193 Cross-Disciplinary Unification via the Axiom of Maximum Information Efficiency: Murray's Law, Polyhedron Formula, and Fermat's Principle

Author: Zhang Suhang, Luoyang

Abstract: The Axiom of Maximum Information Efficiency (MIE), established in previous works, has provided a complete deductive chain from the principle of least action to stability. This paper applies the axiom to three seemingly unrelated fields: biological transport networks (Murray's law), combinatorial topology (Euler's polyhedron formula), and geometric optics (Fermat's principle). We demonstrate that these three classical laws, belonging to different disciplines, can all be derived uniformly from the MIE axiom: Murray's law corresponds to the maximization of information efficiency via energy minimization, Euler's formula corresponds to the extremal structure of topological information compression, and Fermat's principle corresponds to the optimal information efficiency via the shortest optical path. This cross-disciplinary synthesis demonstrates the universality of the MIE axiom as a meta-principle and provides a paradigm for extending MIE to further fields.

Keywords: Axiom of Maximum Information Efficiency; Murray's law; Euler's formula; Fermat's principle; cross-disciplinary unification

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1 Introduction

In works 190 [1], 191 [2], and 192 [3], we established the theoretical framework of the MIE axiom and completed the full deductive chain from the principle of least action to stability. However, the true value of an axiomatic system lies not only in its internal consistency but also in its explanatory power and unifying capacity across disciplines.

This paper aims to demonstrate, through three representative case studies, how the MIE axiom can subsume classical laws from different fields under a single framework:

Field Law Core Content MIE Interpretation
Biofluid dynamics Murray's law r₀³ = r₁³ + r₂³ Energy minimization ≡ Information efficiency maximization
Combinatorial topology Euler's formula V − E + F = 2 Extremal structure of topological information compression
Geometric optics Fermat's principle δ∫ n ds = 0 Shortest optical path ≡ Optimal information efficiency

These three laws have long been rigorously proven within their respective domains and have generally been regarded as unrelated. This paper does not claim to "re-prove" them — their original proofs remain valid. What we do is to show that they can all be re-derived from the MIE axiom, thereby revealing their shared deep foundation.

The structure of this paper is as follows: Section 2 derives Murray's law; Section 3 derives Euler's formula; Section 4 derives Fermat's principle; Section 5 summarizes the unified structure among the three; Section 6 concludes.

2 Murray's Law: MIE Derivation of Biological Transport Networks

2.1 Problem Description

In biological transport networks such as circulatory systems and plant vascular bundles, the radii of parent and daughter vessels at a branching node obey Murray's law:

r₀³ = r₁³ + r₂³

The traditional derivation is based on the principle of minimum energy dissipation: for laminar flow in a tube at a given flow rate, energy dissipation is proportional to Q²/r⁴, and volume is proportional to r²; minimizing total dissipation under a volume constraint yields the above relation.

2.2 MIE Reformulation

Consider the transport network as an information–energy system:

· Information content: the "material information" carried by the flow distribution, quantifiable as I ∝ ln Q, or more precisely, the flow entropy S = –∑ p_i ln p_i.
· Energy dissipation: the energy consumed by the fluid overcoming viscous drag; dissipation per unit length ∝ Q²/r⁴.
· Information efficiency: J = (amount of information processed) / (energy dissipation).

The MIE axiom requires that a long-term stable network structure must extremize J. For a dissipative system, efficiency maximization is equivalent to energy minimization.

2.3 Derivation

Consider a branching node: parent vessel radius r₀, flow rate Q₀; two daughter vessels radii r₁, r₂, flow rates Q₁, Q₂, satisfying Q₀ = Q₁ + Q₂.

Total energy dissipation per unit length:

P = Q₀²/r₀⁴ + Q₁²/r₁⁴ + Q₂²/r₂⁴ (ignoring constant factors)

Total volume (material cost):

V = r₀² + r₁² + r₂² (ignoring length factors)

The MIE extremal condition is equivalent to minimizing P given V (or vice versa). Introduce a Lagrange multiplier λ:

L = (Q₀²/r₀⁴ + Q₁²/r₁⁴ + Q₂²/r₂⁴) + λ(r₀² + r₁² + r₂²)

Taking partial derivatives with respect to r_i and setting them to zero:

∂L/∂r₀ = –4Q₀²/r₀⁵ + 2λ r₀ = 0 ⇒ 2Q₀²/r₀⁶ = λ
∂L/∂r₁ = –4Q₁²/r₁⁵ + 2λ r₁ = 0 ⇒ 2Q₁²/r₁⁶ = λ
∂L/∂r₂ = –4Q₂²/r₂⁵ + 2λ r₂ = 0 ⇒ 2Q₂²/r₂⁶ = λ

Hence:

Q₀²/r₀⁶ = Q₁²/r₁⁶ = Q₂²/r₂⁶ ⇒ Q₀/r₀³ = Q₁/r₁³ = Q₂/r₂³

Using flow conservation Q₀ = Q₁ + Q₂, we substitute:

r₀³ = r₁³ + r₂³

This is Murray's law.

2.4 MIE Interpretation

· Murray's law is a special case of the MIE axiom in biological transport networks.
· Maximization of information efficiency (equivalently, minimization of energy dissipation) drives the geometry of branching networks.
· Branching structures that deviate from this law either have higher energy dissipation or lower information-carrying efficiency, and are eliminated by evolution.

3 Euler's Formula: MIE Derivation for Convex Polyhedra

3.1 Problem Description

For any convex polyhedron, the numbers of vertices V, edges E, and faces F satisfy:

V − E + F = 2

Traditional proofs use planar graph expansion and induction, or the Euler characteristic in homology theory.

3.2 MIE Reformulation

Consider a convex polyhedron as an information network:

· Information content: the minimum number of binary digits required to identify the topological structure.
· Energy dissipation: the number of edges E (each edge being an information transmission channel).
· Information efficiency: J = (topological information content) / (energy dissipation).

The MIE axiom requires that a long-term stable polyhedral structure (such as crystals, foams, viral capsids) must extremize the information efficiency.

3.3 Quantifying Information Content

For a planar graph (embedded on a sphere), a classical measure of combinatorial entropy is:

· The number of independent cycles (first Betti number) β₁ = E − V + 1.
· The number of faces F = E − V + 2 (this is precisely Euler's formula, which we aim to derive).

To avoid circularity, we adopt a more fundamental measure: the minimum number of bits required to identify the graph structure is proportional to ln(number of spanning trees). However, a more direct approach is to note that maximizing information efficiency drives the system toward a maximal-edge configuration.

3.4 Extremal Argument

Given a fixed number of vertices V, a stable polyhedron tends to maximize its information-processing capacity. For convex polyhedra, the number of edges E has an upper bound: for a triangulation (every face a triangle), E_max = 3V − 6 (for V ≥ 4). Then:

F_max = 2V − 4

Computing the Euler characteristic:

V − E_max + F_max = V − (3V − 6) + (2V − 4) = 2

Thus, the information efficiency extremal state (maximal number of edges) automatically satisfies Euler's formula. Any polyhedron with fewer edges (e.g., predominantly quadrilateral faces) has a lower E/V ratio, lower information efficiency, and is therefore not the preferred stable structure.

3.5 General Derivation from MIE Extremum to Euler's Formula

A more rigorous derivation is based on an MIE reformulation of the discrete Gauss–Bonnet theorem. For a convex polyhedron embedded on a sphere, curvature is concentrated at vertices, and the total curvature is 2πχ = 2π(V − E + F). The MIE extremum (here, maximizing the efficiency of topological information compression) requires maximizing flatness, i.e., minimizing the absolute value of the total curvature. On a sphere, χ = 2 is the unique positive integer that minimizes the absolute curvature, yielding:

V − E + F = 2

3.6 MIE Interpretation

· Euler's formula is a manifestation of the MIE axiom in combinatorial topology.
· For a given number of vertices, the maximum information efficiency of a convex polyhedron corresponds to a triangulation.
· The topological invariant of a triangulation is exactly χ = 2.
· Hence, Euler's formula is not an "accidental coincidence" but a necessary consequence of information efficiency extremization.

4 Fermat's Principle: MIE Derivation in Geometric Optics

4.1 Problem Description

Fermat's principle (1657) is historically the first explicitly stated extremal path principle: the path taken by light traveling from point A to point B extremizes (usually minimizes) the optical path length:

δ∫_A^B n(r) ds = 0

where n(r) is the refractive index.

4.2 MIE Reformulation

Consider light propagation as an information–energy system:

· Information content: the phase information carried by the light wave. The phase difference between neighboring paths determines the interference pattern.
· Energy dissipation: the travel time T (or optical path length S). A fixed energy budget is consumed per unit time.
· Information efficiency: J = (amount of information transferred) / (energy dissipation).

The MIE axiom requires that the long-term stable propagation mode (i.e., the macroscopically observed ray) must extremize the information efficiency.

4.3 Path Integral Perspective

In the path integral formulation of quantum electrodynamics, the probability amplitude for light to travel from A to B is:

A = ∫ exp(i(2π/λ) S[γ]) Dγ

where S[γ] = ∫ n ds is the optical path length. For macroscopic scales, λ → 0, the path integral is approximated by the stationary phase method: only paths near the extremum of S contribute significantly; other paths have rapidly oscillating phases and cancel destructively.

MIE interpretation:

· The system "tries" all possible paths.
· However, only the extremal paths produce constructive interference, concentrating the light energy.
· This is equivalent to: the amount of information (coherence) transferred per unit energy is maximized.
· Therefore, Fermat's principle is a necessary consequence of the MIE axiom in the limit of wave optics.

4.4 Deriving Snell's Law from the Extremal Condition

Consider a ray refracting at the interface between medium 1 (refractive index n₁) and medium 2 (refractive index n₂). The angles of incidence and refraction are θ₁ and θ₂. The optical path length is:

S = n₁√(x² + a²) + n₂√((d−x)² + b²)

The extremal condition dS/dx = 0 yields:

n₁ sin θ₁ = n₂ sin θ₂

This is Snell's law. It is a direct consequence of Fermat's principle and a concrete manifestation of the MIE axiom in geometric optics.

4.5 MIE Interpretation

· Fermat's principle is the historical precursor of the MIE axiom in geometric optics.
· The shortest optical path ensures maximum information transfer efficiency for a given energy budget.
· Any ray deviating from the extremal path either takes longer or has reduced coherence.
· Thus, light "choosing" the extremal path is not anthropomorphic teleology but a necessary manifestation of the MIE axiom.

5 Unified Structure of the Three Laws

5.1 Common Mathematical Form

All three laws can be expressed as δF = 0, where:

Law Functional F Variables
Murray's law Energy dissipation P = ∑ Q_i²/r_i⁴ Radii r_i
Euler's formula Topological entropy H_top (or −E) Number of edges E
Fermat's principle Optical path length S = ∫ n ds Path γ

In the MIE framework, these functionals are equivalent forms of the information efficiency J under specific constraints.

5.2 Common MIE Core

Step Content
1 Define information content I (flow entropy / topological complexity / phase coherence)
2 Define energy dissipation E (viscous dissipation / number of edges / time)
3 Construct efficiency J = I/E
4 MIE axiom: δJ = 0
5 Under specific constraints, δJ = 0 reduces to δF = 0
6 Solve the Euler–Lagrange equations to obtain the classical law

5.3 Significance of Cross-Disciplinary Unification

· Murray's law: demonstrates that MIE applies to continuous dissipative systems (biophysics).
· Euler's formula: demonstrates that MIE applies to discrete topological structures (combinatorics).
· Fermat's principle: demonstrates that MIE applies to wave/optical systems (physics).

Together, they illustrate the unifying power of the MIE axiom across fluid dynamics, topology, and optics.

6 Conclusion

This paper has completed a cross-disciplinary synthesis using the MIE axiom:

1. Murray's law: From the MIE extremum (energy minimization) we derived r₀³ = r₁³ + r₂³.
2. Euler's formula: From the MIE extremum (maximizing topological information compression efficiency) we derived V − E + F = 2.
3. Fermat's principle: From the MIE extremum (shortest optical path) we derived Snell's law.

These three laws originally belong to biology, mathematics, and physics, each with its own independent body of proofs. This paper does not challenge those proofs but reveals that they share a common MIE axiom foundation.

Thus, the MIE axiom series is now as follows:

Number Content Role
190 From least action to MIE Foundational
191 Conservation laws and symmetry Middle layer of the chain
192 Extremal criterion for stability Terminal link of the chain
193 Cross-disciplinary unification Horizontal demonstration

The MIE axiom not only provides a unified deduction from least action → conservation → symmetry → stability but also extends this framework beyond classical mechanics, demonstrating its potential as a cross-disciplinary meta-principle.

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References

[1] Zhang Suhang. 190 Axiom of Maximum Information Efficiency (I): From Least Action to MIE.
[2] Zhang Suhang. 191 Axiom of Maximum Information Efficiency (II): Derivation of Conservation Laws and Symmetries.
[3] Zhang Suhang. 192 Axiom of Maximum Information Efficiency (III): Extremal Criterion for Stability.
[4] Murray, C. D. The Physiological Principle of Minimum Work. PNAS, 1926.
[5] Euler, L. Elementa doctrinae solidorum. Novi Commentarii academiae scientiarum Petropolitanae, 1758.
[6] Fermat, P. de. Synthese ad refracciones (1657).
[7] Landauer, R. Irreversibility and Heat Generation in the Computing Process. IBM J. Res. Dev., 1961.
[8] Shannon, C. E. A Mathematical Theory of Communication. Bell Syst. Tech. J., 1948.

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Appendix: Comparative Table of MIE Derivations for the Three Laws

Step Murray's law Euler's formula Fermat's principle
System Vascular branching Convex polyhedron Light in media
Information content I Flow entropy –∑ p_i ln p_i Topological complexity ln(# spanning trees) Coherence degree
Energy dissipation E Viscous dissipation Q²/r⁴ Number of edges E Time T
Efficiency J I/P I/E I/T
MIE extremum δJ = 0 δJ = 0 δJ = 0
Equivalent form Minimum energy dissipation Maximum edges (triangulation) Minimum optical path
Derived law r₀³ = r₁³ + r₂³ V − E + F = 2 n₁ sin θ₁ = n₂ sin θ₂