2170: Coordinate Precision
Title text: 40 digits: You are optimistic about our understanding of the nature of distance itself.
This cartoon gives increasingly precise latitude and longitude coordinates for a location on the planet Earth. However, a given pair of coordinates covers a trapezoidal region of land, and thus leaves some ambiguity; therefore, greater precision requires an increasing count of decimal places in your coordinates. This comic uses this information to roughly identify how precise a given coordinate length might be.
The increasing precision of coordinates in this cartoon are similar to the increasing magnification in the short documentary "Powers of 10," which can be found here. (Also parodied in #271:Powers of One).
The coordinates at 28.52345°N, 80.68309°W (in decimal degrees form; in geographic coordinate system form using degrees, minutes, and seconds, 28° 31′ 24.4″N, 80° 40′ 59.1″W) are pointing to the Rocket Garden at the Kennedy Space Center in Merritt Island, Florida —specifically, the tip of the Delta rocket.
The sixth entry in the table, with seven digits of precision, includes the caveat that, while your coordinates map to areas small enough on the Earth's surface to indicate pointing to a specific person in a room, "since you didn't include datum information, we can't tell who". This is a reference to the geodetic datum or geodetic system — different ways of dealing with the fact that the Earth is neither perfectly spherical nor perfectly an oblong ellipsoid. The various datums do not make much difference at six digits of precision, but at seven, there is enough skew depending on which system is in use that the person in a room you are referring to with the coordinates is ambiguous. It is unstated, but the remaining lines in the table with ever-greater precision suffer from this same issue and are equally ambiguous without datum information.
The final entry, with seventeen digits of precision, suggests that either the user is referring to individual atoms in the much-larger-scale whole-Earth coordinate system, or (perhaps more likely) has not bothered to format the values from the GPS module for viewing in the software UI in any way whatsoever, resulting in a value that is meaninglessly precise because the measurement wasn't that accurate to begin with. Even if the value is accurate, locating individual atoms by coordinates is not actually useful in most cases, and the motions of multiple systems within our physical world (continental drift, subtle vibrations, Brownian motion, etc.) would render the precise value obsolete rather quickly.
For the decimal places past the 5th on the latitude, the digits given are actually the first part of the decimal expansion of the constant e (2.7182818284), while for the decimal places past the 5th on the longitude, the digits given are part of the decimal expansion of the constant π (3.14159265358) starting with the second digit (4).
The title text references how at sufficiently small distances, our understanding of reality itself begins to break down. Smaller than the Planck length, which is more than a quintillion times smaller than the diameter of a proton, the ideals of Euclidean geometry no longer apply and space itself may be composed of a quantum foam where the very geometry of spacetime itself fluctuates, meaning coordinate systems based on an assumption that space doesn't change would no longer work. String theory, on the other hand, assumes that at a short enough distance the world is composed of ten space dimensions, which precludes the use of a two-dimensional coordinate system (not that our “normal” three dimensions don't do so in themselves).
The actual number of longitude digits needed to identify a point to a particular precision depends on its latitude. Near the poles, you need fewer longitude digits than at the equator – starting with one digit fewer at around lat. 85°, past all constantly inhabited human settlements, and with two digits fewer at lat. 89.5°, inaccessible to anyone but polar researchers and the occasional guided tour. The number of latitude digits for some particular accuracy stays essentially the same everywhere.
|Decimal places||Resolution*||In the comic||Location||Explanation/notes|
|0||110 km (70 mi)||Something space-related||Somewhere near the east coast of Florida||This resolution is enough to point out a large-scale feature like a country, a mountain range, a large lake, or a significant island on a map of the world. It can also be used to tell if certain celestial phenomena are visible from a given location.|
|1||11 km (7 mi)||A specific city||Cape Canaveral||Cities typically span a couple kilometers/miles in diameter and are far enough from each other to distinguish them at this resolution. There are exceptions though, and the veracity of this statement depends greatly on the definition of a “city”, which varies by location and history.|
|2||1.1 km (¾ mi)||A neighborhood||Kennedy Space Center Visitor Complex|
|3||110 m (360 ft)||A suburban cul-de-sac||The Rocket Garden at the Kennedy Space Center|
|4||11 m (36 ft)||A particular corner of a house||Somewhere near the center of the Rocket Garden|
|5||1.1 m (3½ ft)||A specific person in a room (given geodetic datum information)||The Thor-Delta rocket in Rocket Garden|| As the comic notes, the differences between geodetic datums – different ways to map geodetic coordinates to specific points on the Earth's surface – become large enough that one needs to specify the one in use when supplying coordinates to this degree of precision (or greater, of course). Since the Earth is not a perfect ellipsoid, different parts of the planet conform best to ellipsoids of slightly different proportions, resulting in different coordinates for a specific location; not to mention that locally used datums have local reference points, which means that the local and global standards are slowly drifting away from each other with the tectonical plates.
Note that the comment in the comic concerns only the NAD 1983 datum which is fairly close to the international, “one size fits all” standard WGS-84. Other datums may be shifted by tens or even hundreds of meters (yards), making geodetic datum specification necessary for less precise coordinates as well.
|7||1.1 cm (⁷⁄₁₆ in)||Waldo on a page||Presumably the very tip of the rocket|| This refers to Where's Waldo?, a series of books and magazines containing various scenes (densely packed with people) where one must find Waldo, a character wearing a red and white striped shirt. In the puzzles, he usually stands less than 2 cm (1 in) tall.
Finding Waldo on a page using satellites was also referenced in #1358.
|9||0.11 mm (4⅜ thou)||A specific grain of sand||N/A|
|15||110 pm (1.1 Å)||Raw floating point precision or an individual atom|| A double-precision (64-bit) floating point variable stores 52 significant bits (with an implicit 1 in front), so that 180.00000000000000 and 179.99999999999997 may be represented as distinct values. (This is only 14 decimals, however; the larger the integral part, the fewer bits remain to represent the fractional part.) This level of precision is useful for mitigating rounding errors in computations, but this advantage only shows if the last few digits are treated as non-significant and thus, ideally, hidden from view. To work with data that is actually this precise – like tracking individual atoms or representing continental drift up to the second –, one must make allowance for these additional non-significant digits and store the coordinates in quadruple precision.
To track atoms, however, one needs very sensitive (and expensive) equipment with a severely limited range (according to our current understanding of science and technology). Using a global-scale coordinate system when a micrometer-scale would fit much better is either an abuse of the system and a great waste of memory and computing power, or it means that a significant portion of the Earth's surface has been blanketed by quantum microscopes, which would be an abuse and a waste of many other things as well.
|40||1.1 × 10–11 ym (1.1 × 10–35 m)||Near (or past) our current understanding of the nature of distance||This is where the resolution reaches the Planck length (1.6 × 10–35 m). At this scale, the very structure of spacetime (and thus, the notion of distance) may be different than what we know; measuring anything to Planck length precision would necessitate such tremendous amounts of energy in one place that would create minuscule black holes, warping spacetime further (in addition to wreaking havoc with whatever you were trying to pinpoint).|
*Since the Earth is not exactly spherical, the actual length of one degree of latitude varies between 110.574 km (68.707 mi) at the equator and 111.694 km (69.403 mi) at the poles, while one degree of longitude is 111.320 km (69.171 mi) at the equator, 55.800 km (34.673 mi) at lat. 60°, and 0 km (0 mi) at the poles.
- [Single panel containing a table with two columns for "Lat/Lon Precision" and "Meaning" and a caption above the table.]
- Caption: What The Number of Digits in Your Coordinates Means
- [Row 1]
- Lat/Lon: 28°N, 80°W
- Meaning: You're probably doing something space-related
- [Row 2]
- Lat/Lon: 28.5°N, 80.6°W
- Meaning: You're pointing out a specific city
- [Row 3]
- Lat/Lon: 28.52°N, 80.68°W
- Meaning: You're pointing out a neighborhood
- [Row 4]
- Lat/Lon: 28.523°N, 80.683°W
- Meaning: You're pointing out a specific suburban cul-de-sac
- [Row 5]
- Lat/Lon: 28.5234°N, 80.6830°W
- Meaning: You're pointing to a particular corner of a house
- [Row 6]
- Lat/Lon: 28.52345°N, 80.68309°W
- Meaning: You're pointing to a specific person in a room, but since you didn't include datum information, we can't tell who
- [Row 7]
- Lat/Lon: 28.5234571°N, 80.6830941°W
- Meaning: You're pointing to Waldo on a page
- [Row 8]
- Lat/Lon: 28.523457182°N, 80.683094159°W
- Meaning: "Hey, check out this specific sand grain!"
- [Row 9]
- Lat/Lon: 28.523457182818284°N, 80.683094159265358°W
- Meaning: Either you're handing out raw floating point variables, or you've built a database to track individual atoms. In either case, please stop.
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