The Expanding Universe

Posted on 2016-09-10 by Randy Bancroft

By The Metric Maven

After I learned it, I’ve made good use of the Whole Number rule in my technical work. But over the last eight years, I’ve found that now and then, I need to present data with a large dynamic range. What this means is that the numbers involved vary from a very small value to a very large value and span many metric prefix ranges. I’ve spent a lot of time trying to find a way to represent these values as intuitively as possible, and until recently all the options were, in my view, unsatisfactory.

Not long ago I was reading an essay by Isaac Asimov entitled The Figure of The Farthest. The essay discusses the increase in the estimated size of the known universe from antiquity to 1973, when this essay was written.

Asimov starts with Hecataeus of Miletus (c.550 BC — c.476 BC) who wrote a world survey. The extent of the world he traveled was about 5000 miles across and thought to be a flat disk.

Hecataeus of Miletus Reconstructed World Map — Wikimedia Commons

Greek philosophers came to the realization around 350 BC that the Earth is probably a sphere. Eratosthenes of Cyrene (c.276 BC — c.195/194 BC) was the first to devise a method to measure the dimensions of the spherical Earth. He realized that incident sunlight at separate locations arrive at different angles. Using the distance between two points with a known measured angular difference, he estimated the Earth to have a diameter of 8000 miles.

The Greek astronomer Hipparchus of Nicaea (c.190 BC — c.120 BC) used trigonometric methods to compute the separation from the Earth to the Moon. His estimate was equal to 30 times the diameter of the Earth. Using Eratosthenes estimate for the Earth’s diameter, this distance is ~~480,000~~ 240,000 miles.

Eighteen centuries passed before more refined astronomical estimates of distances were computed. The invention of the telescope in 1608 allowed astronomers to measure celestial values with much greater precision and accuracy. In 1671 French astronomer Giovanni Domenico Cassini (1625–1712) would be the first to measure the parallax of Mars with reasonable accuracy. The model of the solar system provided by Johannes Kepler (1571–1630) was used to determine the distance to numerous astronomical bodies. The furthest distance measured was Saturn at 1,800,000,000 miles.

Edmund Halley (1656–1742) would compute the orbit of his eponymous comet to have a maximum distance from the Sun of 6,000,000,000 miles.

Measuring the distances to the stars proved much more difficult. A race was on to measure stellar parallax. A succession of star distances were finally measured. The distance to the star Vega was determined in 1840 by Friedrich Wilhelm von Struve (1783–1864). It proved to be the furthest at 54 light-years.

Countless stars remained without measurable parallax values. Clearly the Universe was much larger than the distance to Vega. William Herschel (1738–1822) surveyed the number of stars in different directions and realized that it varied. He suggested they formed a flattened lens-shaped distribution that we now call our Galaxy. It would not be until 1906 that Dutch astronomer Jacobus Cornelis Kapteyn (1851–1922) was able to use photographic techniques to estimate the dimensions of this stellar distribution. The largest dimension of the Galaxy was computed to be 55,000 light-years.

Harlow Shapley (1885–1972) would use a variable star called a Cepheid to determine the extent of our galaxy with increased accuracy. It was now thought to be about 100,000 light-years across. Shapley further demonstrated that the Magellanic Clouds are outside of our Galaxy. The farthest extent of our Universe increased to 330,000 light-years by 1920.

The Andromeda Nebula became a source of scientific controversy. Was it inside of our Galaxy or outside? In 1923 Edwin Powell Hubble (1889–1953) showed the Andromeda Nebula is indeed a Galaxy outside of our own, and the furthest extent of the known Universe expanded to 5,400,000 light-years.

Other fuzzy glowing patches could be seen, which suggested the Universe was much, much larger than the distance to Andromeda. By 1940, the maximum distance measured for the size of the Universe was around 400,000,000 light-years.

This value seemed to be a measurement limit and no further progress would be made. Then objects much brighter than galaxies were discovered. They were originally called quasars, but are now thought to be black holes which are swallowing nearby matter. The information they provided ballooned the maximum extent of the known Universe to 2,000,000,000 light-years. The year that Asimov wrote his essay, 1973, the farthest known quasar increased the size of the Universe to about 24,000,000,000 light-years.

In his essay, Asimov does not provide a table of these values. This is unusual, as Dr. Asimov had no reluctance to present numerous tables in his other essays and full length books. Each succeeding estimate of the Universe is separated with the explanations I’ve summarized above. I decided to create a table of Dr. Asimov’s data to use for illustration:

The values of the known Universe in miles become larger and larger and then Dr. Asimov shifts them to light-years. In my view this data has a perceptual discontinuity, and is not an acceptable way to present this data.

So what to try? Often in other tables distances are kept in provincial Kilometers, but clearly always starting with Kilometers would restrict an optimum starting value. I decided to try to categorize the data with a set of metric prefixes:

Clearly the situation is worse, the numerous sets of prefixes produce at least four perceptual discontinuities in place of the single one in Asimov’s data. I gave it another try where I attempted to minimize the number of prefixes and separate each set with a line skip:

Again, this is just a perceptual mess, with three rather than one discontinuity. I then thought about representing the lengths using a logarithm, and the knee-jerk way to do this is to use decibels. (Yes, I know that it is not usual to represent lengths as decibels, please humor me for a moment.) This yielded:
While this is continuous, it is also laughably decanted of all perceptual interpretation for most people. It simply seems to hide the magnitudes involved.

I started from scratch, and decided that it might be best if one chose a metric prefix which produces the smallest integer value possible in the data set. In this case it is eight Megameters. I then used only Megameters with standard three digit separations. This seems to be a useful way to present the large dynamic range data. One can see a large magnitude jump from 150 BC to 1671. The size of the universe was refined from 1671 to 1840 until another large magnitude jump occurred in 1906. The values increased without another large magnitude jump until 1940. From 1940 onward the increase was again without a quick jump in magnitude.

Still the table seemed to be missing something that might increase numerical clarity. I showed the above table to Sven for some brainstorming, and he immediately had a suggestion. One could place the appropriate metric prefix at about a 30 degree angle above each of the three digit separations. I thought that using the metric prefix-base unit abbreviation might be best. Sven also thought that some light separation lines might be a good idea. My sense, from what I’ve learned from the book The Visual Display of Quantitative Information, was that this would distract from the data. When I implemented my thoughts, I ended up with:
My eye seems to be drawn to the metric prefixes at the top, which then act as a distracting interpretive boundary while I’m looking at the data. It struck me that a better alternative might be to put the metric prefixes at the bottom:

While this is not perfect, it seems to help allow one to concentrate on the numbers with less distraction. My best suggestion for large dynamic range data is to:

Use the smallest metric prefix that produces the smallest integer value possible for the smallest value in the data set.
Tabulate the data with three order of magnitude separations spelling out the units at the right.
Place the metric prefix-base unit abbreviations below each appropriate column.

I’ve pondered this problem for a long time. This is the first instance where a satisfactory form for large dynamic range data was obtained. This format may very well have been used before, but I don’t have an example (I’m sure my readers will let me know). I’m going to implement this format going forward, and continue to evaluate it. The use of spaces between the metric magnitude triads allows this format to work aesthetically. The column separation is immediately apparent. If commas are inserted as triad separators, the columns merge and become very difficult to cognitively distinguish. Independent of whether this is an optimum choice for large dynamic range data, it is simply not possible to create a table of this form using Ye Olde English units. It illustrates once again the superior nature of the modern version of the metric system’s units and methods.

If you liked this essay and wish to support the work of The Metric Maven, please visit his Patreon Page and contribute. Also purchase his books about the metric system:

The first book is titled: Our Crumbling Invisible Infrastructure. It is a succinct set of essays that explain why the absence of the metric system in the US is detrimental to our personal heath and our economy. These essays are separately available for free on my website, but the book has them all in one place in print. The book may be purchased from Amazon here.

The second book is titled The Dimensions of the Cosmos. It takes the metric prefixes from yotta to Yocto and uses each metric prefix to describe a metric world. The book has a considerable number of color images to compliment the prose. It has been receiving good reviews. I think would be a great reference for US science teachers. It has a considerable number of scientific factoids and anecdotes that I believe would be of considerable educational use. It is available from Amazon here.

The third book is called Death By A Thousand Cuts, A Secret History of the Metric System in The United States. This monograph explains how we have been unable to legally deal with weights and measures in the United States from George Washington, to our current day. This book is also available on Amazon here.

It’s About Time

Posted on 2016-08-10 by Randy Bancroft

By The Metric Maven

Now and then after I’ve given a lecture on the metric system, someone will invariably ask me about metric time. My standard answer is “The only base unit of time in the metric system is the second. It’s up to you to divide them up.” This may sound a bit flippant, but there is a reason for the statement.

Sven once related an interesting anecdote about measuring time. It is from the book Keeping Watch by Michael O’Malley. In 1826, New Haven Connecticut gave Eli Terry, a celebrated clock-maker, $200 to install a clock in their town hall. This would be a clock with the utmost accuracy possible. It was to be a symbol of punctuality and reliability. All seemed well with the clock, but after a while it began to diverge from the other standard of civic time, the Yale College clock. The new clock fell further and further behind until it lagged by about 15 minutes. It then began to catch-up with the Yale clock. There were hopes that the new clock only needed to “break itself in” and that it would, in time, settle down and match the Yale timepiece. To the great concern of the civic authorities, the new clock approached, equaled and then overtook the time displayed by the Yale clock, until it was almost 15 minutes in advance of the college standard. This advancing and retarding of the new clock with respect to the Yale clock continued in a predictable manner. Terry was a competent clock-maker, and the Yale timepiece was also considered to be a good standard, so which clockwork was wrong?

The answer is that each clock represented a different definition of time. Yale’s clock followed the Sun, but Terry’s device was designed to compute mean time, which is an average of the Sun’s daily variation. This difference caused a local controversy about what the proper definition of time should be.

There is still a global brouhaha about the definition of time.^[1] 2016 will be one second longer than ~~last~~ an ordinary year. At the end of this year, clocks will read 2016-12-31 23:59:60 before greeting 2017. This is done to keep Coordinated Universal Time (UTC) synchronized with changes in the Earth’s rotation. Since 1972 twenty-six leap-seconds have been added to deal with the slowing rotation of the Earth. Current standard clocks are more stable than the time kept using the Earth’s rotation. There are some who would abolish the leap-second. This would slowly allow the Sun to drift in its celestial position with respect to time. Those that would abolish the leap-second often cite the difficulties they can create for computer systems.

I recall the concerns that were raised about the dangers of the Y2K bug. Just before the year 2000, there was much concern that because of the abbreviated way time was stored in computers, that 2000 would be the same as 1900. This was a possible error that could affect computer code in an adverse manner it was argued. There were hand-bills posted on trees offering, for a price, to give advice on how to survive the digital Armageddon which was about to visit itself upon us. I went to the most computer knowledgeable and trusted engineer I knew and asked him if there was any reason for concern. He dismissed the idea that a PC would have a problem as its system time is computed from an offset in seconds from January 1, 1980. Wikipedia indicates that the internal clock resolution is 10 milliseconds. Unix time is the number of seconds elapsed since the start of the Unix epoch on 1 January 1970 00:00:00 UT, with exceptions for leap seconds. There are many examples found in the Wikipedia entry for system time, and they are not consistent. System time can be converted to calendar time with months days and include leap seconds.

In his fragmentary, and in my view irrelevant examination of time, John Bemelman’s Marciano (JBM) writes in Whatever Happened to The Metric System?

The correlation of these measures [Earth distance and time] had been a fundamental aim of the metric system, but its creators were inconsistent in their method; they split the earth into 400 degrees and its circumference into 40 million parts but divided the day by ten. A plan to correct this original error was proposed to the American Metrological Society by one of its most fiercely pro-metric members, Fredrick Brooks, who advocated splitting the day into 40 parts. (pg 171)

I’m not sure why time would be included in JBM’s book, other than as a polemic device to impugn the originators of these ideas and then connect them with the modern metric system. JBM claims that the introduction of a ten hour day was an “original error.” What does Wikipedia have to say about metric time?:

When the metric system was introduced in France in 1795, it included units for length, area, dry volume, liquid capacity, weight or mass, and even currency, but not for time. Decimal time of day had been introduced in France two years earlier, but was set aside at the same time the metric system was inaugurated, and did not follow the metric pattern of a base unit and prefixed units.

If you are not comfortable with Wikipedia, Isaac Asimov wrote this in his 1960 book Realm of Measure (pg 100):

The original committee that established the metric system never attempted to do anything with time measurement. They realized that the day and the year were fixed by the rotation of the earth and by its revolution about the sun. Nothing could be done with either. The repetition of day and night, and the seasons was too basic and fundamental to be tampered with, and these simply could not be fitted into the decimal system.

The originators of the metric system did not introduce “metric time” and JBM’s “original error” is more of a contemporary error on the part of the vacuous author of an anti-metric polemic. In her informative PhD thesis: The Role of Five Eighteenth-Century French Mathematicians in The Development of the Metric System (tip of the hat to Amy Young) Ruth Inez Champagne quotes Lagrange concerning metric time:

I observe that, in the measurement of time, the decimal system is much less important for the needs of everyday life than all the other units of measure; since, with the exception of astronomers, no one ever needs to do long calculations with hours, minutes and seconds, ….

JBM indicates that this “original error,” which is so obvious to him, but does not actually exist, made it past the mind of Joseph Louis Lagrange (1736-1813)? Lagrange contributed to the calculus of variations. Lagrange also worked on systematizing mechanics (a lot of algebra) and worked out the mathematics of our solar system. Perhaps you’ve heard of Lagrange Points? But poor Lagrange allegedly could not realize the problem with a ten hour day divided into a 400 degree circumference? My mind reels at the juvenile hubris.

But why on Earth, so to speak, would dividing a circle into 400 degrees make any sense? Ken Alder in his book The Measure of All Things has this to say:

A 400-degree circle (with a 100 degree right angle) would not only ease calculation, it would synchronize astronomy and navigation. In a world where the quarter meridian was 10 million meters long, each degree of latitude would then measure 100 kilometers. This would simplify maps and assist sailors.” pp 141-142

Ferdinand Hassler (1770-1843) also realized the utility of the metric system for surveying. Andro Linklater in Measuring America has this to say:

..the exacting standards he set had become a part of the United States Coastal Survey. As a result it took another fifty-five years to survey the entire coastline from Maine to New Orleans, and every yard of it was measured in meters. Later the Coast Survey was extended to cover the entire United States. The words and Geodetic were added to this title, and the whole landmass was mapped in the same careful, metric fashion.

JBM’s chronology is completely askew in my view. He impugns brilliant mathematical scientists like Lagrange and others for an error that never occurred in 1795, yet his fractional titled chapter 11/16 opens in the 1870s? JBM has chapters designated 1/16, 2/16, 3/16 …..16/16 in the table of contents. Chapter 8, as most of us who like integers might call it, is designated 8/16 or One Half on page 115. Sven rightly points out that one is not half-way through the book until one has read all of this chapter (i.e Chapter 8/16, 1/2, or 8), so perhaps the next chapter should be 9/16 or One Half?—to properly divide the book into halves.

How, by the way, does any of this have anything to do with why the US does not have the metric system? As the old beer commercial used to nihilistically inquire: “why ask why?” The “metric time” section of JBM’s work is just one more part of a monograph who’s origin seems to not rest on its merit, and so if it was not written to explain why the US does not enjoy the metric system, one must wonder why and how it was published at all.

[1] “Leap Second Ahead” New Scientist 2016-07-16 pg 6.

If you liked this essay and wish to support the work of The Metric Maven, please visit his Patreon Page and contribute. Also purchase his books about the metric system:

The Metric Maven

Exploring US Metrication Issues – The Metric System: It's Better By 1000

Category Archives: Metric Reform

The Expanding Universe

It’s About Time