To understand the origin of the observed angle diversity, a violation of rule 3 (Fig. 1f), we assume that each link i is characterized by its unique circumference constraint w i . Without a loss of generality, we set w 1 = w 2 = w and w 3 = w′, and vary the ratio ρ = w′/w, to obtain the minimal manifold that connects nodes 1, 2 and 3 (Fig. 4a,b). Although Steiner’s solution posits a constant steering angle Ω 1→2 ≈ 0.3π, surface minimization predicts two distinct regimes separated by a threshold value ρ th (Supplementary information Section 7). (1) For ρ > ρ th , we predict the steering angle Ω 1→2 ≈ k(ρ − ρ th ) (Fig. 4e,f), that is, a linear dependence on ρ (Fig. 4g). This regime can therefore account for the wide range of angles observed in Fig. 1f. (2) For ρ < ρ th , surface minimization makes an unexpected prediction: if links 1 and 2 have comparable diameters, they are expected to form a straight path (that is, continue with solid angle of Ω 1→2 = 0), whereas the thinner link 3 is predicted to emerge perpendicularly at Ω 1→3 ≈ Ω 2→3 , consistent with an orthogonal sprouting behaviour (Fig. 4c,d). Note that a geometric approach predicted as early as 1976 (refs. 28,29) that the branch angles converge to 90° in the ρ → 0 limit (Supplementary information Section 7). By contrast, our framework predicts that the 90° solution is optimal for any ρ < ρ th (Fig. 4g). Hence, orthogonal sprouts are not singular solutions that emerge only in the ρ → 0 limit28,29. Rather, they are stable solutions of surface minimization that remain minimal for a wide range of parameter values and hence they should be not only observable but prevalent in real physical networks.
Fig. 4: Branching versus sprouting bifurcations. a, We start from a triangular node configuration, with w 1 = w 2 = w and w 3 = w′. b, We construct the minimal-surface manifold connecting the three nodes. c,d, For small ρ = w′/w, the link of node 3 is thin and the optimal manifold favours a sprouting structure: nodes 1 and 2 linked through a straight line and node 3 by means of a perpendicular link. e,f, For large ρ, we find a linear relation between ρ and the three-dimensional steering angle, Ω 1→2 , related to the branching angle θ (Fig. 1f) through Ω 1→2 = 4πsin2((π − θ)/4). As ρ increases, the bifurcation point approaches the triangle centre and the bifurcation gradually resembles a symmetric branching. g, Ω 1→2 versus ρ. We observe a transition from sprouting (Ω = 0) to branching (Ω > 0) at ρ ≈ 0.6. The symmetric branching observed by Steiner appears near ρ = 1. h, In the human connectome, 92% of the observed sprouts end on synapses, suggesting that neuronal systems use surface minimization to form direct synaptic connections to adjacent neurons with minimal material cost. i–n, According to g, cumulative \(| {\int }_{\rho }^{{\rho }_{{\rm{th}}}}\varOmega (\rho ){\rm{d}}\rho | \) should follow approximately (ρ th − ρ)1 for ρ < ρ th and approximately (ρ − ρ th )2 for ρ > ρ th , predictions closely followed by real physical networks. Band thickness represents one standard error of the fitting. Full size image
To test these predictions, we identified all bifurcation motifs in each network in our database and then searched for branches that satisfy w 1 = w 2 = w. We then measured Ω(ρ) = Ω 1→2 as a function of the empirically observed ρ, finding that almost all bifurcations for ρ < ρ th are sprout-like, characterized by small Ω(ρ) (Supplementary information Section 7). In Fig. 4i–n, we show the cumulative value of the observed angles in the two regimes, offering evidence that the cumulative \(| {\int }_{\rho }^{{\rho }_{{\rm{th}}}}\varOmega (\rho ){\rm{d}}\rho | \) follows approximately (ρ th − ρ)1 for ρ < ρ th and a quadratic behaviour approximately (ρ − ρ th )2 for ρ > ρ th , in line with the predictions of Fig. 4g.
The key outcome of surface minimization is the predicted prevalence of the orthogonal sprouts, expected to emerge each time ρ < ρ th . To falsify this prediction, we ask: are such sprouts really present in physical networks? Note that the excess of sprouts over the expectations of length or volume optimization was already noted in arterial systems as early as 1976 (ref. 29). This abundance remained unanswered and it also remains unclear whether sprouts represent a generic feature across all physical networks or are unique to blood vessels. To address this, we first identified all bifurcations with w 1 ≈ w 2 in blood vessels, confirming that, in 25.6% of the cases, the third branch, independent of ρ, is perpendicular to the main branches, representing an abundant sprouting behaviour. Yet, we find that sprouts are not limited to the circulatory system but are present in all studied networks, representing 12.9% of the w 1 ≈ w 2 cases in the tropical trees, 52.8% in corals, 11.2% in arabidopsis, 13.8% in the fruit fly neurons and 18.4% in the human neurons. Most importantly, some systems have learned to turn sprout behaviour to their advantage, assigning it a functional role. Indeed, in the human connectome, we identified 4,003 sprouts, finding that 3,911 of these (98%) end with a synapse (Fig. 4h). In other words, neuronal systems have adapted to rely on surface minimization by using orthogonal sprouts as dendritic spines that allow them to form synapses with nearby neurons with minimal material cost. Similarly, roots in plants46 and hyphae branches in fungi47 are known to sprout perpendicularly, allowing plants and fungi to explore a larger volume of soil for water and nutrients with minimal material expenditure.
The predicted relation between Ω(ρ) and ρ in Fig. 4g leads to further falsifiable predictions for the P(Ω) angle distributions, conditioned on the empirically observed ρ values. In the sprouting regime (ρ < ρ th ), we predict Ω = 0, independent of ρ, hence we anticipate a sharp peak of P(Ω) at Ω = 0, in agreement with the empirical data (left side, sprouting regime in Fig. 5a–f). In the branching regime (ρ > ρ th ), however, P(Ω) is predicted to exhibit a broad distribution with high variance, rooted in the linear behaviour of Fig. 4g. The empirical data support this prediction as well (right side, branching regime in Fig. 5a–f). By comparison, the Steiner prediction posits a sharp peak of P(Ω) independent of ρ (thin grey lines in both sprouting and branching regimes in Fig. 5a–f).