This is Plateau’s problem You dip the rings into soap, lift them out, and the film quietly organizes itself. No equations are solved on paper, yet the shape that appears is the solution to a Calculus of variations problem: Among all surfaces spanning the same boundary, it picks the one with minimal surface area. #PlateausProblem #CalculusOfVariations #MinimalSurfaces #Catenoid #SoapFilmPhysics #GeometricOptimization

Gradient Descent in Curve-Space Mirrors Reality Running gradient descent on curves (instead of parameters) is not a numerical trick it mirrors how physical systems relax. Reality explores configuration space and discards unstable geometries until a fixed coherent structure remains. This is why equilibrium looks inevitable after the fact. #NMSI #EmergentDynamics #3IATLAS #GeometricOptimization

#FrequencyWaveTheory #SpatialCoherence #WaveInterference #ResonanceHarmonics #DynamicFields #QuantumEntanglement #HolographicPrinciple #TopologicalStability #GeometricOptimization #ScientificVisualization Here is the artistic visualization of the universe through frequency wave theory, depicting the interconnected waves, quantum fluctuations, cosmic inflation, wave-particle duality, dark matter, dark energy, gravitational waves, and consciousness interacting with the cosmic field. The image represents the dynamic equilibrium and interconnectedness of all phenomena with an abstract and scientific aesthetic. @joerogan @terrencehoward

Final product: tiny.cc/flat-earth-society Activate layers on the left one by one by clicking on the circles and interact with the tangent vectors - we had fun interacting with this during the lecture! #VNAV #GeometricOptimization #OptimizationOnManifolds