# Motivation

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)

# Questions

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...)

# Footnotes

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).

Why would the clothing equation have to be more complex than...

(Surface of Area of wearer measured in fractional square yards) S

times

(Culturally or Environmentally mandated coverage area, expressed as a percentage of wearer total body surface area) P

times

(Weight of preferred (potentially multi-layered) fabric measured in fractional square yards) W

with a small percentile addition for buttons, buckles and insignia.

...?

( so my Hard Science mandated equation would be... Fabric Yardage Weight, Y = S * P * W )

I think the inverse square law only applies to solid objects. Not their wrappers. The private parts of a giant don't require thicker fabric to maintain decency... only more of it.

• How would that work to e.g. scale a west? The typical human west does not satisfy (any?) typical human P. How would it work with a necklace (P~0)?
– Zsar
Sep 9, 2018 at 18:43
• @Zsar, what definition of the "West" are you referring to? Here are the ones that I know about and none of them seem to apply to clothing? As for a necklace, the strength of the wearer would influence its scale more than its size. Notice how even at our fixed scale, only large men where gold medallions larger than coin size. Sep 9, 2018 at 18:57
• Also, it seems like you believe that fabric toughness is somehow related to the scale of the wearer. I don't need to wrap a giant in 10oz canvas just because he is big. A good denim will contain him just as it would a smaller person. If the Incredible Hulk's clothing was properly tailored to his body, it wouldn't rip as he moved, even if it was finest linen. Sep 9, 2018 at 19:00
• Persistent typo: I meant "vest". Fixed in the OP.
– Zsar
Sep 9, 2018 at 19:15
• Okay, for a vest, measure the percentage of a normal size body which is covered by the vest. That becomes P. W stays fixed. So now you can just feed in values of S for different sized characters to get Y which is the weight of that character's vest. Sep 9, 2018 at 19:21