Mathematik
Interaktive Visualisierungen mathematischer Konzepte
105 visualizations
IFS Iterated Function Systems - Classic Fractals Collection
Comprehensive IFS visualization featuring 8 classic fractals: Sierpinski Triangle/Carpet, Koch Snowflake/Curve, Dragon Curve, Levy C Curve, Barnsley Fern, and Cantor Set. Supports both chaos game (random iteration) and deterministic iteration methods with adjustable parameters, multiple color schemes, fractal dimension display, and export functionality. Explore self-similarity and contraction mapping principles
Attractor Basin - Multi-System Visualization
Comprehensive attractor basin visualization showing convergence domains for multiple dynamical systems on the complex plane. Features two system types: (1) Newton Fractals - Five polynomial presets including z^3-1 (3 roots), z^4-1 (4 roots), z^5-1 (5 roots), z^6-1 (6 roots), and z^3-2z+2 (custom roots) using Newton's iteration z_{n+1} = z_n - f(z_n)/f'(z_n). (2) Quadratic Maps - Three complex parameter values exploring Julia set dynamics: c = -0.7+0.3i, c = -0.8+0.156i (period-3), and c = -0.4+0.6i, using iteration z_{n+1} = z_n^2 + c. Three coloring modes: by basin (attractor identity), by iterations (convergence speed), and smooth coloring (gradient interpolation). Five color schemes: rainbow, pastel, neon, thermal, ocean. Adjustable parameters: max iterations (20-500), convergence tolerance, quality presets. Interactive features: zoom with mouse wheel, pan by dragging, trace mode showing convergence trajectories from clicked points, attractor highlighting, and animation. Real-time displays: cursor position, convergence target, render time, and trace information (start point, final attractor, iterations used). Educational content covers historical background (Poincaré, Mandelbrot, Feigenbaum), mathematical principles of basins of attraction, fractal boundary properties, system comparisons (Newton vs quadratic dynamics), applications in numerical analysis, physics, biology, engineering, and art, exploration tips, and comprehensive control guide. Multi-language support (zh, en, es, fr, de, ru, pt).
Complex Exponential Mapping - Periodic Spiral Fractals
Interactive exploration of complex exponential mapping fractals using z_{n+1} = e^{z_n} + c. Features Web Worker-powered rendering for smooth performance, zoom with mouse wheel/touch pinch, pan by dragging, adjustable max iterations (50-500), escape radius (2-20), complex parameter c (real and imaginary parts), 7 color palettes (rainbow, fire, ocean, psychedelic, grayscale, neon, sunset), periodicity markers showing 2π periodic behavior, equipotential lines display, preset views (spiral region, periodic region, origin). Discover the unique spiral structures, periodic patterns, and rapid escape behavior of exponential fractals. Compare with Mandelbrot set and explore Euler's formula e^(iy) = cos y + i sin y in action
Fifth Consumption Era - 第五消费时代 (7S Framework)
Interactive visualization of Atsushi Miura's Fifth Consumption Era theory (2021-2043), exploring the fundamental shift from material accumulation to spiritual fulfillment and well-being in aging societies. Features the comprehensive 7S Framework: (1) Slow - from fast trends to handmade and vintage; (2) Small - 15-minute walkable communities; (3) Sociable - real human connections; (4) Soft - walking and cycling lifestyle; (5) Sustainable - green and long-term responsibility; (6) Sensuous - from stimulation to spiritual massage; (7) Solution - social problem solving over profit. Seven interactive modules: (1) 7S Framework Radar Chart with adjustable weights (0-100) for each dimension, real-time radar visualization, total score calculation, and three preset scenarios (Japan Current, China Trend, Ideal State); (2) Interactive Timeline displaying all five consumption eras (1912-1937 First Era: Middle Class Emergence, 1955-1974 Second Era: Mass Consumption, 1975-2004 Third Era: Individualization, 2005-2020 Fourth Era: Sharing & Minimalism, 2021-2043 Fifth Era: Spiritual Fulfillment), click-to-select detailed era information with characteristics and consumption patterns; (3) Demographics Trends with household type evolution (single vs nuclear family 1990-2040), aging population projections, birth rate decline, interactive year slider for future predictions (2025-2040); (4) Well-being Dashboard featuring multi-dimensional happiness assessment (relationships, social stability, inner peace, self-actualization), interactive bar charts, radar visualization, and comparison with Maslow's hierarchy of needs; (5) 7S Interactive Explorer with expandable cards for each dimension showing real-world examples and business strategy recommendations, covering topics like slow food movement, micro-apartments, community spaces, walkable cities, circular economy, mindfulness practices, and social enterprises; (6) Comparison Analysis Tools with 7S comparison across Fourth vs Fifth eras, consumption pattern evolution (material vs experiential), and value shift over time (material/social/spiritual values from 1980-2030); (7) Educational Content covering Atsushi Miura's background and contributions, definition and characteristics of the Fifth Consumption Era, well-being philosophy, transition from tool-based to instant gratification consumption, lonely society impact, real-world case studies (Tachikowa Green Springs Japan, Pang Dong Lai China, Lying Flat movement), and important disclaimer that this is a sociological framework not a mathematical model. Uses Canvas API for dynamic chart rendering. Responsive design with mobile support. Multi-language support (zh, en, es, fr, de, ru, pt).
Narrative Economics - 叙事经济学 (SIR Model)
Interactive visualization of how economic narratives spread like epidemics using the SIR (Susceptible-Infected-Recovered) model from epidemiology, pioneered by Nobel laureate Robert Shiller. Features the fundamental differential equations: dS/dt = -c·S·I (narrative spreads through contagion), dI/dt = c·S·I - r·I (active spreaders), dR/dt = r·I (people lose interest), where c is contagion rate (how quickly narrative spreads), r is recovery rate (how quickly people stop spreading), and basic reproduction number R₀ = c/r determines epidemic potential (R₀ > 1: narrative grows, R₀ < 1: narrative dies out). Five interactive modules: (1) SIR Simulation with real-time animated curves showing S(t) declining, I(t) bell curve, R(t) rising, adjustable parameters c (0-1), r (0-1), population N (100-10000), initial infected I₀, optional SIRS model for narrative recurrence, and play/pause/reset controls; (2) Phase Plane Analysis with 2D S vs I trajectory visualization, streamlines showing vector field, nullclines (dI/dt=0 vertical line at S=N·r/c, dS/dt=0 horizontal axes), click-to-set initial conditions, multiple trajectory comparison, and current state display; (3) Parameter Exploration with four preset scenarios (Viral Spread: c=0.8, r=0.2, R₀=4.0; Quick Fade: c=0.1, r=0.5, R₀=0.2; Sustained: c=0.3, r=0.3, R₀=1.0; Recurrent Narrative: SIRS with periodic resurgence), side-by-side scenario comparison chart, and one-click scenario application; (4) Real Historical Cases featuring Laffer Curve narrative (1970s-80s supply-side economics), Great Recession narrative (2008-2009 Great Depression comparison), Bitcoin "Get Rich Quick" narrative (2017 crypto boom), each with peak timing, duration estimates, and stylized intensity curves; (5) Educational Content covering mathematical derivation of SIR equations, R₀ interpretation and economic significance, contagion parameter factors (emotional resonance, simplicity, social media amplification, authority endorsements), recovery parameter factors (attention span, competing narratives, counter-evidence, narrative fatigue), social media impact on c (network effects, algorithmic amplification, echo chambers), and policy applications (predicting market bubbles, designing communications, countering harmful narratives). Uses Runge-Kutta 4th order numerical integration for accurate solution of differential equations. Visualizes narrative bell curve characteristic, epidemic threshold behavior, and recurrent patterns. Color coding: blue (Susceptible), red (Infected), green (Recovered). Real-time R₀ calculation with meaning display (High/Low/Critical). Responsive design with touch support for mobile. Multi-language support (zh, en, es, fr, de, ru, pt).
Multibrot Set - 多项式迭代族分形
Interactive exploration of the Multibrot set, a generalization of the Mandelbrot set with the iteration formula z_{n+1} = z_n^p + c where p is a continuously adjustable parameter. When p=2 it becomes the classic Mandelbrot set, p=3 corresponds to the Tricorn set, and other p values create diverse fractal shapes. Real-time fractal rendering with complex power operation z^p = e^{p(ln|z| + i·arg(z))} using the principal branch. Interactive controls: power parameter p (1.5-10 with 0.1 step), zoom (mouse wheel/pinch), pan (drag), adjustable iterations (50-1000), multiple color palettes (rainbow, fire, ocean, psychedelic, grayscale). Quick preset buttons: Mandelbrot (p=2), Tricorn (p=3), Quartic (p=4). Live viewport display showing current power p, center coordinates, and zoom level. Dynamic formula display updates with current p value. Smart viewport scaling adapts to different p values to keep fractals visible. Touch gesture support for mobile devices. Keyboard shortcuts (arrow keys for pan, +/- for zoom, R for reset). Educational content covers Multibrot definition, complex power operation mathematics, characteristics of different powers (integer vs non-integer), mathematical insights on symmetry breaking, rotation symmetry for integer powers, exploration tips for boundary regions, and applications in complex dynamics, fractal geometry, chaos theory, and computer graphics. Multi-language support (zh, en, es, fr, de, ru, pt).
Burning Ship Fractal - 爆裂火焰分形
Interactive exploration of the Burning Ship fractal, a famous variant of the Mandelbrot set discovered in 1992. Features the absolute value iteration formula z_{n+1} = (|Re(z_n)| + i|Im(z_n)|)² + c creating unique flame-like self-similar structures. Real-time fractal rendering using Web Workers for non-blocking computation. Interactive controls: zoom (mouse wheel/pinch), pan (drag), adjustable iterations (100-1000), multiple color palettes (fire, ice, rainbow, ocean, psychedelic, grayscale), and coloring modes (linear, logarithmic, square root). Famous Burning Ship region with coordinates centered at Re: -1.75, Im: -0.03. Quick location presets: main ship body, detail areas, deep structures, and extreme details. Live viewport display showing center coordinates, zoom level, and iteration count. Touch gesture support for mobile devices. Keyboard shortcuts (arrow keys for pan, +/- for zoom, R for reset). Save rendered images as PNG. Educational content covers Burning Ship formula derivation, absolute value transformation effect, comparison with Mandelbrot set, 'origami' folding geometry, exploration tips for the ship bottom region (negative imaginary), applications in complex dynamics, fractal geometry, computer graphics, generative art, and data visualization. Multi-language support (zh, en, es, fr, de, ru, pt).
Replicator Dynamics Visualizer - Evolutionary Game Theory
Interactive visualization of Replicator Dynamics in evolutionary game theory. Explore the fundamental equation ẋᵢ = xᵢ[(Ax)ᵢ - xᵀAx] on the 2D simplex with three strategies. Features: (1) Simplex Visualization - triangular representation of all possible strategy distributions with barycentric coordinates, click-to-set initial conditions, and trajectory trails; (2) Phase Portrait - vector field showing evolution direction with arrow length indicating speed, toggleable display options; (3) Time Evolution - real-time plots of strategy frequencies over time; (4) Interactive Controls - play/pause/reset/clear buttons, adjustable speed and time step (dt); (5) Payoff Matrix Editor - editable 3×3 matrix with real-time updates; (6) Preset Games - Rock-Paper-Scissors (cyclic dominance, center equilibrium), Hawk-Dove (aggression vs passivity), Coordination (multiple pure equilibria), Stag Hunt (safety vs cooperation), and custom mode; (7) Equilibrium Analysis - automatic detection of pure, edge, and interior equilibria, stability classification, Nash equilibrium and ESS markers; (8) Current State Display - real-time frequencies, individual payoffs, and population average. Educational content covers replicator equation derivation, simplex geometry, equilibrium types (pure, mixed, interior), ESS definition, classic game analysis, and applications in biology, economics, social science, and computer science. Multi-language support (zh, en, es, fr, de, ru, pt)
Entropy / KL Divergence
Interactive teaching demo for entropy, cross-entropy, and KL divergence with Bernoulli/Gaussian controls, log-base switching, and Monte Carlo estimation bias exploration.
Markov Chain & Stationary Distribution - 马尔可夫链与平稳分布
Comprehensive interactive visualization of Markov chains and stationary distributions with probability flow animation, convergence analysis, and mathematical properties. Features graph visualization with interactive state nodes (3-5 states), directed edges with transition probabilities, draggable node positioning, and real-time probability flow particles. Transition matrix editor with stochastic validation (rows sum to 1.0), preset matrices (regular chain, absorbing states, periodic chain, random walk, cyclic), and editable probability values. Probability distribution view showing current distribution p_t with animated bars, stationary distribution π overlay, and heatmap coloring on nodes. Convergence visualization with total variation distance chart δ(t) = 0.5 × Σ|p_t(i) - π(i)|, iteration counter, and mixing time estimate. Initial distribution controls (uniform, single state, random, custom) with click-to-concentrate on nodes. Mathematical analysis panel computing stationary distribution via power iteration, eigenvalues of transition matrix P (λ₁=1 guaranteed), chain properties (irreducible, aperiodic, absorbing states, spectral gap), and convergence rate based on second-largest eigenvalue magnitude. Educational content covers Markov property (memoryless), transition formula p_{t+1} = p_tP, stationary equation π = πP, Perron-Frobenius theorem, convergence conditions (regular/periodic/reducible chains), mixing time, spectral gap, and detailed balance. Real-world applications: PageRank algorithm, random walks on graphs, MCMC sampling methods, Hidden Markov Models, queueing theory, and board games analysis. Step-by-step execution with auto-run animation, adjustable speed, and keyboard shortcuts (Space: step, R: run, Esc: reset). Multi-language support (zh, en, es, fr, de, ru, pt).
PCA, Eigenvectors and Covariance Ellipses - PCA、特征向量与协方差椭圆
Comprehensive interactive visualization of Principal Component Analysis (PCA) demonstrating dimensionality reduction through geometric interpretation. Features bivariate Gaussian data generation with adjustable correlation coefficient ρ (-1 to 1), noise level σ (0 to 2), and sample size n (100 to 1000). Covariance matrix computation and eigendecomposition showing eigenvalues (variance) and eigenvectors (principal directions). Visual elements include scatter plot with data points, covariance ellipses at 1σ, 2σ, 3σ levels, eigenvector arrows for PC1 and PC2 with color coding (PC1: green #22c55e, PC2: orange #f97316), mean point indicator, and projected/reconstructed points. Analysis panel displays covariance matrix Σ, eigenvalues λ₁ and λ₂, eigenvectors v₁ and v₂, explained variance ratio bars, and reconstruction error (MSE). Interactive features include preset scenarios (uncorrelated, strong positive/negative correlation, high noise), data centering toggle, projection visualization, dimensionality reduction slider (k components), and view mode switching between original and PC-transformed space. Educational content covers covariance matrix formula Σ = (1/n)XᵀX, eigendecomposition Σv = λv, PCA transformation z = Qᵀ(x-μ), reconstruction x̂ = Q_k z_k + μ, covariance ellipse parametric equation, step-by-step PCA process (centering, covariance computation, eigendecomposition, projection, optional dimensionality reduction and reconstruction), and practical applications in data visualization, feature extraction (Eigenfaces), noise reduction, image compression, anomaly detection, and handling multicollinearity in regression. Uses KaTeX for mathematical formula rendering with formulas for covariance matrix, eigendecomposition, PCA transform, reconstruction, explained variance ratio, and covariance ellipse. Multi-language support (zh, en, es, fr, de, ru, pt).
Wavelet Transform & Multiresolution Analysis - 小波变换与多分辨率分析
Comprehensive interactive visualization of continuous and discrete wavelet transform demonstrating time-frequency localization, multi-resolution analysis, and comparison with FFT. Features four mother wavelets: Haar (piecewise constant), Daubechies 4 (compact support orthogonal), Morlet (complex exponential modulated Gaussian), and Mexican Hat (second derivative of Gaussian). CWT module with formula W(a,b) = (1/sqrt(|a|))*integral(x(t)*psi*((t-b)/a)dt), scale parameter a (1-128), translation parameter b, and real-time wavelet shape preview. DWT decomposition tree showing multi-level structure: approximation coefficients A (low-pass) and detail coefficients D (high-pass) with adjustable decomposition levels (1-5). Scalogram visualization displaying time-frequency heatmap with logarithmic Y-axis (scale/1/frequency) and color-coded coefficient magnitude. Six test signals: sine wave (pure frequency), step function (edge detection), impulse (time localization), chirp (frequency sweep), noisy signal (denoising demo), and custom drawable signal. Interactive features: animate wavelet scan across signal, step-by-step decomposition visualization, coefficient thresholding for compression/denoising, and reconstruction quality metrics (MSE, SNR, compression ratio). FFT vs Wavelet comparison mode showing STFT spectrogram side-by-side with CWT scalogram to demonstrate adaptive multi-resolution vs fixed window trade-offs. Educational content covers Heisenberg uncertainty principle in time-frequency localization, multi-resolution analysis theory, orthogonality and energy preservation, compact support properties, JPEG2000 image compression, signal denoising applications, edge detection, ECG biomedical analysis, and when to use FFT vs wavelets. Uses KaTeX for formula rendering. Multi-language support (zh, en, es, fr, de, ru, pt).