Matemática
Visualizações interativas de conceitos matemáticos
105 visualizations
Gödel's Incompleteness Theorems - 哥德尔不完备性定理
Interactive visualization of Gödel's Incompleteness Theorems - exploring the fundamental limits of formal mathematical systems through axioms, proof trees, and undecidable propositions. Features comprehensive formal system exploration: axiom toolbox with drag-and-drop interface (Zero Axiom: 0 ∈ ℕ, Successor Axiom: ∀n(S(n) ∈ ℕ), Induction Axiom: P(0) ∧ ∀n(P(n) → P(S(n))) → ∀nP(n), No-Cycle Axiom: ∀n(S(n) ≠ 0)), interactive derivation tree visualization using SVG with hierarchical structure showing axioms → intermediate steps → theorems, color-coded nodes (blue: axioms, green: proved theorems, gold: undecidable propositions, red: contradictions, gray: in progress), and smooth step-by-step proof construction animation with play/pause/step/reset controls. Undecidable propositions showcase: Gödel sentence ('This statement cannot be proved'), Goodstein's theorem, Continuum Hypothesis, with interactive explanations demonstrating why these are true but unprovable. Consistency checker allows adding custom axioms with automatic contradiction detection, visual feedback showing system status (consistent/inconsistent), and demonstration of 'ex falso quodlibet' principle when system becomes inconsistent. Proof assistant workspace for building simple proofs using drag-and-drop axioms and inference rules with real-time validation feedback. Four educational tabs: Theory (formal systems, completeness vs consistency, Gödel numbering, first and second incompleteness theorems, implications for mathematics), History (Hilbert's Program, Gödel's 1931 paper, mathematicians' reactions, Church-Turing thesis, modern impact), Examples (Peano Arithmetic, ZFC set theory, true but unprovable statements, computability perspective), and Experiments (build formal system, hunt for undecidability, break the system with contradictions, Gödel sentence explorer). Interactive experiments: start with minimal axioms and build up, attempt to prove undecidable propositions, add contradictory axioms to watch system 'explode', and interactive walkthrough of Gödel's self-reference construction using arithmetization of syntax. Statistics display: theorem count, axioms used, proof depth, undecidable propositions discovered, and system consistency status. View controls: zoom in/out, fit to screen, pan canvas, and export visualization as JSON. Responsive design with dark academic theme, clean mathematical typography, and smooth animations. Perfect for logic foundations courses, computability theory education, mathematical philosophy, and understanding the fundamental limits of formal reasoning and computation. Multi-language support (zh, en, es, fr, de, ru, pt).
Brownian Motion & Random Walk - 布朗运动与随机游走
Comprehensive interactive visualization exploring Brownian motion and random walks from physics to finance, bridging statistical mechanics and mathematical finance. Features multiple simulation modes: Physics mode (standard Brownian motion with Einstein diffusion theory, particle trajectory visualization, ensemble average calculation), Finance mode (geometric Brownian motion for stock price simulation, Black-Scholes option pricing, volatility and drift analysis), and Math mode (simple random walk, Central Limit Theorem demonstration, Wiener process properties). Real-time particle tracking with 1-100 particles, adjustable parameters: diffusion coefficient D (0.1-10), drift rate μ (-2 to 2), time step dt (0.001-0.1), and volatility σ for finance. 2D/3D view switching with interactive rotation for 3D trajectories. Statistical analysis including mean square displacement ⟨x²⟩ with linear regression to verify Einstein relation ⟨x²⟩ = 2Dt, position distribution histogram with theoretical Gaussian overlay, variance σ² calculation, and ensemble average plots. Financial instruments: stock price paths using geometric Brownian motion dS = μS dt + σS dW_t with analytic solution S_t = S_0 exp((μ - σ²/2)t + σW_t), European call/put option pricing via Black-Scholes formula C = S·N(d₁) - K·e^(-rT)·N(d₂), realized volatility tracking, maximum drawdown calculation, and risk-neutral valuation demonstration. Initial conditions: single particle at origin, multiple particles, grid layout, or random distribution. Preset scenarios: Einstein's diffusion verification (1905), stock price simulation, pollen grain observation (Brown 1827), option pricing, and CLT demonstration. Mathematical theory coverage: simple random walk S_{n+1} = S_n + ξ_n, continuum limit to Brownian motion, Fokker-Planck diffusion equation, Itô calculus with (dW_t)² = dt, scaling law x ~ √t, and Wiener process properties (W_0=0, independent increments, continuous paths). Historical context timeline: Robert Brown (1827), Louis Bachelier (1900), Albert Einstein (1905), Jean Perrin (1908, Nobel 1926), Norbert Wiener (1923), Black-Scholes-Merton (1973, Nobel 1997). Educational experiments: verify Einstein relation, observe Gaussian emergence from discrete steps, drift effect analysis, stock market scenarios comparison (bull/bear markets), and Monte Carlo option pricing. Keyboard shortcuts: Space (start/pause), S (step), R (reset), P (pause). Multi-language support (zh, en, es, fr, de, ru, pt).
Tent Map Visualization - Piecewise Linear Chaos Theory
Explore the tent map x_{n+1} = r · min(x_n, 1 - x_n) - a piecewise linear dynamical system exhibiting chaotic behavior, exact Lyapunov exponent λ = ln(r), ergodicity, and topological conjugacy to logistic map at r = 2
Musical Intervals & Frequency Ratios - 音程与频率比
Interactive exploration of musical intervals, frequency ratios, and the harmonic series with real audio playback using Web Audio API. Features comprehensive music theory visualization: (1) Piano Keyboard UI - Interactive keyboard showing all notes with clickable keys, visual highlighting of root note (green) and interval note (red), 2-octave range (C3 to B5), support for selecting any root note and interval up to 2 octaves. (2) Frequency Spectrum Analyzer - Real-time FFT-style display showing fundamental frequencies and harmonics of both notes, color-coded bars (green=root fundamental, red=interval fundamental, orange=harmonics), harmonic series comparison table showing first 8 harmonics with frequency ratios, visual representation of harmonic alignment creating consonance. (3) Waveform Visualization - Three separate canvas displays: root note waveform (single frequency), interval note waveform (single frequency), combined waveform showing interference pattern and beating, real-time animation during playback, multiple wave types supported (sine, triangle, square, sawtooth). (4) Interval Calculator - Automatic frequency ratio calculation (r = f₂/f₁), cent value using formula ¢ = 1200×log₂(r), just intonation vs equal temperament comparison showing difference in cents, interval name identification (unison, octave, perfect fifth, major third, etc.). (5) Interval Presets - One-click access to common intervals: Unison (1:1, 0¢), Octave (2:1, 1200¢), Perfect Fifth (3:2, ~702¢), Perfect Fourth (4:3, ~498¢), Major Third (5:4, ~386¢), Minor Third (6:5, ~316¢), Major Sixth (5:3, ~884¢), Minor Sixth (8:5, ~814¢). Adjustable parameters: root note selection (all 12 chromatic notes), root octave (3, 4, 5), interval in semitones (0-24), wave type selection, volume control. Real-time audio playback using Web Audio API with dual oscillators (one for root, one for interval), independent gain control, master volume, support for all four standard waveforms, 60fps canvas rendering. Mathematical foundation section covering: Frequency Ratio Formula r = f₂/f₁ (ratio of higher to lower frequency), Equal Temperament Formula f = f₀×2^(n/12) (frequency n semitones above f₀, basis of modern Western music), Cents Formula ¢ = 1200×log₂(r) (logarithmic interval measure, 1200¢ = octave), Common Just Intonation Ratios: octave 2:1, perfect fifth 3:2, perfect fourth 4:3, major third 5:4, minor third 6:5, major sixth 5:3, minor sixth 8:5, Harmonic Series f, 2f, 3f, 4f, 5f... (integer multiples explaining instrument timbre and consonance). Comprehensive educational content: What are Musical Intervals? (definition as distance between two pitches, consonance vs dissonance based on frequency ratio simplicity, psychoacoustic perception of intervals), Just Intonation vs Equal Temperament (just intonation uses pure ratios from harmonic series for perfect consonance but makes modulation difficult, equal temperament divides octave into 12 equal semitones enabling key changes but slightly compromising interval purity except octave, historical context and practical trade-offs), The Harmonic Series (fundamental frequency + overtones at integer multiples, explains why certain intervals are consonant - their harmonics align, gives each instrument unique timbre, basis of timbre and sound color), Real-World Applications (Music Theory: intervals are foundation of melody, harmony, and chord construction in all musical traditions, Instrument Design: piano and fretted instrument manufacturers use equal temperament calculations for string/fret placement, Tuning Systems: different cultures/historical periods use various systems - just intonation, meantone, well temperament, Audio Engineering: harmonics understanding helps with EQ, compression, avoiding phase cancellation, Choral Singing: professional singers drift toward just intonation for purer harmonies), and Listening Guide (start with unison and octave - most consonant, waves align perfectly, perfect fifth 3:2 - most stable after octave, foundation of Western harmony, compare major third 5:4 bright/happy vs minor third 6:5 sad/dark - small ratio changes dramatically affect mood). Interactive features: Click keyboard keys to select root note, preset buttons instantly change interval with auto-play, semitone slider for custom intervals, wave type buttons change timbre, stop button terminates audio immediately, real-time updates of all calculations and visualizations. Responsive layout with keyboard section (120px height), controls panel with preset grid and note selection, three waveform displays (80px each), spectrum analyzer (250px height), harmonics comparison table, formula cards grid, and educational sections. Canvas-based rendering with smooth 60fps animations, color-coded keys (green=root, red=interval), accurate piano keyboard geometry with black key positioning, professional music theory appearance. Multi-language support (zh, en, es, fr, de, ru, pt, ja) with complete translations of musical terminology, interval names, formulas, and educational content.
PID Controller Visualization - Interactive Control Theory Tool
Interactive visualization of PID control algorithm with real-time parameter tuning and physical system animation. Explore proportional (P), integral (I), and derivative (D) control components through dynamic visualizations including: (1) Real-time response curve plotting with setpoint tracking, error area visualization, and time-axis scrolling display. (2) Physical system animation showing ball/cart tracking target position on track,直观展示振荡、超调、稳态误差 with velocity indicators and position markers. (3) PID components breakdown chart displaying individual P, I, and D contributions over time. (4) Interactive parameter controls with Kp (0-10), Ki (0-5), Kd (0-5) sliders and real-time value display. (5) Multiple test scenarios: Step response (sudden setpoint change), Disturbance response (external interference), and Sine tracking (dynamic reference following). (6) Real-time metrics: current error, integral term, derivative term, PID output, overshoot percentage, and settling time calculation. (7) Educational content covering PID theory with mathematical formula u(t) = Kp·e(t) + Ki·∫e(t)dt + Kd·de(t)/dt, component explanations (P: responds to current error, I: eliminates steady-state error, D: predicts future error), tuning guidelines (start with Kp, add Ki for steady-state error, add Kd to reduce oscillation), and practical applications (temperature control, motor speed, robotics, drones, cruise control, HVAC systems). (8) System settings: adjustable setpoint (0-100), noise level simulation (0-10), and start/pause/reset controls. PID algorithm implementation: error = setpoint - measured_value, integral += error * dt (with anti-windup clamping), derivative = (error - last_error) / dt, output = Kp * error + Ki * integral + Kd * derivative. Physical simulation using second-order mass-spring-damper system with Euler integration. Canvas-based rendering at 60 FPS with smooth animations, gradient-filled error regions, and responsive design for mobile devices. Multi-language support (zh, en, es, fr, de, ru, pt) with complete translations of theory, controls, metrics, and educational content. Perfect for engineering students studying automatic control principles, embedded developers, robotics enthusiasts, and anyone interested in feedback control systems.
Huygens Clocks - Coupled Pendulum Synchronization
Explore the fascinating synchronization phenomenon discovered by Christiaan Huygens in 1673 - observe how coupled pendulums spontaneously synchronize using the Kuramoto model, with adjustable coupling strength, phase difference heatmap, Fourier spectrum analysis, and real-time visualization of multiple pendulums on a shared beam
Hénon Map - Chaotic Attractor Visualization - 埃农映射
Interactive exploration of the Hénon map, a classic 2D discrete dynamical system exhibiting chaos and fractal structure. Discovered by Michel Hénon in 1976, this simple mapping demonstrates how deterministic nonlinear systems produce complex chaotic behavior. Features: Strange Attractor visualization with density-based coloring showing the iconic banana-shaped attractor, adjustable parameters a (0.8-1.5) and b (0.1-0.4) with classic chaotic values a=1.4, b=0.3, up to 500,000 iterations with real-time rendering, animation mode with parameter oscillation. Fractal Structure Explorer with interactive zoom (drag to pan, scroll to zoom) and auto-zoom animation, multiple color modes (velocity, iteration count, distance), self-similarity demonstration showing Cantor-set-like structure, box-counting dimension ≈1.26. Lyapunov Exponent calculation displaying chaos measure in real-time (λ>0 chaotic, λ≈0 periodic, λ<0 stable), bifurcation diagram visualization across parameter range showing period-doubling route to chaos. System Comparison tab contrasting Hénon map (discrete 2D, difference equations) vs Lorenz attractor (continuous 3D, differential equations) with side-by-side visualizations and feature comparison table. Mathematical Principles section covering iteration equations x_{n+1}=1-ax_n²+y_n, y_{n+1}=bx_n, Jacobian matrix analysis, fixed point calculation, bifurcation sequence (period-2 emerges at a≈1.06, chaos at a≈1.4), applications in chaos theory, cryptography, and signal processing. Educational content on chaos theory, strange attractors, sensitivity to initial conditions (butterfly effect), and fractal geometry. Multi-language support (zh, en, es, fr, de, ru, pt).
Fractal Dimension: Box Counting Method - 分形维数估计：盒计数法
Interactive exploration of fractal dimension estimation using the box counting method. Real-time visualization of covering fractals with grid boxes and calculating dimension from the scaling relationship. Features dimension formula: D = lim(ε→0) [log N(ε) / log(1/ε)], where ε is box size and N(ε) is the number of boxes containing fractal parts. Interactive demonstrations include: (1) Classic fractal gallery - Koch curve (D ≈ 1.262), Sierpinski triangle (D ≈ 1.585), Cantor set (D ≈ 0.631), and fractal trees with adjustable parameters. (2) Grid overlay visualization - dynamic display of different box sizes covering the fractal pattern with real-time box counting. (3) Adjustable box size slider - control ε values from 2 to 100 pixels with immediate grid and N(ε) updates. (4) Log-log plot - live plotting of log(N(ε)) vs log(1/ε) scatter points with automatic linear regression fit showing the slope (dimension estimate). (5) Dimension calculation - real-time computation of log N(ε) / log(1/ε) with convergence process visualization across multiple ε values. (6) Algorithm steps - six-step educational walkthrough: choose box size, overlay grid, count boxes, record point, repeat for different ε, fit line to find slope. (7) Data table - complete tracking of ε, N(ε), log(1/ε), log(N(ε)), and dimension ratios. (8) Practice problems - prediction exercises, manual counting practice mode, and exploration of different generators. (9) Custom drawing mode - upload images or draw simple graphics for dimension analysis. Educational content covers box counting algorithm, log-log linearity relationship, theoretical vs estimated dimension comparison, convergence as ε→0, and practical applications in image analysis, coastline measurement, and natural pattern characterization. Mathematical foundation rendered with KaTeX. Multi-language support (zh, en, es, fr, de, ru, pt).
Law of Cosines - Interactive Triangle Visualization
Interactive visualization of the Law of Cosines with draggable triangle vertices. Features four comprehensive sections: (1) Explore - Drag vertices A, B, C to change triangle shape in real-time. Displays side lengths (a, b, c) and angles (A, B, C) with live measurements. Shows the Law of Cosines formula c² = a² + b² - 2ab·cos(C) with substituted values. Automatically identifies triangle type (acute/right/obtuse) and provides step-by-step verification showing both sides of the equation are equal. (2) Proof - Animated 5-step derivation from Pythagorean Theorem. Step 1: Drop altitude h from B to AC, creating two right triangles. Step 2: Apply Pythagorean theorem to both triangles (h² = a² - x² and h² = c² - (b-x)²). Step 3: Equate expressions and expand (a² - x² = c² - b² + 2bx - x²). Step 4: Solve for x ((a² - c² + b²) / 2b). Step 5: Use cosine definition (x = a·cos(A)) to derive final formula c² = a² + b² - 2ab·cos(C). Each step includes visual diagram updates and mathematical explanations. (3) Examples - Pre-configured triangle demonstrations: Acute (all angles < 90°), Right (one angle = 90°, reduces to Pythagorean theorem), Obtuse (one angle > 90°, cos(C) is negative), SAS case (given two sides and included angle), SSS case (given three sides, find angles). Each example includes detailed explanations of how the Law of Cosines applies. (4) Practice - Two modes: Find Missing Side (given SAS) or Find Missing Angle (given SSS). Random problem generator with valid triangle constraints. Interactive canvas visualization of each problem. Input validation with answer checking. Step-by-step solution reveal feature. Practice history tracking showing recent attempts with correctness indicators. Real-time formula substitution and calculation display. Educational content covers relationship to Pythagorean theorem, behavior in different triangle types, and practical applications. Multi-language support (zh, en, es, fr, de, ru, pt) with comprehensive translations. Responsive design with color-coded measurements, smooth animations, and mobile-friendly touch controls.
Six Thinking Hats - Edward de Bono's Parallel Thinking Method
Interactive visualization of Edward de Bono's Six Thinking Hats parallel thinking method (1985). Features five view modes: (1) Overview - Circular display of all six hats with color-coded symbols and click-to-select interaction. (2) Explore Hats - Larger interactive view with connection lines showing relationships between thinking modes. (3) Sequential Mode - Step-by-step progression through thinking sequences with animation support, progress tracking, and phase indicators. Common sequences: Initial Ideas (Blue→White→Green), Evaluation (Yellow→Black), Development (Blue→Green→Red), Full Process (Blue→White→Red→Black→Yellow→Green→Red→Blue). (4) Practice Mode - Four real-world scenarios (Business Meeting Decision, New Product Launch, Team Conflict Resolution, Strategic Planning) with guided questions for each hat. (5) Comparison View - Grid layout showing all six hats side-by-side with focus areas and key questions. Interactive elements: Click hats to reveal detailed information (focus area, key questions, examples), problem-solving tool for user-specific issues, animate sequence button for automatic progression with 3-second intervals, scenario selector for practice exercises. Hat definitions: White (📊 Facts & Information - What do we know? What do we need to find out?), Red (❤️ Emotions & Feelings - How do I feel about this? What's my gut reaction?), Black (⚠️ Risks & Caution - What could go wrong? What are the risks?), Yellow (☀️ Benefits & Optimism - What are the benefits? Why will this work?), Green (💡 Creativity & Alternatives - What are the alternatives? What's possible?), Blue (🎯 Process & Control - What's our thinking process? Where should we start?). Color scheme: White (#FFFFFF with gray border), Red (#E74C3C), Black (#2C3E50), Yellow (#F1C40F), Green (#27AE60), Blue (#3498DB). Educational content: Edward de Bono biography (1933-2021, Maltese physician, lateral thinking pioneer), parallel thinking vs adversarial thinking comparison, when to use Six Thinking Hats (strategic planning, decision making, problem solving, innovation, meeting facilitation, conflict resolution, team building), eight key benefits, common thinking sequences for different situations, five real-world case studies (NASA space projects - risk assessment and mission planning, Shell Oil strategic planning - 70% meeting time reduction, Singapore education - critical thinking curriculum integration, IBM innovation teams - breakthrough products, medical decision making - complex case discussions), comprehensive comparison table of all six hats with color, focus, and key questions, beginner tips (start/end with Blue Hat, stick to one hat at a time, time each phase, use physical hats, practice simple sequences first, honor Red Hat feelings, balance Black and Yellow, Green Hat needs space without criticism). Canvas-based hat rendering with hover effects, selection highlighting, and smooth animations. KaTeX formula support for traditional vs parallel thinking notation. Responsive three-column layout with side panel for hat details, problem-solving tool, sequence progress tracking, and practice scenarios. Multi-language support (zh, en, es, fr, de, ru, pt) with comprehensive translations of all hat definitions, questions, examples, scenarios, and educational content.
Lateral Thinking - Visual Learning Lab
Interactive teaching page for lateral thinking: digging-well paradox, reverse thinking, extreme assumptions, moderate chaos, quantity-first ideation, and real-world innovation cases.
Two-Sided Markets Economics - 双边市场经济学
Interactive visualization of two-sided market theory and platform economics, founded by Jean-Charles Roché and Jean Tirole (2014 Nobel Prize). Five comprehensive modules: (1) Network Externalities Visualization with interactive bipartite network graph showing Group A (blue nodes) and Group B (orange nodes) users, dynamic connections demonstrating N = n_A × n_B formula, real-time connection count updates, node animation with physics-based movement, add/remove users interactively, and platform value calculation V = α(n_A · n_B)^β. (2) Growth Flywheel Animation illustrating positive feedback loops with four phases (Group A growth → Group B value increase → Group B growth → Group A value increase), logistic growth simulation n(t) = K/(1+e^(-r(t-t₀))), viral coefficient adjustment, critical mass detection and milestone indicators, momentum percentage display, and play/pause/speed controls. (3) Skewed Pricing Simulator exploring asymmetric pricing strategies based on demand elasticity, with preset cases (Taobao: buyers free/merchants pay, Uber/Didi: dual subsidies, Visa/Mastercard: cardholders free/merchants pay discount rate, PlayStation/Xbox: hardware loss/software 30% royalty, Google/Facebook: users free/advertisers pay), interactive price and elasticity sliders, demand function Q = Q₀(1 - ε·P/100), revenue/profit calculations, real-time bar charts showing user distribution and revenue breakdown, and pricing principle: price inversely proportional to elasticity. (4) Platform Type Comparison displaying three platform categories: Marketplace Platforms (facilitate direct transactions - Taobao, Uber, Airbnb, OpenTable), Attention Platforms (content attracts users sold to advertisers - Google, Facebook, TikTok, YouTube), and Ecosystem Platforms (cross-market reinforcement - WeChat, App Store, Steam, PlayStation), with clickable case cards showing detailed modal popups for each platform's structure, pricing strategy, and growth mechanics. (5) Educational Content covering theoretical origins (Roché & Tirole contributions, Nobel Prize background, evolution from bazaars to digital platforms), core concepts (direct vs indirect network externalities, positive vs negative effects, utility function U_i = f(n_j)), pricing strategies (why not cost-based pricing, elasticity determines subsidy direction, subsidy selection criteria), growth mechanisms (chicken-and-egg problem, critical mass importance n_A×n_B≥C, viral growth models, logistic S-curves), and practical case studies with success factors and failure analysis. Mathematical formulas rendered with KaTeX including network connections, utility function, critical mass condition, platform profit Π = P_A·n_A + P_B·n_B - C(n_A,n_B), growth differential equations, and logistic growth formula. Responsive three-column layout (controls, canvas, statistics) with dark mode support. Multi-language support (zh, en, es, fr, de, ru, pt).