The “Square Cube Law”
In: Mathematics
When you make things bigger / smaller, their 1D, 2D and 3D characteristics get bigger / smaller at different rates.
This has big practical consequences. Often an exact bigger or smaller version of a thing won’t work exactly the same (or at all).
Shrinking or enlarging humans or animals is a popular theme in fiction — Gulliver’s Travels, Magic School Bus, Honey I Shrunk the Kids, and so on. In reality, the grown or shrunken subjects’ bodies would badly malfunction, and they’d [quickly die in gruesome ways](https://www.youtube.com/watch?v=MUWUHf-rzks).
Let’s say you want to fill a small box with M&M’s as a present for your friend.
First, you try a 1″ x 1″ x 1″ red box.
– Your scoop is just the right size, the box takes precisely 1 scoop of M&M’s.
– The back of the wrapping paper is marked with 1″ x 1″ squares
– To wrap it, you need 6 squares cut out in a T shape [like this](https://en.wikipedia.org/wiki/Cube#/media/File:Hexahedron_flat_color.svg)
Then you decide the 1″ x 1″ x 1″ red box is small. You try a green box instead, its measurements are 2″ x 2″ x 2″.
How much bigger is the green box? It depends on whether you measure its 1D, 2D or 3D characteristics [1]:
– (1D) The green box’s sides are 2x as long as the red box.
– (2D) The green box uses 4x as much wrapping paper as the red box.
– (3D) The green box uses 8x as many scoops as the red box.
[1] Here’s some explanation to help you see why this is so:
– (2D) The green box’s wrapping paper use the same T shape, but each side is a 2″ x 2″ square, which is made up of four 1″ x 1″ squares. Since the 6 sides each have 4 squares, the total squares are 6 x 4 = 24, or 4x the red box’s 6 squares.
– (3D) Think about filling the green box with red boxes. You can put 4 in the bottom layer, and 4 more in the top layer.
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