Editing 2170: Coordinate Precision

The coordinates at [https://tools.wmflabs.org/geohack/geohack.php?pagename=Cape_Canaveral&params=28.5234_N_80.6830_W_type:landmark_region:US-FL_scale:10000 28.5234°N, 80.6830°W] (in {{w|decimal degrees}} form; in {{w|geographic coordinate system}} form using degrees, minutes, and seconds, 28° 31′ 24.24″N, 80° 40′ 58.8″W) are pointing to the {{w|Rocket Garden}} at {{w|Kennedy Space Center}} in {{w|Merritt Island, Florida}} (usually referred to as '''{{w|Cape Canaveral}}''')—specifically, the tip of the [https://www.kennedyspacecenter.com/-/media/DNC/KSCVC/Blog-Images/Rocket-Garden/rocket-garden-with-labels.ashx?h=860&w=1173&la=en&hash=7B9ADC7AFF5370E462AC98D9651945B806B77B2C 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 probably a reference to the fact that persons are animate, and different persons can occupy the same position at different times. It might also be a reference to the various ''{{w|geodetic data}}'' or ''geodetic systems'' — different ways of dealing with the fact that the earth is neither perfectly spherical nor perfectly an oblong ellipsoid. The various data 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 {{w|False precision|meaninglessly precise}} because the measurement wasn't that {{w|Accuracy and precision|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, 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 decimal expansion of the constant ''e'' (2.7182818284), while for the decimal places past the 6th on the longitude, the digits given are the decimal expansion of the constant ''pi'' (3.14159265358).

The title text references how at sufficiently small distances, our understanding of reality itself begins to break down. Smaller than the {{w|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 {{w|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.

| 110 km (70 mi)

| 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 what constitutes a “city”, which varies greatly by location and history.

| A neighborhood

| Kennedy Space Center Visitor Complex

| A suburban cul-de-sac

| As the comic notes, different persons can occupy the same position at different times. It might also point to the differences between {{w|geodetic datum}}s – 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.

|

| 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 less 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 actually track individual atoms (which may be as close to one another as 32 pm) or to represent continental drift up to the second, one would need to work with coordinates in ''quadruple'' precision.

Tracking atoms, however, 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.

| This is where the resolution reaches the Planck length. 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 it would inevitably create minuscule black holes.

<nowiki>*</nowiki>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 (64.673 mi) at lat. 60°, and 0 km (0 mi) at the poles.