| EASY MATHEMATICS

This morning, as I idly scrolled through my Twitter feed while awaiting the first lesson of the day, I came across the following infographic:

Qatar Airways, justifiably proud of their new record-breaking route from Doha to Auckland have offered this statistical elucidation of its gruelling 1065 minutes (equivalent to 426 viewings of a plot summary of Wagner's Ring Cycle).

One of the comparisons, in particular, caught my eye. It just seemed a little too neat to be true.

Isn't it an extraordinary coincidence that the length of the flight, to five significant figures is equivalent to the length of the coastline of the country to whose largest city you're flying?

You might have thought that it would be an easy fact to refute in the age of the internet. Let's just look it up on Google:

About 15000km? About? What sort of answer is that?

If I asked it how many strings a cello has would it say, "about four"?

No!

In fact it gets worse for the fact-checking fraternity (don't tell Donald Trump - he'll tweet some outlandish claims about the length of coastline on the border between USA and Mexico) because non-Google sources contradict the 15000km figure.

The World Resources Institute puts the figure at 17209km, suggesting that Qatar Airways needs to up its game and find a longer flight. That's both more precise and a fairly major disagreement with Google. Should we accept it because it's less vague?

Why is it that Google isn't sure about the length of New Zealand's Coastline but is pretty bloody certain about its area?

The answer is, of course, mathematical (which is why I'm bothering to write about it) and lies with the Coastline Paradox.

Here's a map of New Zealand:

If you have extraordinarily good eye-sight, you will see that there is a length labelled "200km" in the bottom right hand corner. Suppose I were to use it to try and measure the coastline:

Eleven of these lengths make up the south island, and twelve make up the north island. Twenty three lengths of 200km gives a total coastline of 4600km (equivalent to a flight from Doha to Landstuhl, Germany).

I'm about 10000km short of Qatar Airways' figure.

A second glance at my fairly ropy measurement should illustrate why. Owing to the 200km base length, I have had to take certain liberties with the geometry of the coastline; certain liberties that include consigning the capital city to the Pacific Ocean!

If I were to halve my unit I could follow the coastline more accurately:

Here, the north island now has 2500km of coastline. We've gained an extra 100km just by altering the smallest unit of measurement. Suppose we were to measure in 10km chunks, or 1km chunks; every time we increase the detail, we find more coastal nooks and crannies and the total length increases.

This means that Qatar Airways can happily justify their coincidence by choosing how precisely they'd like to inspect the inlets and outcrops surrounding the country.

In fact, the closer you look, the more length you find.

This might be worth bearing in mind for any airline that tries to beat Qatar's record with a longer commercial flight from Auckland to, say, the moon. Given a small enough ruler, it will be true to claim, "Solar System's longest flight; Sea of Tranquility to Auckland; 384400km (equivalent to the full New Zealand coastline".