Blood vessels, neurons, and tree branches often form junctions that look irregular, yet they follow a shared geometric rule.
Using three-dimensional scans from six kinds of organisms, researchers in New York linked those shapes to surface costs.
Branching tubes and fibers have thickness, so their outer surfaces must merge smoothly where they meet.
At the Rensselaer Polytechnic Institute (RPI), Dr. Xiangyi Meng led a project that treated branches as surfaces.
Dr. Meng’s research focuses on the geometry of complex networks, where physical constraints can change what counts as efficient.
A rule that governs branching networks
Many older models followed a wiring economy idea, the idea that shorter connections cost less, and they minimized total length.
“We were treating these structures like wire diagrams,” noted Dr. Meng, and that one-dimensional view ignores how real surfaces must connect.
That model predicts mostly two-way splits with neat angles, and it struggles when branches have thickness and need smooth joins.
A physical branch carries an outside boundary, and building more boundary takes extra molecules and energy over time.
That surface minimization, a search for the least total surface area, treats each link as a sleeve that must join smoothly.
“There seems to be a universal rule governing the formation of biological networks,” said Meng.
Applying the string theory
Some of the necessary math came from string theory, a physics framework where particles act like strings, moving through space.
When the strings travel, the equations track the surfaces they trace, and the smallest-area surface becomes a natural solution.
At RPI, the team used that string-theory math to build a model that values smooth surfaces over short lines.
Real networks often use three-way junctions, especially when branches thicken locally near a meeting point.
The model predicts a transition to trifurcation, a junction where one link meets three, once thickness makes extra joints costly.
That prediction fits why trees and vessels show plenty of three-way nodes, without blaming measurement noise or biology alone.
Branching networks in the brain
Many neurons grow thin side branches that leave the main shaft at right angles, then stop at a nearby target.
In human brain data, 98 percent of those sprouts ended in a synapse, a junction where one neuron passes signals.
The geometry favors that right-angle stub, because it lets a cell reach a close neighbor while adding little surface area.
Six living systems, same rule
The RPI team tested the model across networks spanning humans, insects, plants, and reef builders.
They used high-resolution 3D reconstructions of neurons, blood vessels, tropical trees, corals, and Arabidopsis thaliana.
Across that mix, the branching patterns matched surface-based predictions better than older one-dimensional wiring models.
Angles tell a quieter story
Classic network math points to a Steiner tree, a shortest-connection solution with extra branch points, creating only two-way splits.
Once the team measured angles in real tissue, they found wide spreads that simple length rules could not reproduce.
Surface costs allow asymmetric angles, because the model balances area against smooth joining at the junction.
As a branch thickens, its surface grows faster than its length, so the cheapest layout can change abruptly.
The paper shows that thicker links can merge two nearby two-way junctions into a single higher-degree node.
That mechanism sets a clear boundary, because very thin links still behave like lines and follow older rules.
Biology does not chase perfection
Real organisms build networks while growing, healing, and competing, so they rarely reach the strict mathematical minimum.
Across their samples, total network length averaged about 25 percent longer than the Steiner tree solution.
That gap leaves room for biology’s extra demands, such as carrying fluids, staying robust, or following developmental cues.
In the brain, a connectome, a map of neural wiring, depends on local branching choices that shape who connects to whom.
Blood vessels follow similar branching rules, and odd junctions can signal disease or injury when growth goes wrong.
A geometry-based rule can guide analysis, but it cannot replace biology-specific studies of flow, growth factors, and repair.
Engineering learns from living networks
Engineers already build branching systems in soft materials, particularly when they print channels that must carry fluids.
Surface-based design rules could help them place junctions that stay smooth, so flow meets less resistance and leaks.
The same logic may also inform transportation or power layouts, where extra thickness and clearance matter in the real world.
Across blood, brains, and plants, the model suggests that surface costs in three dimensions explain branching patterns better than length alone.
If later studies extend the model to denser networks and loops, designers could gain a practical tool, not a final law.
The study is published in the journal Nature.
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