## Sunday, September 26, 2010

### Some basics about the Square-cube law

I've had to learn a lot about the Square-cube law just to survive. Actually, though, it's fairly simple.

We'll start w/ a regular cube, and a daz 3d woman. To make things simpler we'll be growing. The same principles work in reverse for shrinking.

Now we'll double the height. Just the height. This is scaling on the line.

Okay, see how tall and thin she is. And notice that there are two boxes, so the weight has been doubled. To make her look more like a person and less like a Na'vi we have to double the width, too. This is scaling on the square.

Now we have four boxes, so the weight is four times our starting weight.  Our model, seen from the front, looks like she has the same proportions as when she started. From the side, however...

She's actually sort of squashed flat So we have to double the breadth as well. We have to scale on the cube.

Now she's twice her starting height, w/ her normal proportions in all directions. But see the boxes? We needed eight boxes to double the boxes in all three  dimensions. That means that girl weighs eight times her starting weight. Diet time!

My height is one/tenth normal. On the line. My surface area, shoe size, and some other things is one one-hundredth normal, or on the square. My weight is on the cube. One one-thousandth what it had been the night before.

More information can be found at Square-cube law on Wikipedia and Square Cube Law on TVTropes, The Biology of B-Movie Monsters, and On Being the Right Size by J. B. S. Haldane.

You can also look at The Physics of Superheroes: Spectacular Second Edition by James Kakalios and Insultingly Stupid Movie Physics: Hollywood's Best Mistakes, Goofs and Flat-Out Destructions of the Basic Laws of the Universe by Tom Roger.

Yeah, it's messed up, and I'm still not used to it.