Unification Under the Maximum Information Efficiency Axiom: Homology Between Murray’s Law and Polyhedron Law

Author: Suhang Zhang, Luoyang

Abstract

Murray’s law describes the cubic scaling relation of vessel diameters in biological branching networks, originating from the principle of minimum energy dissipation. The polyhedron law (Euler’s formula V - E + F = 2) characterizes the combinatorial invariant between vertices, edges, and faces of convex polyhedra, with traditional proofs relying on induction or planar graph theory. Long separated into biological fluid mechanics and discrete topology, the two laws have been regarded as independent natural laws. Within the framework of the Maximum Information Efficiency (MIE) axiom, this paper proves that the two are essentially manifestations of the same extremal principle—maximizing information-processing efficiency per unit energy expenditure—at different constraint levels and in different physical media. By defining a unified information-efficiency functional and imposing boundary conditions of continuity (fluid transport) and discreteness (topological networks) respectively, we simultaneously derive the power-law exponent of Murray’s law and the characteristic number of Euler’s formula. This unification not only reveals the deep homology between leaf veins, vascular networks, and polyhedral structures, but also provides strong evidence for the MIE axiom as a cross-domain meta-principle.

Keywords: Maximum Information Efficiency Axiom; Murray’s Law; Euler’s Formula; Information-Energy Manifold; Extremal Principle

 

1 Introduction

Branching networks and closed polyhedral structures are widespread in nature, from plant leaf veins and animal circulatory systems to polyhedral units in crystals and foams. For a long time, laws describing these structures have been compartmentalized into different disciplines. Murray’s law (1926) states that in optimally branched networks under laminar flow, the radius of the parent vessel r_0 and the radii of daughter vessels r_1, r_2 satisfy r_0^3 = r_1^3 + r_2^3, derived from the minimization of fluid energy dissipation. The polyhedron law (Euler’s formula, 1758) asserts that any convex polyhedron satisfies V - E + F = 2, proven via local combinatorial operations or topological invariance. On the surface, the two appear unrelated.

However, recent interdisciplinary research at the interface of information theory and dynamics suggests that many seemingly disparate natural laws may share a common underlying extremal principle. The Maximum Information Efficiency (MIE) axiom [1,2] proposes that any long-term stable dynamical system necessarily extremizes (maximizes or minimizes, depending on boundaries) the information-processing efficiency functional per unit energy consumption. This paper aims to prove that Murray’s law and the polyhedron law are particular solutions of the MIE axiom in continuous transport networks and discrete topological networks, respectively.

 

2 Basic Form of the MIE Axiom

Let the system be described by a state space \mathcal{S} and an evolution operator \Phi. Define the information content I(x) as the minimum binary length (or entropy) required to identify state x. Each evolutionary step consumes a fixed energy \delta E = 1. The information change from state x to y = \Phi(x) is \Delta I = I(y) - I(x). The long-term average information-efficiency functional is

\mathcal{J}[\Phi] = \lim_{T\to\infty} \frac{1}{T} \sum_{t=0}^{T-1} |\Delta I_t|.

MIE Axiom: In the absence of external driving, the system necessarily extremizes \mathcal{J} (maximizes or minimizes, according to boundary conditions).

For conservative systems (total information conserved), maximization is typically chosen; for dissipative systems (e.g., fluid networks), minimizing energy dissipation maps to maximizing information efficiency (as energy acts as the denominator). In the following, we adopt a unified treatment: under given constraints, the Euler–Lagrange equations from the extremal condition determine the structure of the system.

 

3 Continuous Case: Derivation of Murray’s Law

Consider a branching node: parent vessel radius r_0, flow rate Q_0; branching into two daughter vessels with radii r_1, r_2 and flow rates Q_1, Q_2. The fluid is incompressible Newtonian fluid in laminar flow, with energy dissipation per unit length proportional to Q^2 / r^4 (Poiseuille’s law). Meanwhile, information efficiency requires maximizing the information transmitted per unit energy consumption. Information content can be related to flow entropy I \propto -\sum p_i \ln p_i, where p_i = Q_i / Q_0. A more direct approach is: under fixed total flow Q_0, the network structure must minimize energy dissipation, which is equivalent to maximizing information efficiency (more information processed with the same energy). The extremal problem is thus

\min_{r_1, r_2} \left( \frac{L_0 Q_0^2}{r_0^4} + \frac{L_1 Q_1^2}{r_1^4} + \frac{L_2 Q_2^2}{r_2^4} \right),

subject to Q_0 = Q_1 + Q_2, with vessel lengths L_i treated as fixed. Mass conservation requires Q_i \propto r_i^2 v_i, while under laminar flow the average velocity v_i is independent of radius (Poiseuille’s law gives Q \propto r^4 \Delta p/L; classical optimization uses either “minimum total volume” or “minimum energy dissipation”, which are equivalent). Using the method of Lagrange multipliers:

\mathcal{L} = \sum_i \frac{L_i Q_i^2}{r_i^4} + \lambda \sum_i L_i r_i^2.

Differentiating with respect to r_i and setting to zero yields -4 L_i Q_i^2 / r_i^5 + 2\lambda L_i r_i = 0 \Rightarrow Q_i^2 \propto r_i^6 \Rightarrow Q_i \propto r_i^3. From Q_0 = Q_1 + Q_2, we immediately obtain r_0^3 = r_1^3 + r_2^3, i.e., Murray’s law.

Reinterpretation under MIE: The system’s information-processing efficiency \mathcal{J} is proportional to information flow per unit energy dissipation. Information flow can be taken as \sum Q_i \ln Q_i (entropy rate). Minimizing energy dissipation under fixed total information generation rate maximizes \mathcal{J}, leading to the same power law. Thus Murray’s law is a necessary corollary of the MIE axiom in continuous laminar flow networks.

 

4 Discrete Case: Derivation of the Polyhedron Law (Euler’s Formula)

Consider a convex polyhedron viewed as an information transmission network: vertices as information nodes, edges as communication channels, and faces as feedback loops. Define information content I as the minimum number of bits required to identify the entire topological structure. Let the degree of each vertex be d_v and the number of edges per face be s_f. Classical combinatorial topology gives the Euler characteristic \chi = V - E + F = 2. We now derive it from the MIE axiom.

We construct the information-efficiency functional as follows: each edge (channel) consumes one unit of energy per information transfer. The total information processing of the system can be measured by the number of reachable paths between vertices or the independence of all cycles. A simpler approach uses average information entropy: let each face be an independent information-processing unit, with information exchange between faces via edges. Total system information I_{\text{total}} can be approximated as F \cdot \langle s \rangle (average edges per face), minus redundancy. A well-established alternative uses the graph Laplacian spectrum, though it is overly complex. Here we use a known result: for any planar graph (embeddable on a sphere), Euler’s formula is equivalent to the discrete Gauss–Bonnet theorem:

\sum_v \left(2\pi - \sum_{\text{faces at }v} \theta_{v,f}\right) = 2\pi \chi,

where \theta_{v,f} is the internal angle of a face at vertex v. In convex polyhedra, the total angle sum at each vertex is less than 2\pi, concentrating curvature. From the MIE perspective, maximizing information efficiency is equivalent to maximizing topological entropy. It is known that triangulations (3F = 2E) yield the maximum combinatorial entropy among all planar graphs on the sphere. More directly, a convex polyhedron can be seen as an extreme information-compression system: given vertex count V, how to arrange edges and faces to minimize the information required to describe the whole structure? Minimizing description length is equivalent to maximizing efficiency, with description length related to \ln E and \ln F. The extremal condition enforces V - E + F = 2.

We present a concise variational argument: let total system energy dissipation be proportional to edge count E (one unit per edge transmission). Total information processing is proportional to the number of independent cycles (first Betti number) \beta_1 = E - V + 1 (for a sphere, face count F = E - V + 2, so \beta_1 = F - 1). Information efficiency \mathcal{J} = \text{(information content)} / \text{(energy dissipation)}. Information content may be taken as \ln \mu, where \mu is the number of distinct edge walks; for simplicity, maximizing information efficiency requires maximizing independent cycles per edge:

\frac{\beta_1}{E} = \frac{E - V + 1}{E}.

For fixed V, differentiating with respect to E:

\frac{d}{dE}\left(1 - \frac{V-1}{E}\right) = \frac{V-1}{E^2} > 0,

implying E should be maximized. However, convex polyhedra have an upper bound: for triangulations E_{\max} = 3V - 6 (except V=4). Substituting gives F_{\max} = 2V - 4, and at this extremum V - E + F = V - (3V-6) + (2V-4) = 2. Thus, driven by MIE maximization, the system naturally reaches an extremal graph (maximum edges), which exactly satisfies Euler’s formula. Non-extremal graphs (polyhedra with fewer edges) have lower information efficiency and are therefore not adopted by stable systems. Euler’s formula is thereby derived as a topological invariant under the MIE extremal condition.

 

5 A Unified Example: Leaf Venation

Leaf vein networks must simultaneously satisfy:

- 2D planar topological constraints (no self-intersections, approximate spherical tiling) → globally consistent with Euler’s formula;
- Optimized fluid transport energy dissipation → branching approximately obeys Murray’s law.

Traditionally viewed as independent, the MIE axiom reveals they both arise from extremizing a single functional

\mathcal{J} = \frac{\text{total information processing}}{\text{total energy dissipation}}.

At the global topological scale, total information is determined by the combinatorial structure of faces, edges, and vertices; at the local branching scale, by flow distribution and radius scaling. The extremization process locks both, causing leaf venation to naturally form “Euler-consistent Murray networks”.

 

6 Conclusion

Within the framework of the Maximum Information Efficiency axiom, this paper unifies Murray’s law and the polyhedron law. We show that:

- In continuous transport networks, the MIE extremal condition leads to minimum energy dissipation, yielding r_0^3 = \sum r_i^3;
- In discrete convex polyhedral networks, the MIE extremal condition (maximizing independent cycles per edge) leads to maximally edge-dense triangulations, which necessarily satisfy V - E + F = 2.

This unification reveals a shared physical origin behind biological branching networks and geometric-topological structures: stable systems always maximize information-processing efficiency per unit energy. Our results provide two cross-domain proofs for the universality of the MIE axiom, and predict that other hitherto unconnected laws (e.g., Kepler’s conjecture, honeycomb optimization) can also be derived from MIE.

Future work will formalize this unification as a variational principle and explore its applications in graph theory and biological morphogenesis.

 

References

[1] Zhang, S. Axiomatic Reconstruction of the Maximum Information Efficiency Axiom and the Collatz Conjecture (I). Preprint, 2026.
[2] Zhang, S. The Maximum Information Efficiency Axiom as a Global Constraint in Discrete Dynamical Systems. arXiv, 2026.
[3] Murray, C. D. The Physiological Principle of Minimum Work. Proceedings of the National Academy of Sciences, 1926.
[4] Euler, L. Elementa doctrinae solidorum. Novi Commentarii Academiae Scientiarum Petropolitanae, 1758.
[5] Katifori, E., et al. Damage and Fluctuations Induce Loops in Optimal Transport Networks. Physical Review Letters, 2010.