Riemann Sphere Projection of Graph Cliques

Mapping exponentially scaled graph structures onto the Riemann sphere using stereographic projection

Mathematical Foundation

S: ℂ → S² where S(z) = (2x/(1+|z|²), 2y/(1+|z|²), (|z|²-1)/(1+|z|²))

The Riemann sphere provides a compact representation of the extended complex plane, allowing us to map infinite graph structures onto a finite surface.

Stereographic Projection:

For a point z = x + iy in the complex plane, its projection onto the Riemann sphere (centered at origin with radius 1) is:

X = 2x/(1+x²+y²), Y = 2y/(1+x²+y²), Z = (x²+y²-1)/(1+x²+y²)

Graph Clique Mapping: We first scale our K₄ cliques in the complex plane using vector projections, then apply stereographic projection to map them onto the Riemann sphere.

This creates a compact representation where infinity corresponds to the north pole of the sphere.

K₄ → ℂ → S² via z ↦ (X,Y,Z)

Interactive Visualization

Clique Count: 4

Projection Type: Stereographic Mercator Orthographic

Rotation Speed: 3

Visualization Notes: The Riemann sphere is shown with latitude/longitude lines. Graph cliques are projected onto the sphere surface. Different projection methods preserve different properties of the graph structure.

Projection Methodologies

Stereographic Projection

Preserves angles (conformal mapping) but distorts areas. Ideal for analyzing local graph structure and connectivity patterns.

z → (X,Y,Z)

Mercator Projection

Preserves angles but severely distorts areas near poles. Useful for visualizing large-scale graph patterns.

z → log(z)

Orthographic Projection

Preserves distances along parallels but distorts shapes. Good for 3D visualization of sphere-embedded graphs.

(θ,φ) → (cosφsinθ, sinφ)

Complexity & Applications

Operation	Complexity	Riemann Sphere Advantage
Clique Scaling	O(n²)	Finite representation of infinite scaling
Projection Mapping	O(n)	Compact visualization of large graphs
Distance Calculation	O(1) per pair	Chordal distance on sphere preserves topology
Neighborhood Analysis	O(k log n)	Spherical geometry simplifies nearest-neighbor search

Applications: Riemann sphere projection enables analysis of graph limits, provides intuitive visualization of network growth patterns, and facilitates the study of graph properties at different scales through Möbius transformations.

Mathematical Properties

Möbius Transformation: f(z) = (az + b)/(cz + d)

Möbius transformations correspond to rotations of the Riemann sphere, allowing us to analyze graph properties from different perspectives:

Key Invariants:

Cross-ratio: Preserved under Möbius transformations, useful for analyzing graph connectivity patterns

Chordal Distance: d(z₁,z₂) = 2|z₁-z₂|/√((1+|z₁|²)(1+|z₂|²))

Spherical Metric: ds² = 4|dz|²/(1+|z|²)²

Graph Theoretical Implications: The Riemann sphere projection allows us to study graph limits as n→∞, analyze scale-free properties, and understand the relationship between local and global graph structure through conformal invariance.

Riemann Sphere Graph Projection | Complex Analysis & Network Theory