Spherical Fibonacci Graphs

• Spherical Fibonacci graphs are defined by placing vertices on a sphere using a golden-angle (Fibonacci spiral) approach and linking points within a chord distance.
• They achieve nearly uniform vertex distribution with a mean degree near 4, incorporating controlled curvature defects reflective of the sphere’s topology.
• These graphs are instrumental in simulations, supporting statistical physics, graph neural network benchmarks, and discretizations in geophysical and machine learning applications.

A spherical Fibonacci graph is a particular class of undirected graph formed by associating vertices to quasi-uniformly distributed points on the 2-sphere, generated using the Fibonacci spiral algorithm, and placing edges between pairs of vertices that are mutually within a prescribed chord (Euclidean) distance. The construction yields a network with exceptional uniformity of vertex degree and spatial distribution, yet necessarily incorporates a small fraction of topological defects arising from the inherent curvature of the sphere. Spherical Fibonacci graphs are widely used as discretizations of the sphere in numerical simulations, particularly for models in statistical physics and machine learning approaches to lattice systems in curved geometries (Song et al., 2021, Zhou et al., 12 Jan 2026).

1. Mathematical Construction of the Spherical Fibonacci Lattice

Consider a sphere of radius $R$. The spherical Fibonacci lattice of size $N$ consists of $N$ points $\{\mathbf{r}_i\}_{i=1}^N$ constructed via an analytic “golden-angle” spiral:

• Let $\varphi = (\sqrt{5} - 1) / 2$ denote the inverse golden ratio.
• The $i$-th point’s coordinates are

$z_i = R \left( \frac{2i-1}{N} - 1 \right)$

$\theta_i = 2\pi i \varphi$

$x_i = \sqrt{R^2 - z_i^2} \cos\theta_i$

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Alternatively, in spherical coordinates, $\alpha_i = \arccos(z_i / R)$ gives the colatitude, and $\theta_i$ is the longitude.

This method yields an “almost perfectly uniform” spiral distribution, sometimes called a Fibonacci spiral, producing local density fluctuations of order $O(N^{-1/2})$. The partitioning of area is nearly equi-areal: each point corresponds asymptotically to a Voronoi cell of area $4\pi R^2 / N$ (Song et al., 2021, Zhou et al., 12 Jan 2026).

2. Graph Construction: Chord-Distance Threshold and Adjacency

The undirected threshold graph, or spherical Fibonacci graph $G=(V,E)$, is formed as follows:

• Vertices: $V = \{1,2,\dots,N\}$.
• Edges: Two distinct vertices $i$ and $j$ are adjacent if the 3D Euclidean (chord) distance

$$d_{ij} = \|\mathbf{r}_i - \mathbf{r}_j\| = \sqrt{(x_i-x_j)^2 + (y_i-y_j)^2 + (z_i-z_j)^2}$$

satisfies $d_{ij} < d_c$, with $d_c$ a fixed cutoff.

• The adjacency matrix $A$ is defined by $A_{ij} = 1$ if $d_{ij} < d_c$ (and $i \neq j$), zero otherwise. The degree of vertex $i$ is $d_i = \sum_{j} A_{ij}$.
• The cutoff $d_c$ is chosen so the mean degree $\bar{z} \approx 4$, maximizing the proportion of four-coordinated vertices (see Section 3).

This prescription allows efficient construction of large, nearly regular graphs on the sphere, scalable to thousands of vertices—substantially surpassing classical arrangements such as icosahedral or dodecahedral partitions (which are limited to $N\leq 20$) (Song et al., 2021).

3. Degree Distribution, Curvature Defects, and Tuning

The statistical structure of spherical Fibonacci graphs is characterized by narrow degree and edge-length distributions:

• Degree Distribution: For $N = 1000$, with $R = 1$, and $r_c \approx 0.11395$, the counts are:
  • 850 vertices of degree 4 ($d_i=4$)
  • 76 with degree 3
  • 74 with degree 5

The average degree is $\bar{z} = 3.998$, and the degree distribution is sharply peaked around 4, with about $85\%$ four-coordinated, and the remainder split between three- and five-fold vertices (Song et al., 2021, Zhou et al., 12 Jan 2026).

• Edge-Length Distribution: For the same example, among the 3998 edges:
  • Average edge length: $1.1455$
  • Standard deviation: $0.0816$
  • Minimum and maximum: $(0.9777, 1.2977)$
  • Median: $1.1194$
• Curvature Defects and Euler Characteristic: The spherical geometry forbids perfectly regular four-coordination everywhere. Curvature is accommodated by “disclinations”: isolated 3-fold (positive charge) and 5-fold (negative charge) vertices, whose total charge sums to +12 (analogous to the twelve 5-fold defects in the 6-coordination setting) (Zhou et al., 12 Jan 2026).
• Tuning $d_c$: Increasing $d_c$ raises the mean degree and introduces vertices of higher coordination. For $N=1000, R=10$, for example:

| $r_c/R$ | $N_3$ | $N_4$ | $N_5$ | $N_6$ | $N_7$ | $N_8$ | $\bar{z}$ | |-------------|-----------|-----------|-----------|-----------|-----------|-----------|------------| | 0.1298 | 76 | 850 | 74 | – | – | – | 3.998 | | 0.1500 | – | 26 | 240 | 734 | – | – | 5.71 | | 0.1700 | – | – | 8 | 722 | 224 | 46 | 6.31 |

4. Scalings, Uniformity, and Topological Properties

• Nearest-Neighbor Scaling: The typical nearest-neighbor spacing scales as $d_{nn} \approx R\sqrt{4\pi/N}$.
• Cutoff Selection: To maintain approximately four neighbors per vertex, set

$d_c = \gamma R \sqrt{4\pi/N}$

with $\gamma \in [0.5, 1.0]$ empirically tuned for the sharpest four-fold degree peak (Zhou et al., 12 Jan 2026).

• Uniformity Properties: The golden-angle spiral arrangement achieves the most uniform known (nonsymmetric) distribution of points on $S^2$ absent explicit symmetry constraints. There are no large voids or high-density clusters, and as $N \to \infty$, the Voronoi areas become increasingly uniform.
• Topological Constraints: The distribution of 3- and 5-fold coordination is dictated by the sphere’s Euler characteristic (genus 0), preventing complete coverage by four-fold coordinated quadrilaterals (Song et al., 2021, Zhou et al., 12 Jan 2026).

5. Applications in Statistical Physics and Computational Science

• Spin Models on Spheres: Spherical Fibonacci graphs support Monte Carlo and machine learning studies of lattice models on curved manifolds. For the XY model and the $q$-state Potts model, they provide a nearly regular discretization for investigating curvature effects, topological defects, and phase transitions in confined geometries (Song et al., 2021, Zhou et al., 12 Jan 2026).
• Graph Neural Networks: Spherical Fibonacci graphs serve as benchmarks for graph-convolutional networks (GCNs) operating on curved, irregular lattices. GCNs trained on regular planar lattices can, by transfer, localize phase transitions in spherical geometries due to the minimal connectivity irregularities (Zhou et al., 12 Jan 2026).
• Other Domains: The structure is relevant for discretizing fields in geophysical and climate simulations, quasi-Monte Carlo integration over the sphere, phyllotaxis and biological modeling, as well as spherical coding and computer graphics sampling (Zhou et al., 12 Jan 2026).

6. Structural Comparisons and Scaling Advantages

• Comparison with Polyhedral Grids: The spherical Fibonacci graph, unlike Platonic solid or geodesic dome constructions, is not constrained by symmetry group size. Consequently, it scales smoothly to arbitrarily large $N$ without clustering or void formation.
• Defects and Homogeneity: The proportion of non-fourfold-coordinated (defect) vertices is $O(1/N)$, vanishing in the large-$N$ limit, ensuring homogeneity required for finite-size scaling, transfer learning, and statistical analyses in computational studies.
• Analytic Construction: The closed-form generation of lattice points enables reproducibility and computational efficiency, bypassing iterative optimization or random sampling.

7. Summary of Defining Properties

A spherical Fibonacci graph $G=(V,E)$ is determined by:

• Vertex set $V=\{\mathbf{r}_i\}$ placed by the Fibonacci spiral on the sphere.
• Edge set $$E = \{\{i,j\} : \|\mathbf{r}_i-\mathbf{r}_j\| < d_c \}$$, with $d_c$ chosen so $\bar z \approx 4$.
• Quasi-uniformity: Nearly uniform spatial sampling and degree, minimal edge-length variability.
• Curvature defects: A controlled, topologically dictated fraction of three- and five-coordinated vertices.
• Scalability: Extensible to large $N$ for numerical and theoretical investigations.

These properties make spherical Fibonacci graphs an essential discretization tool for research at the interface of geometry, statistical mechanics, and data-driven lattice analysis (Song et al., 2021, Zhou et al., 12 Jan 2026).