"Heuristic Association Between Euler’s Polyhedron Formula and the Golden Ratio Under the Maximum Information Efficiency Axiom"

Author: Zhang Suhang, Luoyang

Core Axiom: Maximum Information Efficiency (MIE) Axiom

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Abstract

Based on the Maximum Information Efficiency (MIE) axiom, this paper rigorously derives Euler’s polyhedron formula V - E + F = 2 as a necessary topological invariant of two-dimensional connected planar networks under the extremal constraint of information efficiency. The derivation does not rely on spanning trees or induction; instead, it directly yields triangulation from the MIE condition of “no redundant optimizability,” leading to Euler’s formula. Furthermore, this paper discusses the potential status of the golden ratio \phi = (\sqrt{5}-1)/2 within the MIE framework: the golden ratio emerges from one-dimensional self‑similar ratio optimization, similarly as a product of an extremal principle. However, a rigorous variational derivation from MIE to \phi is currently lacking. This observation is presented heuristically, and it is explicitly stated that the association between the two remains at the level of analogy and conjecture, requiring further rigorous study. This paper aims to provide a unified perspective for information ecological topology by introducing two fundamental structural constants (Euler characteristic 2 and the golden ratio \phi ), while adhering to academic honesty by distinguishing between proven and unproven parts.

Keywords: Maximum Information Efficiency (MIE); Euler’s polyhedron formula; golden ratio; information ecological topology; extremal principle; triangulation

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1. Introduction

The golden ratio \phi \approx 0.618 and Euler’s polyhedron formula V - E + F = 2 appear repeatedly in natural forms and network structures. The former governs phyllotaxis, spiral shells, and artistic compositions; the latter constrains convex polyhedra surfaces, leaf venation, honeycombs, and circuit topologies. For a long time, they have been assigned to distinct fields—number theory/aesthetics and combinatorial topology—without apparent connection. Yet both exhibit a characteristic of “optimal steady state”: any deviation leads to a loss in efficiency. This suggests they may be governed by the same underlying extremal principle.

In previous work, the author established the Maximum Information Efficiency (MIE) axiom: any stable system must render the information efficiency functional \delta \int (dI/dC) \, d\mathcal{V} = 0 extremal. This paper first rigorously proves that, under the MIE axiom, the optimal structure of a two‑dimensional connected planar network must satisfy Euler’s polyhedron formula. Then, it discusses whether the golden ratio can also be derived from the MIE axiom. The conclusion is that a strict derivation—unlike that for Euler’s formula—is currently not available, although strong indications suggest that the golden ratio is the MIE extremal solution in one‑dimensional self‑similar ratio systems. This is presented as a heuristic conjecture, explicitly marked as unproven, and future research directions are indicated.

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2. Maximum Information Efficiency (MIE) Axiom

Axiom (MIE): A stably existing physical system, network, or structure renders the information efficiency functional stationary:

\delta \mathcal{J}_{\text{info}} = \delta \int \frac{dI}{dC} \, d\mathcal{V} = 0,

where dI is the effective amount of information transmitted/expressed, dC is the physical cost incurred (energy, material, time, etc.), and d\mathcal{V} is the spatial integration element. An equivalent statement: the system contains no redundancy or optimizable space that could increase information efficiency.

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3. Rigorous Derivation of Euler’s Polyhedron Formula from MIE

3.1 Problem setup

Consider a connected planar graph (or the net of a convex polyhedron) with V vertices, E edges, and F faces (including the unbounded outer face). The graph is connected, bridgeless, and loops are not counted.

3.2 MIE extremal condition → triangulation

The MIE axiom requires that any edge that can be removed without destroying connectivity is redundant, because its removal reduces cost ( dC decreases) while keeping information transmission capacity ( dI unchanged or only slightly reduced). Hence, in the extremal state, no further edge can be added without violating planarity. That is, the graph is a maximal planar graph. For a maximal planar graph ( V \ge 3 ), every face (including the outer face) must be a triangle. Otherwise, if a face has k \ge 4 sides, one could add a diagonal inside that face, splitting it into two, adding an edge without breaking planarity, while shortening distances between some vertices (improving information transmission efficiency). This contradicts the condition that no edge can be added. Therefore, all faces are triangles.

3.3 Counting relations

· Each triangle has 3 edges, giving a total edge count of 3F .
· Every edge belongs to exactly 2 triangles (under spherical projection, the outer face is also a triangle, so each edge is shared by two faces). Hence:
3F = 2E \quad \Rightarrow \quad E = \frac{3}{2}F.
· For a maximal planar graph, a classic result (derivable from the handshaking lemma and minimum degree at least 3) is:
E = 3V - 6.
This can also be derived directly from MIE: the edge count of a maximal planar graph has reached the upper bound; for planar graphs, the maximum number of edges is indeed 3V-6 . Thus MIE requires attaining this upper bound.

3.4 Solving simultaneously

From E = 3V - 6 and 3F = 2E :

3F = 2(3V - 6) = 6V - 12 \quad \Rightarrow \quad F = 2V - 4.

Substituting into the Euler characteristic:

V - E + F = V - (3V - 6) + (2V - 4) = 2.

Theorem 1 (MIE‑Euler Theorem): Under the MIE axiom, any structure achieving extremal information efficiency in a two‑dimensional connected planar network necessarily satisfies V - E + F = 2 .

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4. Golden Ratio: Heuristic Association with MIE

4.1 Extremal property of the golden ratio in self‑similarity

The golden ratio \phi = (\sqrt{5}-1)/2 \approx 0.618 is the positive root of \frac{1}{\phi} = \phi + 1 . It appears as the unique optimal solution in problems such as:

· Dividing a line segment into two parts so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part (self‑similarity).
· Maximizing the area or some “information density” of a recursive structure under a fixed perimeter.

4.2 Potential connection with MIE

Suppose we have a one‑dimensional continuous system (e.g., a fixed length segment to be divided into two subsegments, and then the subsegments are recursively divided). We wish to describe the entire recursive structure with minimal parameters while maximizing information transmission efficiency at each scale. Such a problem can be formalized as a functional extremum problem, where the cost C is proportional to total length or material, and the information I is proportional to the number of recursion layers or coverage. Intuitively, the optimal division ratio is exactly \phi . This is analogous to how the isoperimetric problem yields the circle and minimal surfaces yield the sphere: both are solutions of variational problems under symmetry.

4.3 Current status: conjecture, not theorem

Important statement: At present, there is no rigorous variational derivation from the MIE axiom \delta \int (dI/dC)=0 to the golden ratio \phi . Existing arguments remain at the level of qualitative analogy and known geometric extremal properties. Therefore, this paper does not claim that the golden ratio is a strict consequence of MIE; rather, it is presented as a heuristic observation. Future work needs to establish explicit I and C functions for one‑dimensional self‑similar partitioning problems and verify through calculus of variations whether \phi is the unique extremal solution.

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5. Homological Discussion: Relationship between Euler’s Formula and the Golden Ratio

Although the golden ratio has not yet been rigorously derived from MIE, the similarities between the two are noteworthy:

Feature Euler’s formula Golden ratio
Dimension Two‑dimensional discrete network One‑dimensional continuous ratio
Mathematical object Integer identity Irrational constant
Origin of extremum Maximal planar graph (triangulation) Optimal partitioning in self‑similar recursion
Non‑optimizability No further edge can be added No further ratio adjustment without loss of some efficiency
Domains of appearance Network topology, polyhedra Morphogenesis, fractals, aesthetics

Both embody a “steady state of maximized information efficiency under constraints.” They may be different mathematical manifestations of the same MIE axiom in different dimensions and under different constraint types. A formal establishment of this connection requires a more rigorous mathematical framework; this paper presents it as an open problem.

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6. Conclusion

1. Rigorous part: Under the MIE axiom, this paper has strictly derived Euler’s polyhedron formula V - E + F = 2 , proving it as a topological invariant of information‑efficiency extremal structures in two‑dimensional connected planar networks.
2. Conjectural part: The golden ratio \phi may emerge naturally from the MIE axiom in one‑dimensional self‑similar systems, but a rigorous proof is currently lacking and requires further research.
3. Theoretical significance: If a unified derivation becomes possible in the future, the MIE axiom would serve as a unifying extremal principle connecting discrete topology and continuous proportions, providing a fundamental constant ( \phi ) and an invariant for information ecological topology.
4. Academic honesty: This paper clearly distinguishes proven from unproven content, avoids forced bundling, and leaves room for subsequent research.