Modular Field Dynamics
Modular Field Dynamics
What You’re Seeing
This visualization represents Modular Field Dynamics – a mathematical framework that explores how complex systems evolve through modular arithmetic and harmonic oscillations. The patterns you see emerge from the interaction between circular motion and modular constraints.
where k ∈ [0, 360] in steps of 3
Key Mathematical Principles
- Modular Arithmetic: Values wrap around fixed boundaries, creating cyclic patterns
- Harmonic Oscillation: Sine waves generate smooth, periodic motion
- Phase Relationships: Each point has a unique phase offset, creating interference patterns
- Color Dynamics: Hue shifts represent different mathematical states in the field
Physical Interpretations
This model can represent various physical phenomena:
- Quantum Fields: Particle interactions in constrained spaces
- Wave Mechanics: Standing waves and resonance patterns
- Cellular Automata: Discrete systems with emergent complexity
- Crystal Lattices: Periodic structures in material science
Technical Implementation
The visualization uses requestAnimationFrame for smooth 60fps rendering. Each frame calculates:
- Radial positions using trigonometric functions
- Dynamic color mapping through HSL color space
- Real-time alpha blending for depth effects
- Rotating coordinate transformations
Connection to Advanced Mathematics
This simple visualization relates to profound mathematical concepts:
- Group Theory: The rotational symmetry represents cyclic groups
- Fourier Analysis: Harmonic decomposition of complex patterns
- Dynamical Systems: Time evolution of state spaces
- Topology: Continuous deformation of mathematical spaces
Ricci Flow
Pseudo-Vector Time Ricci Flow
dgij/dτ = -2Rij + λΦiΦj
Current Parameters:
λ = 0.8 | φ = π/8 | Mode: 2D
What is Ricci Flow?
Ricci Flow is a geometric evolution equation that describes how Riemannian metrics change over time, smoothing out curvature irregularities. It's the mathematical foundation behind Grigori Perelman's proof of the Poincaré Conjecture.
Pseudo-Vector Time Extension
This visualization extends the classical Ricci Flow by introducing a pseudo-vector time component that incorporates directional dependence into the temporal evolution:
Visualization Interpretation
- 2D Mode: Shows cross-sections of the evolving metric tensor field
- 3D Mode: Represents the full geometric evolution in pseudo-time
- Color Mapping: Hue represents different curvature states
- Point Density: Indicates metric tensor component magnitudes
Physical Significance
This extended Ricci Flow model has implications for:
- Quantum Gravity: Geometric evolution in higher dimensions
- Cosmology: Early universe geometry evolution
- Materials Science: Crystal growth and phase transitions
- Fluid Dynamics: Turbulent flow geometry
Mathematical Context
The pseudo-vector time extension introduces:
- Anisotropic Time: Different rates of evolution in different directions
- Memory Effects: Past geometric states influence future evolution
- Singularity Avoidance: Modified flow can avoid certain singularities
- Emergent Symmetry: Hidden symmetries in the extended equations
Technical Implementation
The visualization computes:
- Real-time Ricci curvature approximations
- Pseudo-vector field evolution
- Coordinate transformations between 2D/3D views
- Dynamic parameter updates based on temporal evolution
