The Maritime Approximation

A transcendent measure of coincidence, maths and metrology

A couple of months back I saw Randall Munroe of xkcd fame live in London. The live tour was celebrating the 10th anniversary of the wonderful What If?, a must read for any nerd/geek/STEMophile.

One quote from the show I loved was his description of why he ended up doing what he does:

Not patient enough to be an engineer, but too anchored in the real world to be a mathematician.

But it’s not the live show, the book or this quote that prompted this essay. It’s this:

The Maritime Approximation

That this was published a few weeks after I saw Randall Munroe is just a coincidence. And that’s what this post is about: coincidence (and nerdery/geekery/STEMophilia).

Putting the e into pi

I’ve encountered a number of fun numerical approximations and coincidences, but it was the first time I’d come across this one. There’s a rabbit hole here that leads to a whole warren of interesting interconnected (and disconnected) facts from mathematics, metrology and history. Let’s explore.

The knot (kt, /nɒt/) is a measure of speed — and yes, historically, knots on a rope were used to measure speed at sea. A knot is a measure of nautical miles (NM) per hour, where a nautical mile is defined as 1852 metres. The statute mile (mi) is ~1609 metres in length. (Note that statute mile is the formal name for what many simply call mile, which is also — a little presumptuously — called the international mile… but mostly because everyone else who used other mile systems switched to metric and stopped using them.)

As both π mph and e kt are measures of distance against time, with the unit of time the same in each case (1 hour), we can simply multiply π mph ≈ e kt through by time to make the equivalence one of distance: π mi ≈ e NM.

To confirm, yes, π here is the ratio of a circle’s circumference to its diameter (hence π denotes the p of perimeter or peripheria). π is irrational (3.14159…), but in many cases rationalising it to 22/7 (within ~0.04%) or 3.14 (within ~0.05%) works just fine and, even in the most demanding cases, you’ll never need more than a few digits. e is the base of natural logarithms, which turns up everywhere from fundamental physics to economics. Like π, e is irrational (2.71828…).

In addition to being irrational numbers — they cannot be expressed as the ratio of two integers — π and e are transcendental numbers. Where an algebraic number can be expressed as the solution to a polynomial equation, a transcendental number cannot. All transcendental numbers are irrational, but not all irrational numbers are transcendental. For example, √2 is irrational but not transcendental because it can be found as the solution to x² − 2 = 0.

Returning to the maritime approximation, we can use Python as a calculator to check the figures:

>>> from math import pi
>>> pi * 1609
5054.822579625977
>>> from math import e
>>> e * 1852
5034.257946306151

Although calculated here to a higher precision than is warranted (1 mile is only 1609 metres to 4 significant figures), the two values are within ~0.4% of one another, which, as the original cartoon states, is correct to <0.5%. This tolerance holds (and the difference reduces to ~0.3%) when we use more approximate values of π and e.

>>> 3.14 * 1609
5052.26
>>> 2.72 * 1852
5037.4400000000005

So, yeah, it works. Nice. We’re done.

Unless, that is, you’re interested in learning more about the units themselves…

Miles and miles

Backing up a little, note that I only called out the statute mile conversion to metres as being approximate and not the nautical mile. This is because the modern definition of the nautical mile is exactly 1852 metres in length. For the statute mile, however, it’s a very different story.

Historically, mile comes to us from mille passus (Latin for 1000 paces, where pace here means a full stride) and was 5000 Roman feet in length. The hegemony of the Roman Empire means the term — if not exact measure — of mile (and foot, translated from the Latin pes) was widely adopted across Europe. There was, however, significant variance, e.g., the Roman mile was ~1480 m, but the Italian mile (miglio) was ~1852 m (which might seem familiar), the Welsh mile (milltir) was ~6712 m, and the Danish mile (mil) was ~7532 m, whereas the old Norwegian mile (miil) was ~11295 m. You get the idea. (As well as the motivation for and value of an internationally standardised system of measurement.)

The mile has wandered with time and (especially) geography. It was often used in conjunction with other long units, such as leagues and furlongs, sometimes made to fit them, sometimes not. Indeed, the measures of most regions, nations and professions is a story of unrelated systems and tangled histories, a mish mash of units and contexts with meandering definitions and haphazard alignment.

In Elizabethan England the day-to-day precision and accuracy of long distances was less significant than, say, land area. Long distances were for travel, and travel time was approximate at best. Land area, however, was a matter of law and ownership, a domain that has always enjoyed greater precision and scrutiny. Land was defined in terms of acres and acres were defined in terms of rods (1 rod = 16½ feet, 1 acre = 40 rods × 4 rods) rather than yards (1 yd = 3 ft) or feet, which were not as commonly used for this.

In an attempt to align systems of measure, it was easier to change the mile — a unit that people did not rely on or use consistently — than it was the acre. To be precise, it was easier to standardise the furlong (1 mile = 8 furlongs) as 660 feet (1 furlong = 40 rods). In 1593, as part of the Exchequer Standards, the 5000 feet of the Roman mile (an easy enough figure to work with) became 5280 feet (a nonsense and reminder of why more rational systems started being devised from the 17th century onwards). Various definitions have been used to make the measurement more consistent and precise, the most recent being in 1959 which leads to a statute mile being defined as precisely 1609.344 m.

All for naut

The nautical mile has enjoyed a more rigorous and less divergent history of definition than its land-derived counterpart.

As European cartography and navigation developed rapidly in the 15th and 16th centuries compared to the centuries and millennia beforehand, the measure of the world and, therefore, the sea became clearer. The absence of roads and landmarks means that navigation at sea is more about looking up than down (an observation that remains true for different reasons in the age of GPS). Both the night sky and the land below it are mapped and marked out in degrees, with 360° to a full circle. These days degrees tend to be subdivided by decimal, but historically they were divided into minutes (60' to a degree) and seconds (60'' to a minute), the DMS system. As an aside, Google Maps interprets and displays both DD (decimal degrees) and DMS, as well as parsing DDM (degrees with decimal minutes).

The definition of the nautical mile that emerged from this era was the distance corresponding to one minute around the polar circumference, thus 60 nautical miles to a degree of latitude.

Given this definition, and taking the meridian arc from 0° (the equator) to 90° (the pole) to be about 10 000 000 m, we can calculate how far a nautical mile is.

>>> 10_000_000 / (90 * 60)
1851.851851851852

That would be the end of the story if the Earth were perfectly spherical — it would just be a matter of getting ever more precise measurements of the Earth’s circumference. But it isn’t. The Earth is a little flatter at the poles, so that 1' of latitude is 1861.6 m at the poles and 1842.9 m at the equator, meaning a difference of nearly 20 metres (~1%) between a polar and an equatorial definition of nautical mile.

Such variability would obviously defeat the purpose of the unit. Different nations settled on different fixed definitions, such as defining the nautical mile by 1' of latitude at 0°, or at 45°, or averaged. In 1929 the nautical mile was standardised to be precisely 1852 m.

The nautical mile continues to be used internationally in aviation, seafaring and the definition of territorial waters. This also explains why you will sometimes find untranslated mentions of miles in non-English films and texts (legacy units are normally translated into equivalent metric quantities). For example, in Das Boot, when the submariners talk about miles, they are referring to nautical miles — the film is set in 1941, by which time Germany had officially been using the metric system for nearly seventy years (since 1872).

High seas

So, just to recap, the nautical mile continues to be used internationally in aviation, seafaring and the definition of territorial waters… and space travel.

Although NASA uses SI units for most of its technical work (although back in the 1990s Lockheed Martin did not appear to get the memo), nautical miles are sometimes also used when dealing with matters of astrogation.

But what if we took that original definition of the nautical mile out into space? What if we used the definition of 1' of latitude on other worlds? We can skip past Mercury (hot as Hell), Venus (actual Hell), Jupiter (surface… what surface?), etc., and instead focus on celestial bodies humans actually have some chance of setting foot on to explore.

Let’s start with our nearest neighbour, the Moon, a world humans have already walked on. Using the equatorial radius plus the polar radius (i.e., a + b, the semi-major axis plus the semi-minor axis of an ellipse) as a decent approximation for the mean diameter of an oblate spheroid, we can determine its polar circumference and then divide it into minutes:

>>> pi * (1_738_100 + 1_736_000) / (360 * 60)
505.28736286279167

Meaning that a lunar nautical mile is around half a kilometre (within ~1%), which is both a delightful coincidence and a convenient figure to work with.

Let’s head to Mars:

>>> pi * (3_396_200 + 3_376_200) / (360 * 60)
985.0056521838663

Meaning that a Martian nautical mile is around a kilometre (better than 2%) — an even more delightful coincidence and convenient figure.

While the Moon is a dry world with features named ‘seas’ and Mars is a dry world that once had flowing water, Titan is a world that actually has seas:

>>> pi * (2_574_910 + 2_574_330) / (360 * 60)
748.9265997949401

At ¾ km (within ⅐%), a nautical mile on Titan falls midpoint between that of the Moon and Mars. Of course, Titan’s seas and lakes are of liquid methane and ethane rather than water, and a human landing is exceedingly (and disappointingly) unlikely within my lifetime. But who knows, perhaps one day explorers will navigate Titan using a new nautical mile across new seas.

A question of identity

The only part of the original cartoon left to discuss is Euler’s identity, which is mentioned below the main frame:

It is more conventionally written as

Euler’s identity graffitied on a wall about 10 minutes walk from where I live

A favourite equation of many, often considered the most beautiful equation (or perhaps the pumpkin spiced latte of the maths world), it has it all. Dave Percy observes that

It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants — 0 (additive identity), 1 (multiplicative identity), e and π (the two most common transcendental numbers) and i (fundamental imaginary number).

It also comprises the three most basic arithmetic operations — addition, multiplication and exponentiation.

Given that e, π and i are incredibly complicated and seemingly unrelated numbers, it is amazing that they are linked by this concise formula.

We can calculate it easily in Python, which supports complex numbers as a built-in type:

>>> e ** (1j * pi) + 1
1.2246467991473532e-16j

There are a couple of things to note about this:

1. Python uses the electrical engineering convention of j rather than i for the imaginary unit. Either way, it has to be prefixed by a number to distinguish it from a variable name, hence 1j rather than j.
2. It’s not 0.

With any floating-point calculation we should expect some loss of accuracy, especially where powers are involved. In this case, there is a residual non-zero imaginary amount that lies beyond the precision we have for either e or π.

>>> e
2.718281828459045
>>> pi
3.141592653589793

Even NASA uses no more than 15 digits of π. (Uncoincidentally, 15 digits is also the highest precision of π supported by 64-bit floating-point numbers.)

So, if we take the magnitude of the result and round it to 15 places, we arrive at our expected destination.

>>> round(abs(e ** (1j * pi) + 1), 15)
0.0

Putting the maritime approximation into a similar form, i.e., the distance equivalence becomes π mi − e NM ≈ 0, and rounding somewhat more aggressively (in this case, −2 places), we reach the same destination.

>>> round(pi * 1609.344 - e * 1852, -2)
0.0

Euler’s identity describes an intimate and mathematical relationship between different fundamental constants. And now, thanks to the maritime approximation, we also have a loose and coincidental one.