Start with the number, because the number is the whole story. In April 2026, Tian Dong of the University of Texas Rio Grande Valley and his co-authors made the cover of Science by showing that a rule discovered in 1957 for river tributaries also governs river deltas — the fanned-out distributary channels where a river meets the sea. The rule is Hack’s law, and the number is 0.6.

John Hack, a U.S. Geological Survey scientist, measured streams across Virginia and Maryland and found that a channel’s length scales with its drainage area raised to the power 0.6 — in symbols, L ~ A^0.6. That exponent is the surprise. If you sketched an idealised square field draining to one outlet, you would guess the exponent should be 0.5 (a channel the length of one side, the square root of the area). Nature refuses. As Daniel Rothman of MIT puts it, small basins are “short and squat,” large basins “long and thin” — the network stretches toward the sea instead of staying self-similar.

System. Why 0.6 and not 0.5? Because a river is solving an optimisation problem it was never told about. In seminal papers from the early 1990s, the Italian hydrologist Andrea Rinaldo and colleagues showed that networks obeying the 0.6 exponent are the ones that dissipate the least energy carrying rain to the sea — the optimal channel network. Digging a private channel from every patch of field to the outlet wastes length; a branching trunk that many tributaries share is cheaper. Gravity supplies the potential energy, friction spends it through erosion, and over millennia the landscape reshapes itself until the wasteful configurations erode away and the efficient trunk persists. Hansjörg Seybold of the Austrian Academy of Sciences calls the deeper mechanism “still the big question” — but the direction of the answer is settled: shared infrastructure beats direct routing, and 0.6 is the price of that sharing.

←TODAY: April 2026 — Dong’s team proves Hack’s 0.6 exponent holds even for deltas, where sediment flows the opposite way. →3012: the drainage, MEP and load-path trees of a building are specified against a measured optimal-transport law, not a draughtsman’s habit. Fulcrum: a network that minimises energy dissipation is the same object whether water erodes it over 10,000 years or an engineer draws it in an afternoon — only the timescale differs.

Street. This is not a nature documentary for the desk. Every branching transport tree you draw — stormwater risers, chilled-water distribution, a structural load path fanning to the foundations — is answering Rinaldo’s question. The river’s lesson is blunt: direct-run-everything looks tidy on a plan and is almost always the expensive answer. D’Arcy Thompson said as much in On Growth and Form a century ago — form follows the physics of transport — and rivers are the cleanest proof we have, arising, as Chris Paola of the University of Minnesota notes, from “neither biological evolution nor urban planning” yet obeying the same universal law.

Atelier: The Monday move for a Büro carrying a services model: before you accept the auto-routed riser tree your BIM tool generated, pull the branch lengths and served areas out of the model and check whether the network is trending toward a shared-trunk 0.6 logic or a wasteful direct-run 1.0 — the ratio tells you where the friction (and the pump energy, and the copper) is hiding.

Hack: Fit the exponent yourself before you trust any transport diagram. This computes Hack’s scaling from a handful of gauged sub-basins — the same fit works on channel-length-versus-served-area for a services tree.

import numpy as np
area   = np.array([2.1, 15, 340, 9800])   # basin km^2
length = np.array([1.8, 6.2, 41, 260])    # mainstem km
exp = np.polyfit(np.log(area), np.log(length), 1)[0]
print(round(exp, 2))   # ~0.6 = optimal, not the naive 0.5

An exponent near 0.6 says the network shares its trunks efficiently; drift toward 0.5 and it is over-fragmented, toward 1.0 and it is all private channels. The physics does not care whether the fluid is rain or return air.

Move. Take one branching system already on your board — drainage, HVAC, or a truss load path — extract branch length against served area, and fit the exponent. If it isn’t hugging 0.6, you have found where the design is paying rent to friction, in copper, pump watts and pressure drop.

Source: quantamagazine.org