3023: The Maritime Approximation

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The Maritime Approximation

Title text: It works because a nautical mile is based on a degree of latitude, and the Earth (e) is a circle.

Explanation

Mph (miles per hour) and knots (nautical miles per hour) are both units used to express speed, including that of vehicles. Miles per hour are typically used in the US, UK and some smaller countries for the speed of cars and other similar vehicles, while knots are used by many sailors and pilots to describe the speed of ships and aircraft. Novice sailors or pilots, or those who spend a lot of time on land, may find it helpful to quickly convert between mph and knots, in order to relate to typical ground-surface speeds.

This could be done in the form of 1 knot = 1.2 mph, or 1 mph = 0.87 knots (1 knot = 1.85 km/h and 1 km/h = 0.54 knots for metric navigators). Randall has humorously noticed that π mph ≈ e knots: π mph = 2.72997 knots, while e ≈ 2.71828.

Knots are related to the circumference of the Earth, which can introduce π, but this is only "useful" if you want to express your speed as a fraction of the radius of the Earth: 1 knot = 1 nautical mile per hour = 1/60 of a degree of Earth's circumference per hour = 1/21,600 of Earth's circumference per hour = 2π/21,600 x Earth's radius per hour. However, nowadays this is an approximation, because a nautical mile is defined as exactly 1852 m, which is not exactly 1/60 of a degree of Earth's circumference.

The observation of the interesting near identity between MPH and knots in the comic is misleading, because it is not exact, but only correct to a certain percentage, unlike the identity it is compared to: Euler's Identity, which is exact and expresses a deep mathematical insight, which is what makes the latter truly remarkable. The former is nothing but an unimpressive, if mildly interesting coincidence. This isn't helped by the fact that the comic carries the implication that this neat, easy-to-remember identity is actually useful for sailors, when really, being easy to remember is all it has going for it: it doesn't make calculations any easier, it is impossible to do without a calculator or paper, and doing it on paper is much harder than other conversions, given that π and e are both irrational and transcendental. Finally and most importantly, this conversion between knots and MPH is far far less accurate than the typical conversion factor used, i.e. 1.1508, which is accurate to within 0.00179%; about 280x better than Randall's. This can make a huge difference on shipping routes, which can be hundreds or thousands of miles long.

The title text furthers the joke that this identity between MPH and knots is truly fundamental, but through faulty logic. Whenever π shows up in an equation, the claim made by many mathematicians is that there is a circle hiding somewhere in the math. Randall says that π is coming from the fact that nautical miles are based on the fact that the Earth is round, and shipping routes over its surface are circular. As profound as this sounds, it makes no mathematical sense at all. He also claims that e is in the equation because 'Earth' starts with an E, which is nothing but word play.

The equality shown in this strip consists of several different parts:

1. The mile (1609.344 m) per hour (mph) is a unit of speed common for motor vehicles in a few countries, such as the United States and United Kingdom.
2. The knot is a unit of speed that is one nautical mile (1852 m) per hour, used in nautical contexts.
3. π is a number equal to the ratio of a circle's circumference to its diameter, about 3.14159.
4. e is Euler's number, the base of the natural logarithm, about 2.71828.

π mph × (1609.344 meters/statute mile ÷ 1852 meters/nautical mile) ≈ 2.729969 knots. The result is only about 0.43% larger than e knots ≈ 2.71828 knots.

Randall has in the past made similar observations of different dimensions that equal each other with comics such as 687: Dimensional Analysis, where he compares Planck energy, the pressure at Earth's core, the gas mileage in a Prius, and the width of the English Channel to π. In addition, in What If?, he has compared the mass of Earth to be π "milliJupiters," or π times the mass of Jupiter divided by 1000, and noted that the volume of a cube with side lengths of one mile is roughly similar to the volume of a sphere with a radius of 1 kilometer. In 217: e to the pi Minus pi and 1047: Approximations, Randall gives a lot of similar numerical approximations.

Arguably, as safe operating speeds for particular aircraft/watercraft may bear little relationship to (for example) road vehicle speeds, it might be better just to develop a separate 'air sense' (perhaps mostly at higher velocities, far above any landmark that you might pass by) or 'water sense' (often at lower velocities, and with the particular fluid nature of the water's surface) that is keyed especially to the knots-rated speed of your vessel, without attempting to carry over this aspect of any pre-existing 'road sense'. One hopefully rare exception might possibly be in the event of a plane having to make an emergency landing on a public highway, where it could be useful to know if a (possibly unpowered) plane's final landing speed can be made to be not too far off that of any unsuspecting road traffic that you may have to land in the midst of; but this would never be a trivial endeavour in any case, and even having to attempt such a feat probably means you have few options open to you and very little time to consider many of these finer details.

Transcript

[In a small panel an equation is shown. There is a footnote below the equation:]

π mph = e knots*

*Correct to <0.5%

[Caption below the panel:]

The sailor's version of e^iπ=−1

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Discussion

1.609*3.1416926 looks like 1.852*2.718281828 seems legit 172.71.124.233 (talk) 21:37, 11 December 2024 (UTC) (please sign your comments with ~~~~)

I added the basics of an explanation, it definitely needs some work, but it should do as a starting point. Hope I did well! 172.68.22.92 23:06, 11 December 2024 (UTC)

The knot is exactly 1 nautical mile per hour. Meanwhile π/e ≈ 1.155727, which is close to nm/mi = kt/mph ≈ 1.15078 172.70.134.135 23:26, 11 December 2024 (UTC)

This article says one knot is 1.2 MPH, which is true for the number of digits of precision stated. But in context of the claimed precision of 0.5% it would be more helpful to state that one knot is approximately 1.151 MPH. https://en.wikipedia.org/wiki/Knot_(unit) 172.71.159.7 00:08, 12 December 2024 (UTC)

Transcendental : relating to a spiritual realm. eg "the transcendental importance of each person's soul". Works for me. 162.158.186.248 (talk) 00:09, 12 December 2024 (UTC) (please sign your comments with ~~~~)

Just as a fun fact, "transcendental" in this case is referring to Transcendental number, which are numbers that cannot be expressed as the root of a polynomial, which basically means they cannot be found using algebra alone. I think the two definitions are related, since these numbers "trancend" the "realm" of numbers which can be found with algebra. 172.68.22.82 01:04, 12 December 2024 (UTC)

Another maritime approximation: 1 meter/sec nearly equals 2 knots (actual is 1.94384), perhaps there is an actual explanation for this? 162.158.155.117 (talk) 01:36, 12 December 2024 (UTC) (please sign your comments with ~~~~)

Both the nautical mile and meter derive from measurements of the Earth's circumference, and the number of seconds in an hour is related to the base-60 counting system (as is the number of degrees in a circle), but beyond that it's just how the math works out. 1 nautical mile is (well, was) 1/60 of a degree of latitude. 1 meter is (was) 1/10,000,000 of the distance from the Equator to the North Pole, which is 90°, so that's 9/1,000,000 of a degree of latitude. So 1 m = 27/50,000 nmi. Then, an hour is 3600 s. So 1 m/s = 27∙3600/50,000 nmi/hr. Cancelling, that's 1 m/s = 243/125 nmi/hr, and that fraction is quite close to 2. But there's no real deeper connection.172.70.115.102 15:08, 12 December 2024 (UTC)

A better mnemonic, which I actually use: miles→km is Fibonacci. 2miles≈3km, 3miles≈5km, 5miles≈8km, 8miles≈13km, 13miles≈21km, 21miles≈34km, 34miles≈55km, 55miles≈89km, 89miles≈143.23km, Fibonacchi would predict 144km. But at that point, you can just remove some less significant digits anyway. For everything in between, you can estimate how far it is from the nearest Fibonacci numbers, that works pretty well, too. Fabian42 (talk) 01:54, 12 December 2024 (UTC)

Yes, similar to this comic the ratio of km to miles (1.6093) is very close to the golden ratio (1.6180) or (1 + sqrt(5))/2. 172.68.54.64 04:28, 12 December 2024 (UTC)

My favorite one is that pi squared is approximately the acceleration of gravity (9.8 m/s^2). The best part is that is NOT a coincidence. 172.71.183.174 06:11, 12 December 2024 (UTC)

How is this not a coincidence? (Wowitschris (talk) 20:56, 12 December 2024 (UTC))

It's actually a coincidence, but in a weird sense it almost would not be. When French revolutionaries invented the metric system, they considered defining the meter as the length of a pendulum with a period of one second. If they had done that, g would be exactly pi squared. But it was already known back then that g varies with location, so the actual definition based on earth's circumference was adopted --141.101.105.87 06:39, 13 December 2024 (UTC)

Actually the most common form of Euler's identity is e^iπ + 1 = 0; I find it odd that Randall never writes it that way (see 179 and 2492 for example). --172.69.68.4 12:47, 12 December 2024 (UTC)

The form that you wrote and Randal's preferred form are identical. The equations are slightly different, but they are the same form. Other forms would involve using trigonometric functions, infinite series, integrals or ... something else. Galeindfal (talk) 18:38, 12 December 2024 (UTC)

If you don't like calling it "form" say "appearance" or "arrangement"... anyway the most common is e^iπ + 1 = 0 because it includes the three main operations and 5 fundamental constants, so it is usually regarded as aestetically more satisfying.

--172.71.114.137 12:01, 13 December 2024 (UTC)