The “Square Cube Law”
In: Mathematics
Let’s first explore an example, and then I’ll explain the Square Cube law based on what the example taught you
I am in the business of making cubes for a living. I have three subcontractors who work for me:
* Lino, who puts covers on the edges of the cube (1 dimensional),
* Surfacio, who paints the surface of the cube (2 dimensional, the “square” in the square cube law)
* Volumio, who fills the cube with water (3 dimensional, the “cube” in the square cube law)
I used to commission cubes of a consistent size (2 meter edge size). I pay each of these contractors $100 for their work. For each cube, my contractors would need to do this much work:
* Lino has to put 24m of coverings on (because a cube has 12 edges, each 2m) Effectively, Lino makes $4.61 per meter of covering that he puts on.
* Surfacio has to put on 24m² of paint (because a cube has 6 faces, each 2m by 2m = 4m²) Effectively, Surfacio makes $4.61 per m² of paint that he puts on.
* Volumio has to put 8m³ of water in (because this cube’s volume is 2m by 2m by 2m = 8m³) Effectively, Surfacio makes $12.5 per m³ of water that he puts in.
Everyone is happy. But now I want to ship 4m cubes. That’s twice as big. I’m a generous guy, so I’ll offer my contractors $250 per big cube (instead of the $200 that you’d expect because the cube is two times as large).
But still two of them reject this offer. Why’s that? Well, let’s consider what I’m asking them to do:
* For a big cube, Lino has to put 48m of coverings on (because a big cube has 12 edges, each 4m) Effectively, Lino makes $5.21 per meter of covering that he puts on, which is better than the $4.61 per meter he earns for small cubes.
* For a big cube, Surfacio has to put on 96m² of paint (because a big cube has 6 faces, each 4m by 4m = 16m²) Effectively, Surfacio makes $2.60 per m² of paint that he puts on, which is decidedly worse than the $4.61 per m² that he earns for small cubes.
* For a big cube, Volumio has to put 64m³ of water in (because a big cube’s volume is 4m by 4m by 4m = 64m³) Effectively, Surfacio makes $3.9 per m³ of water that he puts in, which is decidedly worse than the $12.5 per m³ that he earns for small cubes.
What happening here is that even though the cube is twice as big, the workloads don’t just become twice as big. Relative to the scaling factor of the cube:
* Lino, being onedimensional, sees his workload grow by the scaling factor (2). He has to double how many coverings he puts on.
* Surfacio, being twodimensional, sees his workload grow by **the square of the scaling factor** (2² = 4). He has to put on four times as much paint.
* Volumio, being threedimensional, sees his workload grow by **the cube of the scaling factor** (2³ = 8). He has to put 8 times as much water into the bigger cube.
As you can see here, depending on how many dimensions you operate in, your workload grows by a different amount for a given scaling factor.
To generalize: the square cube law tells you that when you increase your base number b by some scaling factor, then its cube (b³) will increase much more than its square (b²) will. In other words, the higher the power, the bigger the impact of your scaling factor is.
(continued in comment reply below)
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