The Square-Cube law is usually cited where questions of scale arise. Actually applying it to real-world objects quickly shows that it is completely unusable to approximate weight differences between (at least) creatures themselves.
Consider some kitten data drawn from Wikipedia (or skip this section if you do not care).
I selected them for a linear progression in head-body length.
Using average values for:
House Cat, Lynx, Jaguar, Lion, Amur Tiger
we get an almost linear increase of average head-body length (in m) of:
0.5, 1.0, 1.5, 2.1, 2.8
for a corresponding increase of average weight (in kg) of:
4, 20, 97, 190, 250
What we can see immediately is that the Square-Cube law would be horribly wrong.
Predicted weights given the House Cat:
4, 32, 88, 296, 702
Mayhap the scale is too big to expect any sort of accuracy, fair enough. But while stepping from i to i+1 (using the statistical value for i, not the one computed in the preceeding step) does improve the prediction, it still breaks down horribly at lion level.
Predicted weights given i for i+1:
4, 32, 68, 266, 450
Even at constant size increases, the Square-Cube law does not hold. Intuitively, this is sensible, as the law requires that everything scale to proportion, which does not hold (presumably, I have not checked) for living bodies.
(E.g. observed via my own eating habits, leg tends to grow faster than liver, unless owner alcoholic.)
(Here I really wanted to repeat the same ballpark calculation for clothing, but sadly neither Wikipedia nor diverse online shops I sampled seem to concern themselves with the weight of such items... if someone could provide some sample data to improve this question, I would be much oblieged.)
Intuitively, clothing ought to suffer from similar causes for the Square-Cube law to be unapplicable as living bodies do. Some reasons to believe this:
- originally: parts of dead bodies draped over living bodies - same stuff, same behaviour
- parts such as buttons, pockets, seams do not scale to proportion
- thickness has a hard lower bound to withstand point forces
- thickness will generally not scale to proportion as "natural" values, e.g. thickness of skin used as material, dictate linear steps (layers)
What is a sensible heuristic function to extrapolate weight of clothing for different scales of wearer, given said weight for one such scale?
(With sensible meaning specifically "superior to Square-Cube law" and generally "as good as it gets".)
Which immediately lends itself to following sub-questions:
- Does such a function exist?
(Presumably producers of items meant to fit various sizes of human have investigated this in the past.)
- How do the bounds of a given function's domain look, outside of which it no longer provides "sensible" heuristics?
(E.g. the Square-Cube law in my example above is already bad at lynx and lion size - wrong by factors >1, but at tiger size it really catches fire.)
- Does a given function also lend itself to extrapolating the weight of items other than clothing, which are likewise required to "fit" their user?
(e.g. backpacks, blankets, furniture, constant lengths of rope...)
It may help that the domain of size I am mainly interested in is about [0.35, 3.5] m head-body length.
I presume (and may be corrected) that within this domain material changes are not yet necessary compared to the well-explored design space of human clothing.
Weight changes between different anatomies are out of scope.
The cats I listed all have a very similar build to each other, similar enough to claim that at least their outward appearances (if not, evidently, their internals) satisfy the Square-Cube law's prerequisite of being "scaled proportionally".
Changes in qualities other than size and weight are mostly out of scope:
Volume of a knapsack, energy of a crossbow do not concern me. Sturdiness however does: A vest is no good if it tears the moment it gets caught in a twig. If such changes are likely to be an issue for predicted items (mostly at the extreme ends of the scale, I would presume), they may be mentioned, but free of the requirements of the "hard-science" tag.
Apologies for the poor display of sample data; as far as I know tables are neither supported via Markdown nor HTML and monospaced font depends on client, so I decided against using an ASCII table.
In engineering the Square-Cube law generally tells us not to "just upscale* any given design. It would be cynical to expect a law showing "how bad an idea this would be" to often correctly predict the actual weight difference between items of different scale - this can only happen, when there is no better way and if the difference is still bearable (or, to be cynical again: when the developer was lazy).