📐 数学

SpirographParametric spirograph patterns — hypotrochoids and epitrochoids with adjustable gear radii. Fractal Explorer热门Mandelbrot and Julia sets rendered with smooth-colouring GLSL shaders. Pan, zoom, switch… Number SpiralsUlam spiral (prime diagonals), Sacks spiral, Fibonacci sunflower and composite highlight… Geodesic DomesInteractive 3D geodesic dome builder. Subdivide icosahedra, octahedra and tetrahedra at… Spherical ProjectionsCompare Mercator, Mollweide, Lambert azimuthal equal-area, sinusoidal and equirectangular… Sierpiński TriangleTwo methods: Chaos Game (random vertex jumps) and recursive subdivision. Hausdorff dimension… 1D Cellular Automata — WolframExplore all 256 Wolfram elementary cellular automaton rules. Watch Rule 30, 90, 110 and 184… KaleidoscopeDraw in one sector and watch it mirror in stunning N-fold symmetry. Choose 3-, 4-, 6-, 8-,… Pythagoras TreeA recursive fractal tree built from the Pythagorean theorem. Adjust branch angle, depth and… Voronoi DiagramInteractive Voronoi tessellation with Lloyd's algorithm for centroidal relaxation. Choose… Barnsley FernInfinite self-similar fern generated by four affine transformations via the Chaos Game… Taylor Series — Polynomial ApproximationNewVisualise how Maclaurin polynomials approximate sin(x), cos(x) and eˣ. Slide N from 1 to 20… Collatz ConjectureFeed any starting number into the 3n+1 rule and trace its hailstone path: peak value,… Riemann IntegralSlice the area under a curve into rectangles and compare left, right, midpoint, trapezoid… Möbius Strip — Topology One-Sided SurfaceTrace a path along a Mobius strip to prove it has one side and one edge, then cut it… Matrix Transformations — 2D Linear Algebra VisualizerType a 2x2 matrix and watch the plane rotate, shear or scale, with the determinant as area… Klein Bottle — Topology of Non-Orientable SurfacesRotate an immersed Klein bottle, follow its self-intersection, and see how two Mobius strips… Torus & Surface Genus — Topology of Orientable SurfacesCompare a sphere, torus and double torus by genus and Euler characteristic, and see how a… Knot Theory — Invariants, Crossings & Reidemeister MovesDeform a trefoil or figure-eight knot with Reidemeister moves and watch crossing number,… Eigenvalues & EigenvectorsInteractive 2D matrix visualizer: watch the unit circle transform into an ellipse,… SVD VisualizerWatch a 2x2 matrix decompose as A=UΣVᵀ: rotate, stretch by singular values, then rotate… Fourier SeriesBuild square, sawtooth and triangle waves from sinusoidal harmonics. Animated epicycles,… Riemann IntegralVisualise Riemann sums converging to the true integral. Compare Left, Right, Midpoint, and… Fourier Series — Interactive Phasor & Harmonic Decomposition SimulatorBuild waveforms from sine harmonics interactively. Watch rotating phasors trace square,… Sieve of Eratosthenes — Animated Prime Number FinderWatch the Sieve of Eratosthenes animate in real time. Cross off multiples of each prime and… Eigenvectors & Eigenvalues — Interactive Linear Algebra VisualizerAdjust 4 matrix entries and instantly see characteristic polynomial, eigenvalues λ₁ λ₂,… Torus & Genus — Topology Euler Characteristic InteractiveGenus 0–3 surfaces (sphere, torus, double-torus, triple-torus). χ = V−E+F. Gauss-Bonnet… Newton's FractalNewton's method applied to z^n − 1 = 0 in the complex plane. Each pixel is coloured by which… Sand Pile ModelBak–Tang–Wiesenfeld self-organised criticality. Drop grains of sand onto a grid; cells with… Complex FunctionsDomain coloring of complex functions. Hue encodes the argument and brightness encodes the… Diffusion-Limited AggregationRelease random walkers that stick where they touch, and watch a fractal dendrite grow; lower… Phase PortraitTune the parameters of a 2D dynamical system and watch trajectories in the vector field… Turing PatternsAdjust feed and kill rates in a reaction-diffusion system and watch spots, stripes and mazes… HarmonographNewSet frequencies, phases and damping for two swinging pendulums and watch their decaying… SVD CompressionNewKeep only the top k singular values of an image and watch how few rank-1 layers are needed… Bézier CurvesNewDrag control points and step the de Casteljau construction to see how a Bezier curve is… Brouwer Fixed Point TheoremNewStir and squash a disc onto itself and see why at least one point always stays put, the… Derivative VisualizerNewSlide the point x0 and shrink the step h to watch a secant line collapse onto the tangent,… DFT & STFT VisualiserGenerate signals and watch their Fourier transforms live. Switch between DFT magnitude… Eigenvalues & Eigenvectors — Linear Transformation VisualiserNewInteractive 2×2 matrix transformation visualiser. Drag matrix entries to see how… Space-Filling CurvesNewWatch Hilbert, Peano and Morton (Z-order) curves fill the plane. Colour by path position to… Ulam SpiralNewLay the integers on a square spiral and mark the primes — striking diagonal patterns emerge.… Fourier EpicyclesNewDraw any closed shape and watch a chain of rotating circles redraw it. The radii,… Mandelbulb 3D FractalNewReal-time 3D Mandelbulb raymarched on the GPU with an honest distance estimator de =… Origami Fold MathematicsNewExplore the mathematics of paper folding — crease patterns, fold sequences and rigid origami… Hypocycloids & Epicycloids — Rolling Circle CurvesNewWatch a circle roll inside or outside a fixed circle and trace roulette curves — deltoid,… Newton's Method Fractal — Basins of AttractionNewEach pixel iterates Newton's method in ℂ toward a root, colored by which root it reaches.… Rose Curves — Rhodonea r = a·cos(kθ)NewDraw rose curves r = a·cos(kθ) in polar coordinates. Set k = n/d, watch the parity rule… 3D Lissajous Curves — Harmonic KnotsNewThree perpendicular oscillators woven into a 3D Lissajous curve. Tune frequencies a:b:c and… Riemann Zeta Function — Critical Strip & ZerosNewVisualise the Riemann zeta function on the complex plane: domain-colour ζ(s), walk the… Continued Fractions — Best Rational ApproximationsNewUnfold any real number into continued fraction [a0; a1, a2,…] and watch convergents p/q… Spirograph — Hypotrochoids & EpitrochoidsNewDraw parametric rolling-circle curves: hypotrochoids trace inside a fixed ring, epitrochoids… Penrose Tiling — Quasi-Periodic PatternsNewGenerate Penrose P3 (thick and thin rhombus) tilings via deflation substitution. These… Eigenvalues — Linear Transformations & Principal AxesNewEdit a 2x2 matrix and watch it transform the unit circle into an ellipse: eigenvectors point… Soap Bubble — Minimal Surface SimulationNewSimulate soap films finding minimal surfaces. Plateau relaxation minimizes surface area.… 3D Cellular Automaton — Survival/Birth RulesNewExplore 3D cellular automata with Moore neighborhood (26 neighbors). Famous rules: Cloud,… Hyperbolic Geometry TilingNewTessellate the Poincaré disk model of hyperbolic plane with {p,q} regular tilings. Möbius… p-adic Number Ultrametric SpaceNewVisualize p-adic numbers as a fractal tree. The p-adic distance |x-y|_p = p^{-v_p(x-y)}… Prime Distribution & GapsNewVisualize prime number distribution: spiral (Ulam), prime gaps histogram, and the Prime… Weierstrass Nowhere-Differentiable FunctionNewThe Weierstrass function W(x) = Σ aⁿ·cos(bⁿπx) is continuous everywhere but differentiable… Buffon's NeedleNewDrop random needles onto a floor ruled with parallel lines and watch pi emerge from the… Dragon CurveNewGenerate the Heighway dragon curve from its recursive L/R turn-sequence doubling rule and… Fibonacci Spiral — Golden Ratio VisualizerNewExplore the Fibonacci spiral, the golden angle, and the golden ratio φ≈1.618034 with an… Ford Circles & Farey SequenceNewEvery reduced fraction p/q between 0 and 1 has a Ford circle tangent to the number line at… Koch SnowflakeNewBuild the Koch snowflake by recursively replacing the middle third of every segment with an… Monte Carlo Pi EstimationNewEstimate the value of pi by scattering random points into a square and counting how many… Pascal's Triangle mod pNewCompute binomial coefficients C(n,k) modulo a small prime and colour each cell by its… Peano Space-Filling CurveNewWatch Giuseppe Peano's 1890 space-filling curve unfold through a recursive 3x3 boustrophedon… Platonic SolidsNewInteractive Platonic solids explorer. Rotate wireframe or solid-shaded tetrahedra, cubes,… Recamán's SequenceNewWatch Recamán's Sequence build itself one term at a time as semicircular arcs above and… Sierpinski CarpetNewRecursively subdivide a square into a 3x3 grid, removing the center square at every level,… Wallpaper Symmetry GroupsNewNine of the 17 mathematically possible wallpaper groups, built by tiling an asymmetric motif…