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.

Character Generation #4 - The Skill Point Alternative

The Class system just outlined is just one option in Explore - there are many reasons why you might want to switch instead to the skill points system, which is what the Class system is built on.

If you want more control over your starting character's skills, or to diversify into new skills, or you wish to create custom classes, or you want to improve skills in a gradual fashion rather than wholesale once per level, then read on.

Skill Points To Spend

You have 1 skill point to spend per 10XP. You start with 80XP, so that means you have 8 skill points to spend at first level. You can spend new skill points as you gain them, you do not need to wait until the next level is attained.

If you wish you can simply buy one of the standard classes with your initial 8 points, and you can spend extra skill points you earn to increase each trained skill in turn by one rank, in line with the usual class progression.

Skill Costs

Skill Ranks cost one for the first rank, one for the second, and then double that per extra rank.
On the character sheet you cross off the points spent, for example at rank five you cross off the first five points:

 1   1   2   4   8  16  32  64  128  256  512

The total points spent is simply the first value not crossed off - in this case you have spent 16 points.

Skill Areas and Primary Spell Area

Skills are divided into Physical (Melee, Athletics etc.), Practical (Scout, Thief etc.), and Magical (Air, Earth, Fire, Water etc.)

Magic is divided into four schools - Elementalism, Wizardry, Enchantment, and Sorcery.
Any spell caster must select one as their school, circle this on the character sheet.
Each school has four different disciplines, each of which gives access to a different set of spells.

Skills

Physical Skills: Melee, Unarmed, Bow, Thrown, Athletics, Acrobatics

Practical Skills: Scout, Thief, Ranger, Healer

Elementalism: Air, Earth, Fire / Ice, Water

Wizardry: Light, Sound, Force, Lightning

Enchantment: Plant Control, Animal Control, Mind Control, Body Control

Sorcery: Divination, Undead, Summoning, Time & Space

Bonus Skills

Each skill area has a "bonus skill" at the bottom of the area. This is Parry/Dodge for Physical, Perception for Practical, and Arcane for Magical. You do not spend points on these skills, you get them for free for the points you spent in each area. If you have spent 6 points in Physical Skills and 2 points in Spells, then you get 6 points in Parry/Dodge and 2 points in Arcane, which means you're level 3 in Parry/Dodge, level 2 in Arcane.

This is managed through how the character sheet is laid out - within each category (Physical / Practical / Spells) sum up all the points you have spent and note this in the gap below the points in the category, which is next to the bonus skill. Then sum up all the three totals at the bottom.

Restrictions

There are a couple of restrictions on how you can spend your skill points:

1. You cannot spend more than half your points in a single skill.
2. If you have any spells then as Magic is such a demanding art you can never spend more than half of your available points in non-spell skills.

Switching to Being a Spell Caster

If you start out with no spells and later decide to learn to cast them, then this is quite a dedicated undertaking. From that point on you cannot learn any more non-spell skills until your non-spell skills cost half your available skill points. At this point you can once again learn non-spell skills (but subject to the usual no-more-than-half restriction).

Examples

All the classes in Explore are examples of how you can spend your eight points at first level and they illustrate some of the rules above.

Warrior

Athletics: 3

Melee: 2
Unarmed, Thrown: 1
(Parry: 4)

This is the simplest option - all points (4 + 2 + 1 + 1 = 8) have been spent in physical, so that's 8 points in physical and hence the maximum value of Parry 4. No points were spent in Practical skills so no ranks in Perception. Athletics at rank 3 is 4 skill points, half of the points available, so rank 3 is the maximum for a starting character.

Rogue

Scout, Thief: 2
Melee, Unarmed, Acrobatics, Athletics: 1

(Parry, Perception: 3)
Here half of the points have been spent in Physical (1 in each of Melee, Unarmed, Acrobatics, Athletics) and half in Practical (2 in each of Scout and Thief). Hence 4 points - rank 3 - in both Parry and Perception.

Aetherius
Time & Space: 3

Divination, Undead: 2
(Arcane: 4)
Here one spell list is at maximum rank 3, and the other 4 points have been split equally among two secondary spell lists (both also Sorcery).

Illusionist

Light, Sound: 3
(Arcane: 4)
The illusionist has simply split their points into two Wizardry spell lists - which is the maximum of rank 3 in both.

Healer

Body Control, Healer: 3
(Arcane: 3, Perception: 3)
Here the split is in one spell list and one practical skill. Note that one skill is a spell list, so at least 4 points had to be spent on spells.

Oracle
Divination: 3
Undead, Summoning, Time & Space, Mind Control: 1
(Arcane:4)
The Oracle would like Divination: 4 but is limited to rank 3 (4 points). The only other thing they think is worth bothering with is Mind Control, but that's Enchantment and they're a Sorcerer, so they have to spend a point in all the other three Sorcery disciplines before they can spend a single point in Mind Control. This has now used up all their 8 points.

Now by the time these characters have gained 160XP for level 2 they have had another 8 points to spend and they could have simply added one rank onto all their trained skills. At some point it is normal to diversify at bit, but over diversification will weaken your character - for example lots of low level spell lists does not make a powerful spell caster.

Character Generation #3 - Levels, XP & Classes

Explore has two complimentary systems for generating characters - classes or skill points. Technically Classes are no more than pre-selected sets of complimentary skills, but they represent typical professions in the world Characters might come from, are a good starting point for new Characters as it roots them in the world, and makes creating a character quicker. It also provides an easy way of generating NPCs, and for this purpose you might create more classes. Finally they illustrate the various options you have when creating a bespoke set of skills.

At any point you can switch from the class system to a custom set of skill advancements without any penalty for sub-optimal choices in character creation.

Levels and Experience Points (XP)

Each character starts off at Level 1 with 80 experience points (XP). You gain XP through playing the game, and each time your XP doubles your level increases.

Whether you are class or skill based, each level increase improves certain bonuses.

Level	XP
1	80
2	160
3	320
4	640
5	1,280
6	2,560
7	5,120
8	10,240

Classes for Humans

Choose a class for your character from the list below. This gives the skill ranks your character will have at first level. For each level you gain every skill given increases by one rank.
Note that skills given in brackets are skills you do not train specifically for but gain as a byproduct of the other skills. You gain Parry from physical skills, Perception from practical skills, and Arcane from spells.

Non Magic Using Classes

Warrior

Athletics: 3

Melee: 2
Unarmed, Thrown: 1
(Parry: 4)
Warriors concentrate entirely upon martial skills, and rely more on brute force than subtlety. Typically they would throw an axe in the first round of combat, then close to fight with swords, relying upon their fists when they are disarmed or in the extremely unlikely event they become involved in a brawl. Warriors may be either Soldiers, Guards, or freelance Mercenaries.

Archer

Bow: 3
Athletics: 2

Melee, Unarmed: 1
(Parry: 4)

Archers spend most of their time training with a longbow. They train hard to improve their strength so they can shoot an arrow a long way from a heavy longbow, but pride themselves primarily on their accuracy, as what does it matter how deadly your arrow is if it misses? Archers also train with sword and their fists so as to be prepared when their foes get a little too close for comfort. Archers are often recruited into Armies at time of war, or work as Guards for Castles and other fortified dwellings, but often find themselves having to work in other capacities when work is short during times of peace. 

Ranger

Ranger: 2
Melee, Bow, Unarmed, Athletics, Healer, Scout: 1
(Parry, Perception: 3)

Rangers are members of an ancient order - warriors who patrol the wilds (ranging), often alone or in small groups. They are employed by Rulers to patrol the edges of the wild and to keep the peace in the borderlands, but pride themselves on their neutrality in war. Their primary skill - ranging  - covers wilderness skills such as tracking, survival, riding; but their job is much more than that. They can scout (sneaking / hiding) and being self sufficient means they have some skill in healing. They can handle themselves in a melee, but in the wilds the bow comes in useful more often for hunting and ambushes. 

Rogue

Scout, Thief: 2
Melee, Unarmed, Acrobatics, Athletics: 1

(Parry, Perception: 3)
Rogues are not members of any recognised profession, it is more a "calling" for urban dwellers looking to better themselves. Scouting (sneaking and hiding) along with Athletics (to scale a wall) or Acrobatics (to vault over it) provide an opportunity to use good plain old-fashioned thievery (picking locks and pockets). Growing up in the slums they also know how to handle themselves in a fight. Some Rogues pride themselves on their ancient and noble art and form Thieves' Guilds which have a complex and strict code of behaviour. Others try and become respectable, taking employment from wealthy benefactors who have certain jobs that need doing, and who can pull strings to get their employees out of jail.

Elementalists

Elementalists specialise in the powers of the four elements: Air, Earth, Fire and Water. Some classes specialise in the element counterpart to Fire - Ice. The names of the five disciplines are misleading - the powers relate to Gases, Liquids, Solids, Heat and Cold - but the old names have stubbornly remained due to traditionalists who prefer the more "mystical" ancient terms.

Aeromancer
Air: 3
Earth, Fire, Water, Melee: 1
(Arcane: 3, Parry: 1)

Aeromancers study all the elements, but concentrate on Air spells. Like all Elementalists they use their magic for practical ends, and find that knowing how to use a sword can come in handy.

Geomancer
Earth: 3
Fire, Water, Air: 1
(Arcane: 3, Parry: 1)
Like Aeromancers, Geomancers study all the elements, but concentrate themselves on Earth spells. They too know learn how to handle a sword.

Pyromancer
Fire: 3

Melee, Athletics: 2
(Arcane, Parry: 3)
In contrast Pyromancers have no time for any element other then Fire, whose power they combine with martial training to become deadly magical warriors. They typically enter combat with a burning "fire blade".

Kryomancer
Ice: 3

Melee, Athletics: 2
(Arcane, Parry: 3)

Kryomancers are a dissenting offshoot of Pyromancers, concentrating on the opposing power of cold not heat. Their trademark is the "ice blade". They are looked down upon by the older school of Pyromancy and overlooked by other schools of magic, and this lack of respect dents their pride.

Hydromancer
Water: 3
Air, Earth, Ice, Melee:1
(Arcane: 3, Parry: 1)
Hydromancers study widely on the elements, concentrating on Water spells, but notable study Ice instead of Fire due to the destructive power of Fire on Water. They too learn some practical melee skills.

Wizards

Wizards specialise in the powers of the four energies: Light, Sound, Force, Electricity.

Traditionally there were two different disciplines - Lightning, and Amber (the study of static electricity that built up when amber is rubbed against wool). Neither was a successful branch of magic until a breakthrough was made and both disciplines combined into a single modern discipline - Electricity. The Lightning Mages joined together with the Amber Mages and lost their name, but in return were allowed to name the new joint discipline, and chose "Electricity" (after the old name Electrum for amber, confusingly also used to mean an alloy of gold and silver of the same colour).

Illusionist

Light, Sound: 3
(Arcane: 4)
Illusionists study deeply in deceptive wizardry, through which they hope to bring about the ends they desire without ever having to engage in any physical confrontations.

Mage
Force, Electricity: 3

(Arcane: 4)
Mages study deeply in offensive wizardry. They have powerful spells, but their lack of fighting ability can put them in a tight spot, so they often band together with (or employ) warriors for protection.

Amber Mage
Electricity: 3

Melee, Athletics: 2
(Arcance, Parry: 3)
Amber Mages believe that Electricity is the most powerful and pure of the Wizarding arts, and combined with Martial prowess they should become more powerful than the Pyromancers. Their trademark is the crackling Lightning Sword, an impressive sight (and sound). Note that they take no more notice of the Kryomancers than the Pyromancers do!

Enchanters

Enchanters specialise in the organic powers of nature: Plant Control, Animal Control, Mind Control, Body Control.

Druid

Plant Control, Animal Control: 3
(Arcane: 4)
Druids are an ancient order of Enchanters who are at one with the wild side of nature. They often form close bonds with animals who accompany them on quests to keep the balance of nature.

Psychic
Mind Control: 3
Animal Control: 2
Melee, Athletics: 1
(Arcane:3, Parry: 2)
Psychics probe into or manipulate the minds of others, and they take more than a passing interest in the closely related but subtly different minds of animals. They are mistrusted and usually hide their true calling. It has proved necessary for Psychics to learn how to handle themselves in a fight in case their art causes offence.

Healer

Body Control, Healer: 3
(Arcane: 3, Perception: 3)

Healers combine both the mystical and practical arts to mend the body. They are useful after a fight, but rarely during it.

Sorcerers

Sorcerers specialise in the occult powers: Divination, Undead, Summoning, Time & Space.

Oracle
Divination: 3
Undead, Summoning, Time & Space, Mind Control: 1
(Arcane:4)
Oracles call upon mystical powers to see future possibilities. They study all the mystical arts as this complete knowledge allows them to expand their studies into the realm of Enchantment and learn Mind Control as an alternate way of viewing or shaping the future (though they keep these studies under wraps).

Exorcist
Undead: 3

Athletics: 2
Melee, Thrown: 1
(Arcane: 3, Parry: 3)
Exorcists devote their lives to tracking down and eliminating spectral forces of evil. They do not shy from a fight, but know that the touch of the Undead can be deadly so arm themselves with missiles to throw at corporeal manifestations (and train hard to improve their range) but also melee.

Conjurer
Summoning: 3
Bow, Athletics: 2
(Arcane: 3, Parry: 3)
Conjurers prefer to get their summoned servants to do their dirty work for them. In a battle situation they are likely to stand at the back and shoot over their conjurations at the enemy.

Aetherius
Time & Space: 3

Divination, Undead: 2
(Arcane: 4)
The Aetherius are a mysterious secretive folk who practice powerful magicks that can bend reality. They have an interest in the arts of Divination and the Undead, but view Summoning as a mis-use of power.

Classes for Elves

Elves can be any non magic-using class - Warriors, Archers, Rangers or Rogues - though they perform a different role in society. Typically most Elves work for the good of the realm, and all four of these classes do. They are typically encountered outside of Elven lands when on a mission for the realm (often in a group), and that is the sole purpose of Elven Rangers and Rogues. These missions are often open-ended and vague in purpose, and may be no more than to improve relationships with a distant land, but there is often an ulterior motive such as to monitor activities of a hostile force in that land (for example, the Elves are dispatched to monitor the activities of a baron in a city because the Elves fear he is under the influence of Orcs)

Elven Magic User

Any one spell list: 3

Melee, Bow, Athletics, Acrobatics: 1
(Arcane: 3, Parry: 3)

Elves who wield magic do not form the cliques that humans do - there is a single magic order which typically combines purity of one magical discipline with general military training. Elven Magic Users often accompany non-Magic users in their missions, but pretend to be Archers.

Classes for Dwarves

Dwarves who leave their homes are generally mercenary Warriors who have lost their ancestral home (but plan of regaining it at some point, naturally). Dwarven adventurers can also be Magic Users:

Dwarven Magic User

Ice OR Fire OR Electricity OR Summoning: 3

Athletics: 2
Melee, Thrown: 1
(Arcane: 3, Parry: 3)

Dwarven interest in Magic is very utilitarian - they are only interested in Magic they can clearly see the use of - which generally limits them to variants of the martial spell casters - Pyromancers, Kryomancers, Amber Mages, and Conjurers. (Exorcists are too sanctimonious for Dwarves).

Note as always there is an accompanying FAQ page for this section of the rules.