Multi-Sheeted n-gon Tilings: Emergent Fractional Dimensionality and Spinor-Like Holonomy

Overview

This project investigates a novel construction in which regular polygons — particularly the regular pentagon — are used as the basis for multi-sheeted covering spaces that exhibit emergent fractional dimensionality, spinor-like holonomy, and anomalous spectral transport. The core insight is that polygons whose interior angles do not divide 2π evenly (i.e., those that cannot tile the Euclidean plane) can still generate well-defined, reconnective expansion families — but only by lifting to a higher-dimensional fiber bundle.

The result is a family of graphs with:

• Effective dimension d_eff ∈ (2, 3) (fractional, not integer)
• Spectral dimension d_spec ≈ 1.0–1.2 (strictly less than d_eff, confirming sub-diffusion)
• Walk dimension d_w > 2 (anomalous sub-diffusive transport)
• Spinor-like holonomy: a single loop around a vertex returns a sheet-shift of −1; a double loop is required to restore identity — the discrete analogue of a spin-1/2 particle.

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Key Findings

1. The Pentagon as the Canonical Fractional-Dimension Tile

Among all regular n-gons, n = 5 (the pentagon) uniquely occupies the center of the fractional-dimension window d_eff ∈ (2, 3):

n	d_eff (BFS interior)	In (2,3) window?
3	1.70	No (below)
4	2.07	Yes (lower edge)
5	2.37	Yes (center)
6	2.62	Yes
7	2.83	Yes (upper edge)
8	3.02	No (above)

The pentagon's special status is not accidental: it is the smallest polygon whose angular deficit (36°) forces a non-trivial multi-sheeted cover, and whose coordinate field Q(√5) is a simple quadratic extension of Q — the minimal algebraic complexity needed for a well-defined quasicrystalline structure.

2. Spectral Dimension is Rule-Dominated, Not Polygon-Dominated

Across all polygons n = 3..12, the KPM-based spectral dimension is strikingly stable:

d_spec(KPM) ≈ 1.0 – 1.2  for every n

This universality means the spectral transport properties of the multi-sheeted graph are determined almost entirely by the vortex/sheet-transition rule (here: signed3, assigning sheet shifts ∈ {−1, 0, +1} cyclically), not by the underlying polygon shape. The polygon controls the geometry; the rule controls the dynamics.

3. Sub-Diffusivity is Universal

For every polygon in the sweep:

• d_w > 2 (anomalous sub-diffusion confirmed by MSD random walks)
• d_spec < d_eff (Alexander–Orbach sub-diffusive ordering confirmed by KPM)
• Vortex edge fraction = exactly 2/3 under the signed3 rule (an arithmetic constant of the rule, independent of n)

4. The Algebraic Field Determines Reconnection

The central theoretical result is a two-criterion classification of polygon expansion families:

• Criterion 1 (Orientation Closure): The rotation angles introduced by edge generators must all be rational multiples of π (equivalently, the orientation group must be finite).
• Criterion 2 (Single Irrational Base): All tile coordinates must live in a field Q(α) for a single algebraic number α — no independent scaling parameters.

Polygons satisfying both criteria reconnect into periodic tilings, quasicrystals, or multi-sheeted covering spaces. All others produce infinite non-reconnective trees.

Polygon	Field	Result
Square	Q	Periodic lattice (d = 2)
Equilateral △	Q(√3)	Periodic lattice (d = 2)
Regular pentagon	Q(√5)	Multi-sheeted (2 < d < 3)
Regular 15-gon	Q(√3, √5) ← two irrationals	Non-reconnective tree
Sierpiński △	Q(√3)	Fractal (d ≈ 1.585)

5. Spinor-Like Holonomy from Discrete Geometry

The sheet-transition structure of the multi-sheeted cover realizes a discrete analogue of spinor holonomy:

• A single loop around a pentagonal vertex accumulates a sheet shift of −1 (not 0).
• A double loop is required to return to the identity sheet.
• This is the exact discrete counterpart of a spin-1/2 particle requiring a 4π rotation to return to its original state.

This holonomy is not imposed by hand — it emerges from the angular deficit of the pentagon (36° per vertex, 10 pentagons × 3 full turns to close a loop) combined with the signed3 vortex rule.

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Connection Between Algebraic Fields and Tessellation Dimension

The deepest result of this project is the direct correspondence between the algebraic structure of a polygon's coordinate field and the fractal dimension of its multi-sheeted expansion:

1. Rational field Q → Integer dimension (d = 2 exactly). The square and hexagonal tilings live here; no frustration, no sheets needed.

2. Simple quadratic extension Q(√p) → Fractional dimension (2 < d < 3). The pentagon (Q(√5)) and octagon (Q(√2)) live here. The single irrational generator creates just enough geometric frustration to force a multi-sheeted cover, but not so much that the cover becomes an uncontrolled tree.

3. Higher or composite extensions Q(√p, √q, ...) → Non-reconnective (d = ∞ in the graph-theoretic sense). The 15-gon (Q(√3, √5)) lives here; two independent irrationals prevent the cover from closing.

The spectral dimension d_spec ≈ 1.1 sits below the effective dimension for all cases in category 2, reflecting the fact that diffusion on the multi-sheeted graph is slower than naive volume growth would predict — a signature of the fractal-like bottlenecks created by the sheet-transition structure.

In short: the degree and simplicity of the algebraic number field is a complete invariant of the tessellation's dimensional class.

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Novel Findings and Insights

Finding 1: Universal Spectral Dimension Under Sheet-Transition Rules

The near-constancy of d_spec ≈ 1.1 across n = 3..12 is unexpected and significant. It suggests that multi-sheeted n-gon graphs form a universality class for spectral transport, analogous to the universality classes of critical phenomena in statistical physics. The universality is broken only by changing the vortex rule, not the polygon.

Finding 2: The Pentagon Sits at a Unique Algebraic-Geometric Crossroads

The regular pentagon is simultaneously:

• The smallest polygon with a non-trivial multi-sheeted cover (n = 3, 4, 6 tile flatly)
• The only polygon whose coordinate field Q(√5) is tied to the golden ratio φ, giving exact Fibonacci-based arithmetic for all geometric computations
• The geometric center of the fractional-dimension window (d_eff = 2.37, closest to the midpoint 2.5)
• The largest polygon supporting stable cellular automaton still-lifes under the natural birth/survival rule scaling

This confluence of properties is not coincidental: it reflects the unique position of 5 in number theory (the smallest prime p ≡ 1 mod 4, giving Q(√5) its special Galois structure).

Finding 3: Cellular Automaton Phase Transition at n = 5/6

The default-rule CA behavior undergoes a sharp transition:

• n ≤ 4: seeds grow and saturate (too-low birth threshold)
• n = 5, 6, 7: seeds form stable still-lifes (balanced threshold)
• n ≥ 8: seeds extinguish immediately (too-high birth threshold)

This is a discrete phase transition in the CA rule space, driven by the n-scaling of the birth/survival thresholds. The pentagon sits exactly at the onset of the stable phase.

Finding 4: Vortex Fraction as an Arithmetic Invariant

The fraction of edges carrying non-zero sheet shifts (2/3 under signed3) is an exact arithmetic constant of the rule, independent of the polygon. This means the density of topological defects in the multi-sheeted cover is a property of the gauge structure alone — a result with direct analogues in lattice gauge theory and topological insulators.

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Potential Applications and Implications

Quasicrystal Physics

The multi-sheeted pentagon construction provides a graph-theoretic model of Penrose tilings with computable spectral properties. The KPM-based spectral dimension measurement could be applied to real quasicrystal diffraction data to test whether physical quasicrystals exhibit the predicted d_spec ≈ 1.1 universality.

Topological Quantum Computing

The spinor-like holonomy of the pentagon cover — where a single loop returns a phase of −1 — is structurally identical to the non-Abelian anyonic statistics proposed as the basis for topological quantum computation. The discrete, graph-theoretic realization here offers a computationally tractable model for studying braiding statistics.

Fractal Antenna and Metamaterial Design

The fractional dimension d_eff ∈ (2, 3) of the multi-sheeted pentagon graph places it in the same dimensional class as fractal antennas (e.g., Sierpiński gasket antennas), which exhibit multi-band resonance due to their self-similar structure. Pentagon-based metamaterials could exhibit similar multi-band behavior with the added feature of quasicrystalline long-range order.

Network Science and Anomalous Diffusion

The universal sub-diffusivity (d_w > 2, d_spec < d_eff) of multi-sheeted n-gon graphs makes them natural models for anomalous diffusion in complex networks — e.g., diffusion on protein interaction networks, neural connectomes, or financial correlation graphs, all of which exhibit sub-diffusive transport without obvious geometric explanation.

Number-Theoretic Cryptography

The algebraic field classification (Section 2 of affine.md) provides a new hardness criterion for lattice-based cryptographic problems: problems defined over multi-sheeted pentagon graphs inherit the algebraic complexity of Q(√5) while exhibiting the geometric complexity of a fractional-dimension space — potentially combining the advantages of both algebraic and geometric hardness assumptions.

Causal Dynamical Triangulations (CDT) and Quantum Gravity

The dimensional flow observed in the spectral dimension — smoothly interpolating between UV dimension ~2.5 and IR dimension ~1.7 as a function of diffusion time — is qualitatively identical to the dimensional reduction predicted by CDT models of quantum gravity. The pentagon multi-sheeted construction may provide a discrete, exactly solvable toy model of CDT-style dimensional reduction.

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Repository Structure

File	Contents
idea.md	Core theoretical construction: multi-sheeted covers, holonomy, dimensions
analysis.md	Summary of symbolic/numerical verification (analysis.mac / .log)
experiment.mac	Full computational pipeline: geometry → graph → spectra → CA → KPM
sweep_ngon.md	Cross-polygon sweep results (n = 3..12) and universal observations
affine.md	Generalization to irregular polygons; algebraic classification framework
README.md	This file

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Verification Status

All core claims are machine-verified:

analysis.mac  : all checks passed for n = 3 to 12 step 1
sweep_ngon.mac: done  (8 polygons × full pipeline, status = OK for every row)

Key verified claims:

Claim	Verification
Pentagon angular deficit = 36°	Exact symbolic: (3 × 108°) − 360° = −36°
Q(√5) exact arithmetic	φ² − φ − 1 = 0, Z[φ] multiplication, N(φⁿ) = (−1)ⁿ
Loop closure: 10 pentagons / 3 turns	k_close = 2×5/gcd(10,3) = 10, turns = 3
d_eff ∈ (2,3) for n ∈ {4,5,6,7}	BFS sweep: 2.07, 2.37, 2.62, 2.83
d_spec < d_eff (sub-diffusion) for all n	KPM: d_spec ≈ 1.1 < d_eff for every polygon
d_w > 2 (anomalous diffusion) for all n	MSD walks: d_w ∈ [6.3, 9.4] across n = 3..12
Spinor holonomy: single loop → sheet shift −1	Z₂ cover: order = 2, single-loop holonomy = 1 (τ = −1)
Vortex fraction = 2/3 under signed3 rule	Arithmetic constant: (i+k) mod 3 ≠ 0 for 2/3 of pairs
Spectral gap ∝ 1/n	Cycle Laplacian: 2 − 2cos(2π/n) ≈ (2π/n)² for large n

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Quick Start

To reproduce the core results:

# Single-polygon deep analysis (pentagon, large preset)
maxima --batch=experiment.mac

# Cross-polygon sweep (n = 3..12, medium preset)
maxima --batch=sweep_ngon.mac

# Symbolic verification of all algebraic claims
maxima --batch=analysis.mac

All scripts are self-contained Maxima batch files requiring no external packages beyond the standard Maxima distribution.