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.
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:
- 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.
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The Metric Maven has published a new book titled The Dimensions of The Cosmos. It examines the basic quantities of the world from yocto to Yotta with a mixture of scientific anecdotes and may be purchased here.