Coordinate Precision
40 digits: You are optimistic about our understanding of the nature of distance itself.
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 oblate 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. See 2696: Precision vs Accuracy. 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.