That famous 'law' for how tree branches and blood vessels grow? Turns out it’s just a total mathematical accident.
arXiv · March 17, 2026 · 2603.13687
The Takeaway
For 100 years, 'Murray's Law' has been used to explain the 'perfect' branching of biological networks like arteries and plant veins. This research proves the law is actually a mathematical fluke that fails when you account for the actual thickness of vessel walls, showing that our anatomy is governed by structural metabolic costs rather than an ideal mathematical rule.
From the abstract
Murray's cubic branching law ($\alpha=3$) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees yield $\alpha \sim 2.7-2.9$. We show that this discrepancy has a structural origin: Murray's universality is an artifact of cost homogeneity, not a biological property. Incorporating the empirical vessel-wall thickness law $h(r)=c_0 r^p$ ($p \approx 0.77$) introduces a third metabolic cost term $\propto r^{1+p}$ that renders the cost function inhomo