Wednesday, 13 May 2015

Squaring the Cube

Doubt Sets In
In Sizing Things Up, I presented a method for working out the weight of a creature when you modify its size. The reason behind wanting such a system is that I like things to be consistent, and to work out a rule once that can then be applied everywhere, instead of having to make lots of little decisions.

Based upon what I read about the inaccuracies of BMI, I made weight go up with the 2.5th power of the height of a creature. That is, if you make the height 4 times as much, the weight goes up 42.5 = 32 times. This seemed to give realistic values, so I didn't worry about it any further.

The other day on Semper Initiativus Unum, Wayne blogged about the weight of Giant Spiders, and he invoked the Square-cube Law. to say that it goes up with the 3rd power. That is, if you make the spider 4 times the leg span, the weight goes up 43 = 64 times. This would indeed hold if the spider kept the same proportions, but does it? (Everyone seems to think so). Or does it only go up by the 2.5th power? Do different families of animals have different rules, or is it the same power for all families? This got me thinking hard about the veracity of my rule...

There is a lot on the internet about Allometry studies, and all the ones I saw proved the cube rule - except - they all had something else in common. They all dealt with a specific species of animal, not a family of animals. That is you might find that a particular species of fish obeys the cube law, but there is no data about fish in general. The variations within a species are often purely due to age or environmental factors or sex. What we're interested in is extrapolating from a small animal to a large one in the same family and then applying the same rule to a giant one, so these findings are not necessarily applicable.

Finding Some Data
Spiders seemed like the obvious place to start. Unfortunately, although the leg span of spiders is fairly easy to find, their weights are elusive. People don't bother to weigh small spiders! On wikipedia I could find the Black Widow Spider (1.5'', 1 gram) and the Goliath birdeater (11'', 6oz). It is 170 times heavier. My rule predicts that as it is 8 times bigger, it would be 82.5 = 181 times heavier, whereas the cube rule suggests 8= 512. Hence the 2.5 power rule seems good in this case, but this limited data hardly constitutes as proof!

Looking further I discovered a paper on the web about flying bird wingspan versus weight, and it gives a graph with a 2.42 power rule:

(It's actually got the axes reversed in the paper). The outlier (top left) is the Black Throated Loon.

This is hopeful, so I thought I should try a few different families of animals.

There's a whole load of data on snakes, which I imported:

So that's a 2.61 power rule based upon length. The outlier (at the top) is the Burmese Python. A lot of other snakes are far thicker than you'd expect, until you see that they are snakes like the Boa Constrictor.

For horses and ponies I went for an ideal weight chart based upon the height of the animal's back, (which looks to be about proportional to the overall height) plus zebras and donkeys from Wikipedia:

That's a 2.55 power rule based upon height.

So what about cats? I found data from wikipedia for Wildcat, Jaguar, Bengal Tiger, Leopard, Asiatic Lion and Feral Cat. Not a great sample size but anyway:

There's a 2.63 power rule. Now for this one I had a choice of minimum and maximum height and weight, so I choose the maximum given for each species, and also I has the data for both the length and height - the length has a 2.33 power rule, whereas I've given the height, in keeping with the horses.

For whales I used wikipedia again for Hector's dolphin, Dwarf Sperm Whale, Pilot Whale, Killer Whale, Beaked Whale, Sperm Whale, Blue Whale:

That's a very nice match for a 2.64 power rule.

For deer I used Wikipedia and the Scottish Forestry commission for Southern Pudu, Northern Pudu, Moose, Elk, Roe Deer, Fallow Deer, and Red Deer (scottish):

Which is a 2.72 power rule.

I also tried collecting data for snails, but had the same problem as spiders for getting reasonable weights. I got three values and a power rule of 2.83, but I spent more time stumbling across disturbing experiments about the force to crush a snail shell, or their weight loss when left to dessicate in hot air... 

Crocodiles looked promising, and I got five values and a power rule of 2.31, but it seems rather tricky to measure the length and weight of a crocodile accurately for some reason. I can't imagine why. The weights and lengths are often estimates from a distance!

Crabs seem to be all different shapes so their weights are all over the place; and I thought I'd have the same problem with fish, so I didn't try with those. If you could find a family of similar shapes you could try the rule, but you're in danger of being selective with your choices just to get the rule to work. Hence you can only really choose families where there's size variation but they all look similar.

I had no luck for scorpions, or for centipedes, both for the same reason as spiders.

The last family I successfully got data for was the Tortoise/Turtle family. The Leatherback Sea Turtle, Alligator Snapping Turtle, Giant Tortoise, Speckled Padloper Tortoise, Marginated Tortoise, and Red Footed Tortoise:

Which is a 2.78 power rule.

So the power rules I've found are 2.42, 2.61, 2.55, 2.63, 2.64, 2.72, 2.78. The average of this is 2.62.

So for the length/height of a family of creatures of similar shape, the 2.5 power rule seems a pretty good approximation - far closer than a cube rule, and close enough for my purposes. So we're on pretty solid ground to use this  rule for the weight of a giant version of any animal. There can be big variety of forms in a family of animals - so for a giant boa constrictor choose a boa constrictor as a starting point. With dinosaurs for example there are three basic shapes of dinosaur - 2 legged, 4 legged, and 4 legged with a long neck - so you'd have three starting archetypes for deriving all the weights.

Squaring This Observation With "The Accepted Wisdom"
So what explanation can there be for why this rule seems to hold so widely instead of the cube rule? I know my data's not great, but the results are certainly very consistent.

That the cube rule holds for deriving the weight of giant animals seems to be accepted wisdom. It's the rule everyone seems to use, and it's even used for disproving the possibility of giant animals. Elephants are used as an example of how big creatures have to become stocky and stumpy to cope with their great weight and they have to have big ears to cope with their low surface area to volume ratio, whilst Giraffes are studiously ignored.

The 2.6 power rule simply says that as you get bigger, you become thinner or elongated. Take a look at photos of a Blue Whale compared to a dolphin, or the body of a cat compared to the body of a lion. It seems clear that, precisely because of the Square-Cube law, your shape has to change to maintain a reasonable surface area. That is, as animals get bigger they don't size up according to the cube rule, precisely because of the Square-Cube law!

So does any of this matter? Perhaps not, but it was fun looking up all these different animals, and now I know that some centipedes look really mean. I'm going to have to show the players a photo next time they meet a giant centipede!

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