How Strong Is An Elephant?

The Question

My game, Explore, is supposed to give (vaguely) realistic results to questions such as how varying the size of creatures affects their strength. I had assumed that the strength of a monster was proportional to its weight - thus a doubling in size doubled the weight it could lift. This then has a knock on effect through other rules of the game (if I want to retain consistency) so it is quite fundamental.

When working through my rules on climbing, where power to weight ratio is important (not raw strength) I noticed that the Olympic Weight Lifting records were odd.

Here are all the world records for each class in Snatch and Clean & Jerk (in kg):

Weight Class	Snatch	Clean & Jerk
56	139	171
62	154	183
69	166	198
77	177	214
85	187	220
94	188	232
105	200	246

You can clearly see that the weight lifted is not linear with the weight of the competitor - it only changes by half the rate expected. The lowest category has snatch being 2.5* the weight, so you'd expect the top class to have 263kg, 124kg more than the lowest class - which is twice the actual increase. With a log-log plot you can see that it's proportional to the weight class to the power 0.57 / 0.55. That is, the weight lifted is proportional to the square root of the weight class.

Nothing explains this result, so I decided to go back to first principles.

Return to Allometry

The fundamental scaling law for Explore is how weight scales with height/length. I previously wrote controversially about how the square-cube law does not appear to hold for animals and I noted that for the species I could find data about the average mass of adult animals in a species varied with the 2.62th power of the length of the animal. This is not a contradiction of Galileo's square-cube law, it is simply that larger animals tend to be thinner. The science behind this is called Allometry, but at the time I couldn't find any studies to back up this specific observation.

Well, last week I stumbled across a paper ALLOMETRIC SCALING OF BODY LENGTH: ELASTIC OR GEOMETRIC SIMILARITY IN MAMMALIAN DESIGN which measured length and weight for 1733 different species of mammals to see if the scaling exponent was 0.333 or 0.250. The scaling exponent is the reciprocal of the power, so my prediction was 1/2.62 = 0.382. The paper showed that it was not close to their two expected results, but unexpectedly much higher at 0.359, and 1/0.359 = 2.79. This is higher than my value, but this is for all mammals, whereas I'm looking for how small animals of a given type scale to large animals of the same type. That is I'm scaling from a pony to a horse, this study is about estimating the weight of an Elephant and a Giraffe from that of a hamster.

BMI (body mass index) bizarrely assumes that ideal weight is proportional to height squared, which self evidently wrong. This quotes from wikipedia says:

However, many taller people are not just "scaled up" short people but tend to have narrower frames in proportion to their height. Nick Korevaar (a mathematics lecturer from the University of Utah) suggests that instead of squaring the body height (as the BMI does) or cubing the body height (as the Ponderal index does), it would be more appropriate to use an exponent of between 2.3 and 2.7

So I'm happy with my 2.62 value, and I'm happy to use it both for estimations of the weight of a small or giant version of a species as well as estimations of the weight of individuals of a species of varying sizes.

Elephant Power

Returning to the question of strength I started to look for how strong different species of animals were. I wasn't going to find snatch & jerk records for animals, so I needed an alternative. The power of Horses is known - a draft horse has one horsepower - and I hoped to see how larger species compared, but for other species it proved remarkably difficult. I decided that as Elephants are used for their strength I might be able to find answers for them, but although I could find how much they could lift with their trunk, how much they could have piled on their backs, or how much they could drag - none of these were particularly helpful (or scientifically measured) or comparable to a horse. I started to wonder what I meant by Strength, and found the answer with the origins of horsepower.

James Watt standardised and popularised horsepower as a means for selling his Steam Engine. He agreed to take royalties of one third the savings in coal that people made by switching to his more efficient Steam Engine - but this could not apply for people who still used horses. He took the idea that a steam engine could do the work of several horses, and standardised it. He calculated the force that a draft horse could pull at a particular speed, and force*distance = energy. This gives you the power output of a horse, in watts, or joules/second.

This power output is what I wanted, and I noted a connection with my reading on Allometry.

Metabolic Rate

Basal Metabolic Rate of an organism is defined as the rate of energy burned by an organism at rest, i.e. it is also measured in watts. It seems reasonable to posit (given the complete absence of data) that on average the maximal power output (the strength) of a creature is a fixed multiple of their basal metabolic rate.

The allometry of metabolic rates is a well know result in Allometry:

In plotting an animal's basal metabolic rate (BMR) against the animal's own body mass, a logarithmic straight line is obtained, indicating a power-law dependence. Overall metabolic rate in animals is generally accepted to show negative allometry, scaling to mass to a power ≈ 0.75, known as Kleiber's law, 1932.

Hence I posit that the strength of a creature, its maximal power output, is proportional to its mass to the power 0.75.

(Note that many sources cite that strength scales with 2/3 the power of weight, which is inferred from the scaling of muscle cross-sectional area, which is a confusion of strength=power with the structural strength of materials. It is precisely this fallacy that lead people to insist for years that Kleiber's law was wrong and the value was 0.67, as their logic had told them it must be).

The Strength of an Elephant

Given a draft horse, 730kg, can pull with one horsepower - how much does a 5000kg Elephant pull with? My suggested scaling gives (5000/730)^0.75 = 4.2 horsepower. That is, although the Elephant is nearly seven times heavier it would only be four times stronger. This seems to accord with general consensus (that a horse is stronger, pound for pound) but the reduction is not very pronounced as the ratio of the weight of a draft horse to an elephant is not actually that big.

The Strength of an Ant

Consider instead the Asian Weaver ant, 5mg in weight, which was photographed carrying a 500mg weight - one hundred times its own weight! I'm 178 times taller than the ant is long, and 15 million times heaver than this ant. The ant compared to me, is like me compared to a 1000ft tall giant! Scaling for its size, the ant's power-to-weight ratio should be 60 times that of our 77kg weightlifter.

Is This Proof?
This is not proof of the formula, since to prove it we'd have to have good data, but we can compare this with the alternate theories - linear scaling, 0.67 scaling.

The linear rule would give ants as having no better power-weight ratio than a human, which is self-evidently wrong.

The 0.67 rule would give the power of an Elephant as only being 2.6 times as that of a horse, which is also obviously far too low.

Hence the rule is not only plausible, it also appears to give far more reasonable predictions than the alternates.

Return to the question

I've calculated how power varies with weight - 0.75, but at the beginning we saw clearly that the weight liftable varied with exponent 0.5. How to reconcile?

Well, now I know I'm talking about Power and Energy, which leads to the answer.

To lift a weight above one's head you have to lift the weight through a certain distance, that is you expend a certain amount of energy - m*g*h in fact.Hence taller weight lifters have a disadvantage - they have to do more work to lift the weight higher.
The average height of the weight lifter, and the height they lift it through, should be proportional to weightCat^(1/2.6).
This additional scaling (factoring in height as well as weight) predicts that the 56kg cat weight lifting records would be 146kg/176kg, and the 105kg cat records would be 214kg/259kg.
So not perfect, but pretty close.

What was the point again?

Firstly I'm interested in these answers for their own sake; trying to understand how the world works. Secondly I'd like values in the game not to be arbitrary if possible. But most interestingly applying the scaling rules in the game allows me to work out answers such as how much damage bonus a giant gets compared to how much damage penalty you get from air resistance at max range with an arrow. Once codified into the rules no-one needs understand how I came up with the values, but they're consistent and scalable.