A branching network, like a river delta or a tree crown, is built on the principle of economy. The classic law, derived back in 1926, stated that the cube of the parent vessel's diameter equals the sum of the cubes of the daughter diameters. This rule worked beautifully for ideal tubes, but real arteries stubbornly gave a different exponent — around 2.7 instead of 3.
The authors of the new study found the reason. The law didn't account for the thickness of vessel walls, which grows nonlinearly with radius: the wall is thicker in larger arteries. Adding this 'material tax,' the scientists showed that the universal branching exponent disappears: it becomes different for different sizes. Murray's law turned out to be not a general principle but a special case where walls are considered infinitely thin.
The new calculation gave a range for the exponent from 2.90 to 2.94 for symmetric branching — closer to the real 2.7, but not fully. The remaining discrepancy, according to the authors, is linked to blood pulsation: the rhythmic thumps of the heart add their own requirements to vessel geometry. Additionally, accounting for wall thickness explained why vessels more often divide into two rather than three or four branches: binary branching saves material under steric constraints.
If vessel walls were infinitely thin, all living transport networks, from leaves to lungs, would have the same branching exponent. But nature always pays for material.