almost uncountable units of the old remnants of the non-system of Olde English and Imperial, or the even more laughable American designation of same as: “standard.”

It is time for all of us in the United States to give up this unnatural, non-system of measurement for the one that nature clearly intended—the meter and the metric system.

* In other words the “astronomical unit” has a nice integer value in the metric system