🌀 3D Rotations with Rotors
Interactive exploration of how geometric algebra rotors provide singularity-free 3D rotations without gimbal lock.
R = cos(θ/2) - sin(θ/2)(n⃗ · B)
- Real-time rotor manipulation
- Gimbal lock comparison
- Smooth interpolation (SLERP)
- Euler angle equivalence
⚡ Electromagnetic Field Visualization
See how electric and magnetic fields unite into a single multivector in geometric algebra, demonstrating Maxwell's equations elegantly.
F = E + I·B, ∇F = J
- Unified field representation
- Lorentz transformations
- Wave propagation
- Field line visualization
🧮 Automatic Differentiation
Experience exact gradient computation using dual numbers, comparing with finite differences to see the precision advantage.
f(a + bε) = f(a) + bf'(a)ε
- Real-time function graphing
- Exact vs. numerical derivatives
- Error analysis visualization
- Neural network gradients
⚛️ Quantum Spin with GA
Visualize quantum spin states and measurements using geometric algebra, making the abstract concrete through Bloch sphere representations.
|ψ⟩ = α|↑⟩ + β|↓⟩, σ = e₁₂
- Bloch sphere visualization
- Spin measurement simulation
- Pauli matrices as bivectors
- Quantum evolution
📈 Verified Optimization
Watch optimization algorithms converge with mathematical guarantees, using exact gradients from dual number automatic differentiation.
x_{n+1} = x_n - α∇f(x_n)
- Multiple optimization functions
- Gradient descent visualization
- Convergence verification
- Exact vs. approximate gradients
🌊 Fluid Vorticity Visualization
Explore fluid dynamics where vorticity becomes a natural bivector quantity, making circulation and helicity intuitive geometric concepts.
ω = ∇ ∧ v, Γ = ∮ v·dl
- Vortex ring simulation
- Circulation visualization
- Magnus effect demonstration
- Flow field interactions