What's the largest pizza the Pope could order?
This was an idea that came about when ordering pizza with friends, where we didn't have a very large table to put all the pizza on. The question arose: how large a pizza could we order and have them all fit on the table? I don't remember the exact answer to that particular question, but I do recall where it went next: what's the limiting factor at a larger scale? How large a pizza could the citizens of a small, densely-populated country order and still have them all fit within the country area? As a fan of Randall Munroe's What If? this sort of 'apply maths to stupid question' problem appeals to me, so I had a go.
The nesting of 2D shapes in an irregular planar area is a tricky subject, so to simplify, we'll consider any country to simply be a hexagon. Why hexagons? Because the most dense arrangement of circles on a plane is hexagonal close packing, or 'HCP'.
A unit cell of close-packed pizzas. Drawn using TikZ for maximum complication.
The simplest way to calculate the number of pizzas we can pack into our hexagonal country is to just use the packing density factor for HCP, i.e. the proportion of the plane that will be covered by pizza. For 2D hexagonal close packing, this is approximately 0.907. This assumes that we can shuffle around the space available (and the pizzas) to achieve this density, which is probably close enough for countries with a population above 10000.
But which countries should we provide our pizza for? Obviously the most entertaining answers will come from the extremes: at one end, Monaco's 38300 residents live in a mere 2.02km2, presumably feeling better about the crowding because the GDP per capita is an impressive $115700, largely due to the country's relaxed attitude to taxing foreign income. Giving every Monégasque a pizza of some 7.80m in diameter would give the city-state crust-to-crust coverage in our idealised hexagonal packing. This is already larger than a typical pizza restaurant can supply, but smaller than the largest pizza ever made, at some 40m in diameter. Pleasingly, this pizza was actually circular, unlike the world's largest commercially-available pizza which has a pragmatic (and disappointing) rectangular shape, although it does manage to look a bit more appetising in the pictures, probably because it can be cooked properly in one go.
At the other end of the population density spectrum is Mongolia, where 3.2 million people share 1.6 million km2, allowing them a pizza of nearly 747m in diameter. This is a lot of pizza - more than 18x the size of the current record-holder. Apparently the Mongolian capital of Ulaanbaatar has a number of pizza restaurants, so if any of them fancy this as a marketing gimmick, then they're welcome to it.
As an aside, so far we've used arithmetic population density to look at each citizen's pizza size. This slightly misrepresents how the local population would perceive how crowded things are, because most countries with a low population density are also urbanized, with most people living in a few large cities and towns. Alisdair Rae's work on the 'lived density' of Europe goes some way to illustrating this. As an example, Spain has a population density of 93 people per 2.02km2 when calculated purely arithmetically, but if you divide the country into 1x1 km squares and calculate the density of only those that are inhabited, each of these contains an average of 737 people. This dramatic increase is because a staggering 87% of given 1km squares in Spain are devoid of people - everyone lives in the cities, leaving most of the country even emptier than the mean density might suggest.
For our Mongolia example, using arithmetic density would require most people to walk a long way into some pretty inhospitable terrain to reach their personal pizza. If we restrict our travel distance to keep people in the city or town where they live, rather than suffocating the entire landscape under dough, we get much more reasonable sizes. In Ulaanbaatar for example, where approximately 45% of Mongolian live, we get pizzas a mere 61m across. This is starting to get into the range where the 12 branches of Pizza Hut in the city (who knew?! presumably a lot of Mongolians did...) could get together and make at least one pizza of this size as a proof of principle.
In the case of Monaco, the lived density and arithmetic density are identical - being a city-state means that there isn't much spare land to go around, so this won't change the pizzas at all.
So far we've stuck with countries that are (more or less in Monaco's case) clearly countries, with a police force and military, and some degree of sovereignty and its own economy. However, there are some territories which are a bit more of a grey area - Macau, for example, where our pizza would be marginally smaller than Monaco's at 7.38m.
However, as the title suggests, Vatican City might be the best bet for pizza ordering - mostly because one could pretty easily find a good restaurant who could deliver from Rome. While based on the actual residents of the territory, the world's smallest country has a density of 924 people/km2 (a if we allow the Pope to invite all the citizens for a pizza party (with citizenship being given primarily to those with ecclesiastical jobs in the Catholic Church, and being revoked on retirement from that role), we have 825 people in a mere 0.49km2, each eating a pizza 26.2m across. This is a lot of pizza, and I'm not completely convinced that the Pope could finish his in one sitting, but on the other hand it's significantly smaller than our world record pizza, made in Rome. It's doable, just about, but I think I'll struggle to convince the Pontiff than the best use of his time is covering the whole of his Kingdom in food. More delightfully, even if he did over-indulge a bit, it turns out that occasional over-eating seems not to have immediate negative consequences for metabolic health. I think I might go get a pizza....
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