Visual Math Lab

Interactive, browser-based visualizations of numerical methods.

Visual Math Lab

A picture is worth a thousand equations.

Mathematics is full of ideas that come alive the moment you can see them move. Visual Math Lab makes them touchable. It is designed to accompany introductory courses in applied mathematics and numerical methods: explore on your own, use a page as a lecture demo, or hand one to your students as a lab. Everything here is open source, so feel free to fork, adapt, and extend.

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Root Finding
Newton, bisection, secant, and fixed-point iteration for solving \(f(x) = 0\).

Piecewise Interpolation
Numerical integration and interpolation by linear and cubic splines.

Differential Equations
Explicit, implicit, and adaptive solvers for ODEs, and the Butcher tableau behind them.

Math Art
Pictures drawn by mathematics: Sperner’s lemma on colored triangulations, and spirograph curves.

Model Fitting
Least-squares vs. robust regression, and polynomial fits at increasing degree.

Probability
Updating beliefs from new evidence, and smoothing noisy signals over time.

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Spirographs

Rolls one circle inside or outside another and traces the curve drawn by a pen fixed to the rolling circle.

Distribution Relationships

A clickable map of how the standard probability distributions turn into one another.

1D Recursive Filters

Compares a moving average, an exponential moving average, and a Kalman filter for smoothing a noisy 1D signal.

Positive Predictive Value

Shows how a test’s positive predictive value depends on disease prevalence, not just its sensitivity and specificity.

L1 vs. L2 Regression

Compares least-squares (L2) and least-absolute-deviations (L1) regression on data with outliers.

Polynomial Fits

Fits polynomials of increasing degree to noisy data and shows the resulting overfitting.

Adaptive Integration

Refines a numerical integral by subdividing only the sub-intervals where the estimate is still inaccurate.

Adaptive Methods for ODEs

Steps an ODE solver with an adaptive step size that shrinks or grows to hold the local error near a tolerance.

Butcher Tableau Explorer

Explores how a Runge-Kutta method’s Butcher tableau determines its stages and order.

Explicit Methods for ODEs

Compares explicit ODE solvers – Euler, midpoint, and RK4 – on the same initial value problem.

Implicit Methods for ODEs

Compares implicit ODE solvers against explicit ones on a stiff initial value problem.

Numerical Integration

Compares Riemann sums, the trapezoid rule, and Simpson’s rule for approximating a definite integral.

Polynomial Interpolation

Compares piecewise linear and natural cubic spline interpolation through a set of points.

Bisection Method

Finds a root by repeatedly halving a bracketing interval where the function changes sign.

Fixed Point Iteration

Solves x = g(x) by repeatedly applying g, and shows when that iteration converges or diverges.

Newton’s Method

Finds roots by repeatedly following the tangent line to its x-intercept, converging quadratically near a simple root.

Secant Method

Approximates Newton’s method without a derivative, using the secant line through the last two iterates.

Sperner’s Lemma - Geometric Proof

Proves Sperner’s Lemma with a discrete degree argument that counts oriented RGB triangles.

Sperner’s Lemma - Combinatorial Proof

Proves Sperner’s Lemma with the ‘doors’ argument: a path that must exit through an odd number of bicolored edges.

Sperner’s Lemma - Statement

States Sperner’s Lemma on a triangulated triangle and lets you color vertices to test the condition interactively.

Barycentric Coordinates

Introduces barycentric coordinates on a triangle as the setting for Sperner’s Lemma.

Built with

Quarto, Observable JS, Plotly.js, math.js, and mathviz.