Математика
Интерактивные визуализации математических концепций
105 visualizations
Superellipse (Lamé Curve) - Shape Morphing Visualization
Interactive exploration of the superellipse (Lamé curve) - a geometric shape that generalizes the ellipse using the formula |x/a|ⁿ + |y/b|ⁿ = 1. Discover how parameter n controls shape transformation: n=1 creates diamond, n=2 creates circle/ellipse, n=4 creates the famous iOS squircle, n>2 forms rounded rectangles, n<2 becomes star-shaped. Features real-time canvas rendering with smooth curves, interactive sliders for n (0.5-10), a, and b parameters, special case preset buttons (diamond, circle, squircle), gradient fill with axis intersection markers, coordinate grid with a/b labels, shape label updates, and comprehensive educational content covering Gabriel Lamé's 1818 discovery, mathematical properties, applications in iOS icon design, architecture (Piet Hein's Sergels Torg), typography, and industrial design. Perfect for understanding the mathematics behind modern UI design patterns
Nash Equilibrium Visualizer - Game Theory Interactive Explorer
Comprehensive interactive visualization of Nash equilibrium concepts in game theory. Explore classic games: Prisoner's Dilemma (unique Nash equilibrium), Coordination Game (multiple equilibria), Hawk-Dove Game (mixed strategies), and custom games. Features editable payoff matrices with best response analysis (arrows, color coding), interactive best response dynamics visualization on canvas, mixed strategy analyzer with probability sliders and real-time expected payoff graphs, automated Nash equilibrium finder (pure and mixed strategies for 2x2 games), and extensive educational content including formal mathematical definition uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*), best response concept, Nash existence theorem, real-world applications (economics, biology, politics, computer science), and limitations discussion. Discover why Nash equilibria are stable and how to find them through interactive exploration
Golden Angle Phyllotaxis - Fibonacci Spiral Pattern Visualization
Interactive visualization of phyllotaxis patterns found in nature (sunflowers, pinecones, pineapples). Explore the golden angle (137.5°) arrangement using polar coordinates: θ = n × α, r = c√n. Features adjustable divergence angle (130-145°), scaling factor c, number of points n (100-2000), 7 color schemes (rainbow, nature, sunset, ocean, fire, purple, monochrome), animated point generation, real-time pattern transformation. Discover how 0.1° angle changes create dramatically different spiral patterns, demonstrating the mathematical precision of natural growth patterns and the golden ratio φ = (1+√5)/2
Bayes' Theorem Visualization - Interactive Bayesian Inference
Interactive exploration of Bayes' Theorem - the elegant formula for updating beliefs based on new evidence. Features classic disease detection case (false positive paradox) with adjustable prevalence, sensitivity, and false positive rate; population visualization (10,000 people grid); probability comparison charts; general Bayesian update with interactive prior, likelihood, and evidence sliders; animated belief update process; Venn diagram showing set relationships; and key insights including prior importance, evidence updates beliefs, likelihood ratio power, and iterative updating. Understand how P(H|E) = P(E|H) × P(H) / P(E) transforms prior beliefs into posterior knowledge
Rose Mathematics - Polar Curves Visualization
Interactive exploration of rose curves (rhodonea curves) in polar coordinates. Visualize r = a × cos(k × θ) with adjustable k parameter (0.1-10) for petal count, amplitude a (1-5), animation with automatic θ rotation, polar grid toggle, cartesian coordinates overlay. Features smooth canvas-based rendering with color gradients based on angle θ, petal count formulas (k odd: k petals, k even: 2k petals, rational n/d: complex patterns), preset buttons for common k values, and real-time coordinate display
Mandelbrot Set - Fractal Explorer with Zoom and Pan
Interactive exploration of the most famous fractal - the Mandelbrot set. Features real-time rendering using the iteration formula z_{n+1} = z_n^2 + c, zoom with mouse wheel/touch pinch, pan by dragging, adjustable max iterations (50-2000), and 5 color palettes (rainbow, fire, ocean, psychedelic, grayscale). Explore infinite self-similar patterns and discover the beauty of chaos theory
Bézier Curves Visualization - Interactive de Casteljau Algorithm
Interactive exploration of Bézier curves - fundamental parametric curves in computer graphics. Drag control points to reshape curves, watch the de Casteljau algorithm construction in action with animated parameter t. Features Linear (2 points), Quadratic (3 points), Cubic (4 points), and Higher Order (up to 6 points) curves. Applications in vector graphics (SVG, fonts), animation, CAD/CAM, and game development
Life Variants - Cellular Automaton Rules Comparison and Exploration
Interactive exploration of Conway's Game of Life variants with different cellular automaton rules. Compare multiple rule sets including Conway (B3/S23), HighLife (B36/S23), Day & Night (B3678/S34678), Seeds (B2/S), 2x2 (B36/S125), and Maze (B34/S34). Features single grid view and side-by-side comparison mode, custom rule creation with birth/survival parameters, 2D grid visualization (20-100 cells), adjustable cell size (4-16px), evolution speed control (1-60 gen/s), initial density setting (10-90%), preset patterns (glider, LWSS spaceship, pulsar, HighLife replicator, block, beehive, random), mouse/touch drawing, grid toggle, edge wrapping, color mode, optimized algorithm for large grids, and real-time statistics. Educational content covering rule notation, variant characteristics, and emergent behaviors. Multi-language support (zh, en, de, fr, es, pt, ru). Applications in complex systems research, artificial intelligence, biology, and computer science
Conway's Game of Life - Cellular Automaton Interactive Simulation
Interactive Conway's Game of Life cellular automaton simulation demonstrating emergence and complexity from simple rules. Features 2D grid visualization (20-100 cells), adjustable cell size (5-20px), evolution speed control (1-60 gen/s), initial density setting (10-90%), preset patterns (glider, LWSS spaceship, pulsar, Gosper glider gun, block, beehive, random), mouse/touch drawing and erasing, grid toggle, edge wrapping option, color mode for visual variety, and real-time statistics (generation count, population, density, growth rate). Educational content covering still lifes, oscillators, spaceships, and guns. Multi-language support (zh, en, de, fr, es, pt, ru). Applications in computer science, biology, physics, and philosophy
Fourier Series Approximation - Harmonic Superposition Visualization
Interactive Fourier series approximation visualization demonstrating how periodic functions are approximated by summing harmonics. Features target waveforms (square, sawtooth, triangle, half-wave rectifier), adjustable number of harmonics (1-50) to observe convergence, amplitude spectrum showing Fourier coefficients, individual harmonic visualization, animated approximation progression, real-time error calculation (MSE), and comprehensive educational content on harmonic addition, convergence rates, Gibbs phenomenon, and applications in signal processing, communications, physics, and engineering. Multi-language support (zh, en, de, fr, es, pt, ru)
Fourier Series - Epicycles Visualization
Interactive Fourier Series visualization showing how any periodic function can be represented as a sum of sine functions of different frequencies. Features rotating epicycles (circles) connected head-to-tail, DFT for custom drawn curves, adjustable number of terms, and real-time animation. Explore square waves, sawtooth waves, or draw your own closed curve
Fourier Transform Family - From Continuous to FFT
Complete interactive journey through the Fourier transform family: Continuous FT, Fourier Series, DTFT, DFT, and FFT. Features time-frequency domain visualization, Fourier series approximation, FFT butterfly diagram, DFT vs FFT performance benchmarking, and real-time audio spectrum analysis. Master the mathematics behind modern signal processing, from theory to O(N log N) algorithm