# Maths of the Pokédex

Sam Flower

### Weighty Matters

Introduced in the Generation 3 games Pokémon Ruby and Pokémon Sapphire, Wailord was at the time the biggest Pokémon ever, with a given ‘height’ of 14.5m, and a weight of 398kg. Yet if you do the maths, it turns out the name ‘the float whale Pokémon’ is very apt…

The Pokédex is rather ambiguous on what counts as a Pokémon’s height. Considering that the ‘heights’ given for snake-like Pokémon such as Arbok appear to measure the length of the Pokémon’s fully outstretched body, it seems reasonable to assume the ‘height’ in fact measures the Pokémon’s longest dimension. This would suggest Wailord is 14.5m in length. But how tall is it?

Based on this image from the Sword and Shield games, the length of Wailord seems to be about 5 times the average height above the water of the Pokémon. The above water, blue bit of Wailord takes up about half of Wailords body, giving a rough height to length ratio of 2:5. Given a length of 14.5m, we thus get that Wailord is about 5.8m high. If we model Wailord as a cylinder of length 14.5m, and diameter 5.8m, we get a rough volume for the Pokémon of 383m3. If we divide Wailord’s mass of 398kg by this 383m3 volume, we get a density of 1.04kg/m3. This is a small number. To see quite how small, consider the fact that air has a density of 1.225kg/m3. Wailord is less dense than air! If it really existed, it would float up into the air like a giant blue blimp. The float whale Pokémon indeed.

Clearly Gamefreak did not run the numbers when they came up with Wailord’s Pokédex entry. But what would be a sensible weight for Wailord? Let’s look at real life whales. In particular, let us compare it with the Sperm Whale, which has a similar body shape. They can also both dive to a depth of almost 10,000 feet. The length and weight vary significantly between male and female sperm whales with a male sperm whale around 16m long, with a weight of 41 tonnes (41658kg), while the females are on average 11m long, and weigh a ‘mere’ 14 tonnes (14225 kg).

If we take the smaller female sperm whale, we get that a Wailord is 1.32 times as long as a female sperm whale, and so by the cubic scaling properties of volume (and hence weight), we get that an equally dense Wailord should weigh 14 tonnes x 1.323 = 32 tonnes (32513kg). That’s 81.7 times more than the Pokédex says it weighs! If we instead use the density of a male sperm whale as a reference, we get a slightly smaller estimate of Wailord’s weight, at about 30.5 tonnes. This is still 77.9 times the weight given in the Pokédex.

In conclusion, Wailord’s current weight is so stupidly low that it is literally lighter than air. A more sensible weight would be around 30 tonnes. Beyond just the silly mental image of a floating whale, I think this example nicely shows that what seems like a big number (almost 400kg!) can actually be woefully small depending on context. Always do the maths!

### Supermassive Black Holes

On the other extreme of the density scale, we have Cosmoem. It’s given height, which we again assume to be it’s horizontal width, is 10cm.  It’s very small. Yet it weighs 999.9kg! It weighs a literal tonne, but is small enough to fit in your hand!

By observing its 3D game model and official art from both directions, I estimate that Cosmoem consists of a sphere of 45cm diameter, intersecting a 10cm by 5.4cm ellipse, with 8 approximately half ellipses 2.85cm by 4.5cm. If we take the brown bits to have a width of half a cm (which feels generous), and work out the maths, we get a (very approximate) volume of 10-4 m3. Which is small. Dividing the 999kg mass by this, we get a density of approximately 107 kg/m3. That is, a 1m by 1m by 1m block of Cosmoem stuff would weigh 10 million kg. For comparison Osmium, the densest material on Earth, has a density of only 22610 kg/m3, over 400 times less dense than Cosmoem. The core of the sun has a density of around 1.5 x 105 kg/m3. The Sagittarius A* black hole had a density 10 x less than Cosmoem. This does not mean Cosmoem will warp space and time around it (it still weighs less than most cars), but it does show the sheer ludicrous weight of what is one of the smallest, yet the joint heaviest, Pokémon ever!

### Feel the heat

Another quite ridiculous Pokémon is Magcargo. The Pokédex says that its body temperature is 18,000 Farenheit. The surface of the sun is only 9940 Fahrenheit. This is silly. But just how silly?

Hot objects radiate thermal energy. Absurdly hot objects radiate absurd amounts of thermal energy. The Stefan-Boltzman Law states that the amount of thermal energy radiated by an object per second per metre surface area is a constant (5.67 x 10-8), times the temperature in Kelvin (here it is 10255K) to the power of four. Unsurprisingly, raising a big number to the power of four gives you an even bigger number. Magcargo thus has a thermal power output of 627 million joules per second per m2. This sounds like a lot.

This however is just the power output per m2 of Magcargo’s surface area. To work out actual power output of the Pokémon, we need Magcargo’s surface area. The Pokédex entry gives a height of 78.7 cm. To make our lives easier, we’ll employ the classic biological approximation that Magcargo is a sphere. It clearly isn’t (it ain’t no Voltorb), but this approximation will give us a ballpark estimate, that will inform us as to the rough scale of the numbers, which is all we really need. A sphere of diameter 0.787m has a surface area of 1.95 m2. Multiply this by 627 million joules per second per m2, and we get a thermal power output of 1200 MW, that is 1.2 billion joules per second. For comparison, the UK’s largest power station, Drax, has an output of 3900 MW, enough to provide 6% of the UK’s power consumption. Three Magcargo is enough to output almost as much as this.

But what comes out must first come in, and Magcargo would need to take in energy in order to be able to dispel it. One Mars bar contains 230 kcal of energy, equivalent to 0.963 MJ. Thus, a Magcargo would have to eat 1267 Mars bars every second in order to balance out its monstrous power output.

So to answer my earlier question, Magcargo’s temperature is very, very silly.