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I am undertaking a serious graphic novel project with a team, and we aim to work upwards of one or two decades (simply on the writing and storyboard.) I use to be a math wizard in high-school; but, it's been too long to know quite where to start. We are trying to create a formula to use real-life averages of animal weighs and convert them into anthropomorphic representations for this comic.

While I enjoyed the recent Disney film, we didn't think species of such vastly varying size would be able to interact in a practical world, and hoped to move them towards a more common size-basing the world on if wolves were around six foot.

The average 'North American Gray Wolf" weighs 43-45kg.

-somewhere around 6ft?

The average 'House Mouse' weighs 0.68oz

-somewhere around 4ft?

The average 'African Elephant' weighs 7,000kg

-somewhere around 12ft?

I considered the square cube law would play an integral part of whatever formula or scale was needed. Something that would take into account increasingly larger weights to keep an animal from becoming an outright giant and incapable of interaction. But, I have no idea where to start. I understand it is an option to arbitrarily just 'choose' heights. However, I turned to the math community to see if we can find a way to use an animal's weight (and possibly height) for a consistency. Any guidance would bee deeply appreciated.

Thank you for your time.

Lorem Ipsum
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  • As a first approximation, the weights are proportional to the cube of the longest dimension. Conversely, the longest dimensions are proportional to the cube roots of the weights. –  Oct 14 '16 at 21:42
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    Take a look at this article:

    http://www.cs.cornell.edu/~ginsparg/physics/Phys446-546/gbwscl99.pdf

     – user8960 Oct 14 '16 at 21:44

1 Answers1

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It looks like you want an animal who weighs several times more to be only a couple of feet higher. This means you probably want a logarithmic formula:

$$\text{height}=a+b\log(\text{weight}).$$

You have to choose the constants $a$ and $b$. Say we want the wolf weighing $44$kg to come out at $6$ft and the mouse weighing $0.019$kg to come out at $4$ft, then we would get $a=5.021$ and $b=0.259$. So we have the formula

$$\text{height}=5.021+0.259\log(\text{weight})$$

(working with base $e$ logarithms and in units of kg and ft). So the height of the elephant would come out as

$$5.021+0.259\log(\text{7000})=7.3\text{ft}$$

which is a bit less than you thought originally. This makes sense: the wolf weighs 2300 times as much as the mouse but the elephant only weighs 160 times more than the wolf. So since 160 is less than 2300, the wolf should be closer in height to the elephant than the mouse.

An ant, weighing only $3$mg, would end up with a height of $1.7$ft.

Oscar Cunningham
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  • This is brilliant, thank you.

    May I ask how you arrived at 5.021 and 0.259? If I am able to adjust 'a' and/or 'b' using your answer, it would enable modifying the rate of scale. Correct?

    I ask so I would be enabled to use any two animals as constants.

     – Lorem Ipsum Oct 22 '16 at 01:29