Science and physics
Scale effects and the square-cube law
Double an object's length and its surface grows four times but its volume eight, so a thing scaled up in proportion is a different thing.
Galileo Galilei, Two New Sciences (1638); J. B. S. Haldane's essay On Being the Right Size (1926).
Galileo set this out on the First Day of Two New Sciences (1638): a larger machine built of the same material and in the same proportions as a smaller one will match it in every respect except that it will not be as strong. The reason is geometry. Double every length and areas go up by two squared, four times, while volumes go up by two cubed, eight times. Weight follows volume; the strength of a bone or a beam follows its cross-sectional area. So the doubled animal carries eight times the weight on four times the bone, twice the stress per square centimetre, and scaled by ten it is ten times the stress. Galileo's conclusion was that nature could not make a horse as large as twenty ordinary horses, or a man ten times taller, without changing the proportions of the limbs.
Haldane's 1926 essay did the arithmetic for a 60-foot giant, ten times human height: a thousand times the weight on a hundred times the bone area, so each square inch of bone carries ten times the load, and since a human thigh bone breaks at about ten times body weight the giant breaks its thighs at every step. That is why an elephant's legs are thick columns and a gazelle's are not. The law runs the other way for surfaces: a mouse has so much surface per gram that it loses heat fast and can survive a long fall, because air resistance scales with area and weight with volume. Any system whose costs and capacities grow at different powers of its size has a size beyond which the old shape fails, and the only fixes are a new shape or a smaller size.
The transfer to organisations and software is real but looser, because the exponents are not two and three. Possible communication paths in a team grow with the square of its size while output grows at best linearly, so processes that worked at 10 people fail at 100 and need structure, not more effort. A monolith's build and test time grows with the code, while the number of changes contending for it grows with the team, so the pipeline that was fine at 50 engineers is the bottleneck at 500. The caution is to measure the exponents rather than assume them. Some things scale sublinearly (a bigger warehouse costs less per unit stored), and a 'scaling problem' is sometimes a slow path that nobody profiled.
Example
In On Being the Right Size (1926), J. B. S. Haldane took the giants Pope and Pagan from an illustrated Pilgrim's Progress, drawn at about 60 feet tall, ten times the height of Christian. Being ten times as high, wide and thick, they would weigh a thousand times as much, which he put at 80 to 90 tons. Their bones would have only a hundred times the cross-section, so every square inch of bone would bear ten times the load borne by a square inch of human bone. Since a human thigh bone breaks under about ten times human weight, Haldane concluded that the giants would have broken their thighs with every step, which, he noted, may be why the picture showed them sitting down.
Where it applies
Before scaling a process, ask which of its costs grow with 'area' and which with 'volume': meeting load, on-call rota, build minutes, rows scanned per query. In data engineering, a daily job that scans the whole table is a square-cube trap, because the data grows with time and so does each scan; partition before the growth, not after. For a team, redesign at each order of magnitude instead of adding people to the old shape.
Think about it
What in your system has grown ten times since its process was designed, and what has quietly grown a hundred times with it?
Applied it to something real? Log it.
Test yourselfIf you double every dimension of an animal, what happens to the stress on its bones and why?
It doubles. Weight follows volume (8 times) and bone strength follows cross-section area (4 times), so each unit of bone area carries twice the load.
Related models
- NonlinearityCause and effect are not proportional: a push can do nothing or everything depending on where the system sits relative to a threshold.
- EmergenceSimple local rules among many parts produce patterns at the level of the whole that no part contains or intends.
- Bottlenecks and the theory of constraintsA system's throughput is set by one constraint; improving anything else is wasted effort until that constraint is fixed.
- Orders of magnitudeEstimate to the nearest power of ten from a few facts you know; being within a factor of three is often enough to decide.
- Brooks's lawAdding people to a late software project makes it later: the new hands cost training and communication before they add output.