7
$\begingroup$

In my world, which has an Earth-like gravity and environment, an adult, fit humanoid male is 3 inches tall. When I do a simple ratio calculation, I come up with him weighing 7.9 pounds. Basically, the weight of a gallon of water. This seems excessive. Any thoughts on how to calculate this would be greatly appreciated. Additionally, I may have to make them 4 inches tall to make the science work, so I'm looking for a way to adjust the height as needed. Thanks!

Here is the math I used 190 pounds/72 inches = x pounds/3 inches

$\endgroup$
  • 20
    $\begingroup$ Your error comes from the fact that you are calculating the weight of a person that's 3" tall, but a couple of feet wide and deep. The visual is quite appealing :) $\endgroup$ – Mad Physicist Sep 21 '18 at 16:42
  • 1
    $\begingroup$ You should specify "weigh on Earth . On the moon, less; on Jupiter, a lot more $\endgroup$ – Carl Witthoft Sep 21 '18 at 17:34
  • $\begingroup$ The keyword is allometry en.m.wikipedia.org/wiki/Allometry $\endgroup$ – Hurda Sep 21 '18 at 20:47
  • 1
    $\begingroup$ Is there a reason we’re only interested in the 3” men? (Kinda sounds like a setup for a penis joke.) what about the women? Are they also 3” tall, or are they relatively gigantic, normal-for-Earth sized women? $\endgroup$ – HopelessN00b Sep 22 '18 at 0:05
  • $\begingroup$ Indeed, normal-sized people, including women, do show up. But, my little men also have little women and even smaller children. As you likely know, Jonathan Swift explored the possibilities of a large woman and a small man decades ago. It was certainly an eye opener for me as a young reader. $\endgroup$ – Adam Orth Sep 22 '18 at 1:07
28
$\begingroup$

Volume is the cube of length. Therefore, if the length of a human drops by a factor of x; the volume drops by a factor of x$^3$.

Therefore, a 3" person is $$\frac{3}{72} = 0.042$$ the length of a regular person, then he would be $0.042^3 = 0.000072$ times the volume (and mass) of that person.

Multiply that factor by 200 lbs to get 0.014 lbs; or 0.2 ounces. Not much!

$\endgroup$
  • 5
    $\begingroup$ Only true if all dimensions shrink proportionately. Not guaranteed. $\endgroup$ – Carl Witthoft Sep 21 '18 at 17:33
  • 7
    $\begingroup$ @CarlWitthoft There are obviously problems with a 1/24 scale human from an anatomical standpoint. But I'm pretty sure the OP just wanted help with a math problem. $\endgroup$ – kingledion Sep 21 '18 at 17:47
  • 2
    $\begingroup$ For reference, an average hamster weighs around 5-6 ounces. So 0.2 ounces sounds pretty extreme. $\endgroup$ – void_ptr Sep 21 '18 at 18:04
  • 3
    $\begingroup$ @void_ptr Hamsters are pretty thick, relative to their length, compared to a human $\endgroup$ – kingledion Sep 21 '18 at 18:20
  • 2
    $\begingroup$ @kingledion Right, that's what I meant, whether the 'perfectly scaled' human would have any issues with thermal regulation, nutrients storage and such. Perhaps this warrants a separate question, something along the lines of "What an anatomically correct 3" human would look like?" $\endgroup$ – void_ptr Sep 21 '18 at 18:39
12
$\begingroup$

Weight varies as the cube of height, not linearly. 190 pounds/373,248 cubic inches means 0.014 pounds/27 cubic inches. Your 3” man weighs a fifth of an ounce.

$\endgroup$
12
$\begingroup$

I believe that I once read in a book about about mathematics that Galileo Galilei (1564-1642) popularized the square cube ratio in relation to biology.

As the dimensions of a shape change, the shape's area changes with the square of the dimensions while the shape's volume changes with the cube of the dimensions.

So a man 10 times as tall as your 6 footer would be 720 inches tall. The area of a horizontal slice through his leg bones would be 100 times the area of such a slice through a 72 inch man, so the leg bones would be 100 times as strong, but the man's volume and weight would be 1,000 times that of a 72 inch man, so each horizontal square inch of leg bone would be subject to 10 times as much weight as was the case for a 72 inch man and would probably break under the stress of standing, let alone walking or running.

A man 100 times as tall as a six foot man would be 7,200 inches tall and his bones would have a horizontal cross section 10,000 times that of a 72 inch man, while his volume and weight would be 1,000,000 times as great. Thus his leg bones would be subject to 100 times as much weight and pressure and would certainly snap.

And the same thing goes for the cross section of his muscles compared to his volume and weight. His muscles would be terribly under powered for the weight they had to move. That includes his lung muscles and his heart muscles.

So Galileo proved that interpreting fossil bones of megafauna or dinosaurs as the bones of human shaped giants tens or hundreds of feet tall was not reasonable and those bones should have belonged to highly nonhuman creatures with body plans capable of supporting their vast weight.

A three inch tall man would be 0.0416666 as tall as a 72 inch man. Thus his leg bones would have 0.0017361 times the cross section of a 72 inch man's, and his volume and weight would be 0.0000723 that of the 72 inch man. Thus his leg bones would bear only 0.041666 the weight per unit of area of cross section, and so would his muscles. He would have about 24 times as much strength for his weight that a 72 inch man would have.

There actually are animals with bipedal locomotion similar in size to your three inch man. They are kangaroo mice and similar tiny rodents.

https://en.wikipedia.org/wiki/Kangaroo_mouse 1

And you will notice that their limbs are very slender in proportions, because they don't need thick limbs to move at their tiny size. I suggest that your tiny 3 inch man will have very slender limbs instead of thick human limbs. Watch a television show or movie with child characters and picture your 3 inch man with the skinny arms and legs of little boys and girls instead of the bulging biceps of a a bodybuilder, and then make his arms and legs much skinnier than that.

And there are a number of other adaptations that your 3 inch man would have due to his tiny size. So he would look a bit different from a normal 72 inch man shrunk by a mad scientist's shrink ray to three inches tall.

$\endgroup$
  • 3
    $\begingroup$ The additional considerations you mention are much more important than the raw weight calculation. $\endgroup$ – Mad Physicist Sep 21 '18 at 16:44
  • $\begingroup$ Good point! I’ll submit an additional question. I must say, however, that 24 times the strength for his weight opens up many interesting doors! $\endgroup$ – Adam Orth Sep 21 '18 at 17:53
5
$\begingroup$

under 1.1 ounces. The smallest primate in the world is Madame Berthe's mouse lemur which is just over the size of your 3" human

the average body length is 9.2 cm (3.6 in) and seasonal weight is around 30 g (1.1 oz).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.