Mathematik
Interaktive Visualisierungen mathematischer Konzepte
105 visualizations
Gray Rhino Theory - How to Recognize and Act on the Obvious Dangers We Ignore
Interactive visualization of Michele Wucker's Gray Rhino Theory: a risk management framework for highly probable, predictable threats that tend to be ignored. Features the five-stage crisis response model (Denial, Muddling, Diagnosing, Panic, Action), comparison with Black Swan and Elephant in the Room concepts, real-world case studies (2008 Financial Crisis, Hurricane Katrina, COVID-19, Climate Change), risk formula visualization (Risk = High Probability × High Impact × Ignored/Delayed), and comprehensive prevention strategies. Learn to identify and respond to obvious dangers before they become crises. Multi-language support (zh, en, es, fr, de, ru, pt).
Principle of Least Action - Lagrangian Mechanics
Interactive visualization of the principle of least action and Lagrangian mechanics. Explore how the true physical trajectory is an extremum of the action functional S = ∫ L dt. Features four scenarios (harmonic oscillator, simple pendulum, free fall, brachistochrone problem), adjustable boundary conditions with draggable start/end points, candidate trajectory generation with perturbation analysis, real-time action calculation for each path, animation of trajectory evolution, and comprehensive mathematical explanation covering the Lagrangian L = T - V, variational principle δS = 0, Euler-Lagrange equations, and the universal nature of the principle across physics. Multi-language support (zh, en, es, fr, de, ru, pt).
Innovation-Driven Sustained Growth (Schumpeterian)
Comprehensive teaching visualization of Schumpeterian growth theory: creative destruction, Aghion-Howitt equations, inverted-U competition-innovation relation, dynamic technology replacement simulation, policy interpretation, historical/modern cases, and curated reading list.
Wave / Heat Equation PDE Grid Visualization
Interactive PDE field simulator for heat equation and wave equation on a 2D grid. Adjust alpha or c, Delta t, Delta x, initial disturbance shape (point/line/Gaussian), and boundary conditions (Dirichlet/Neumann/absorbing) to observe diffusion, reflection, and numerical stability.
Cellular Automata - Rule 30 and Rule 110
Interactive 1D cellular automata visualization demonstrating emergence from simple local rules. Adjust rule number (0-255), initial condition (single/random/periodic), and boundary condition (wrap/fixed) to compare structures such as Rule 30 chaos and Rule 110 complex computation-like patterns.
Julia Set - Complex Plane Iterative Fractal with Mandelbrot C-Selector
Interactive Julia set exploration tool for visualizing the famous complex dynamical system z_{n+1} = z_n^2 + c. Features real-time rendering with escape-time algorithm, interactive Mandelbrot set preview for selecting c parameter values, zoom with mouse wheel/touch pinch, pan by dragging, adjustable max iterations (50-2000), escape radius control (2-10), 5 color palettes (rainbow, fire, ocean, psychedelic, grayscale), quality presets for performance, color cycling animation, and 6 famous Julia set presets (circle, dendrite, Douady's rabbit, spiral, Siegel disk, dragon). Educational content covers Julia-Mandelbrot connection, types of Julia sets (connected, disconnected, dendrite), escape-time algorithm explanation, and comprehensive control documentation. Uses optimized Canvas pixel manipulation for performance. Multi-language support (zh, en, es, fr, de, ru, pt).
Linear Sigma Model - Spontaneous Symmetry Breaking
Interactive visualization of the Linear Sigma Model, a toy model for understanding spontaneous symmetry breaking, Goldstone theorem, and Higgs mechanism. Features adjustable field components n (2-4), coupling constant λ, vacuum expectation value v, and temperature for phase transition simulation. Visualizes Mexican hat potential, field configurations, vacuum manifold degeneracy, and particle mass spectrum showing Higgs mode vs Goldstone bosons. Includes O(n) symmetry demonstration, field decomposition into radial (σ) and transverse (π) modes, and gauge coupling explanation. Educational content covers Nambu-Goldstone theory, Higgs mechanism, and applications in particle physics, condensed matter, and cosmology. Uses KaTeX for formula rendering. Multi-language support (zh, en, es, fr, de, ru, pt).
Wilson Loop Lab - Area Law on a Lattice
Interactive Wilson loop visualization on a 2D lattice gauge field. Drag to select a closed loop, adjust coupling and samples, compare U(1) vs SU(2) approximations, and see qualitative area-law behavior tied to confinement intuition. Multi-language support (zh, en, es, fr, de, ru, pt).
Gauge Theory - The Foundation of Modern Physics
Interactive visualization of gauge theory, the mathematical framework underlying the Standard Model of particle physics. Explore U(1) electromagnetic, SU(2) weak, and SU(3) strong gauge symmetries with real-time field visualizations. Features adjustable coupling constants, field strength visualization, test particle simulations demonstrating covariant derivatives, Wilson loop calculations, and comparison of Abelian vs non-Abelian gauge groups. Includes Yang-Mills action, historical development timeline, and Standard Model structure. Uses KaTeX for formula rendering. Multi-language support (zh, en, es, fr, de, ru, pt).
Heart Curve - Romantic Mathematics
Interactive visualization of the heart curve using parametric equations. Explore the mathematical beauty of romance with adjustable scale, color hue, animation speed, and line width. Features gradient fill with pulse effect, sparkle animations at key points, real-time parameter display, and coordinate grid toggle. The curve is defined by x = 16 sin³(t), y = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t). Perfect for Valentine's Day themed presentations and demonstrating parametric equations in education. Multi-language support (zh, en, es, fr, de, ru, pt).
Cycloid & Trochoid - Rolling Circle Curves
Interactive animation of cycloid and trochoid curves generated by a point on a rolling circle. Adjust the radius (r) and distance from center (d) to explore different curve types: cycloid (d = r) with cusps touching the ground, curtate cycloid (d < r) with wave patterns, and prolate cycloid/trochoid (d > r) with loops and self-intersections. Features real-time animation, adjustable speed, grid toggle, and educational explanations of the mathematical formulas x = r(t - sin t), y = r(1 - cos t). Includes applications in gear design, architecture (brachistochrone), and spirograph art. Multi-language support (zh, en, es, fr, de, ru, pt).
Black Swan Theory - Fat Tails, Extremistan, and Anti-fragility
Interactive exploration of Nassim Nicholas Taleb's Black Swan Theory. Four visualization modules: (1) Distribution Comparison - Normal vs Power Law distributions with adjustable σ and α parameters, showing fat tail probability differences; (2) Turkey Problem - 1000-day feeding animation demonstrating inductive reasoning failure and sudden collapse on Thanksgiving; (3) Extremistan vs Mediocrestan - Wealth distribution scatter plots, Pareto 80/20 law, Gini coefficient calculator, and comparison of height (Mediocrestan) vs wealth (Extremistan); (4) Barbell Strategy - 90% safe + 10% risky portfolio visualization with scenario analysis (normal, crisis, boom, black swan), Monte Carlo simulation, and comparison with balanced allocation. Features Canvas-based rendering, smooth animations, interactive controls, mathematical formulas (Normal PDF: f(x) = (1/σ√2π)·e^(-(x-μ)²/2σ²), Power Law: P(X>x) = x^(-α)), real-time probability calculations, multi-language support (zh, en, es, fr, de, ru, pt), and comprehensive educational content on black swan characteristics, anti-fragility, and real-world examples (2008 crisis, COVID-19, internet boom). Understand why history cannot predict black swans and how to build anti-fragile systems