A* Pathfinding
Finds the shortest path on a grid using heuristics.
- Category: cs
- Added: 2026-03-18
This page lists all interactive math, science, computer science, and machine learning modules published on Riano. Each module includes an interactive visualization and a theory companion article.
Finds the shortest path on a grid using heuristics.
Compare conversion rate distributions to evaluate statistical significance and power.
Visualize and compare common neural network activation functions.
Visualize transformation consisting of a linear map followed by a translation.
Visualize how integrals are used to calculate the length of curves and the surface area of solids of revolution.
Attention Mechanism
Visualize the encoding and decoding process of data through a bottleneck layer.
Visualize self-balancing binary search trees through balance factors and rotations.
Visualize B-Tree node splitting, key insertion, and tree balancing with customizable degree parameters.
Visualize the gradient flow and weight updates in a neural network.
Visualize the effects of batch normalization on neural network activations.
A fundamental theorem for updating probabilities based on new evidence.
Visualize the shortest path algorithm that handles negative edge weights by relaxing all edges in V-1 iterations.
Visualize how the leading digits of many real-life numerical datasets follow a logarithmic distribution.
Explore the Beta distribution and the effect of its shape parameters.
Parametric curves for computer graphics.
Visualize how model complexity affects the tradeoff between bias (underfitting) and variance (overfitting).
Finding elements in sorted data by halving the search space.
Birthday Paradox
Visualize the spectral radiance of a blackbody across different temperatures using Planck's Law.
Visualize Bloom Filter element insertion, hash function mapping, and false positive probability.
Visualize statistical bootstrapping through resampling with replacement to estimate distributions.
Exploring the inverse relationship between pressure and volume in an ideal gas at constant temperature.
Simulate the random walk of a suspended particle due to microscopic fluid collisions.
Visualize Archimedes' principle, calculating buoyant force and examining floating objects.
Visualize the Carnot cycle with an interactive PV diagram and moving piston engine.
Visualize how every square matrix satisfies its own characteristic equation through vector transformations.
Sample means converge to a normal distribution regardless of the population shape.
Simulate centripetal force, velocity, mass, and radius in uniform circular motion.
Visualizing how vector coordinates transform between different coordinate systems.
Visualize the Chi-Squared goodness-of-fit test by comparing observed and expected frequencies.
Visualize the factorization of a symmetric positive-definite matrix into a lower triangular matrix and its transpose.
Complex Number Visualization
Observe the inelastic scattering of photons by electrons to visualize the particle nature of light.
Visualize a two-stage process and calculate joint probabilities using a tree structure.
Visualize how confidence intervals capture the true population mean.
Adjust predictions and true values to see how accuracy, precision, recall, and F1 score react.
Visualize consistent hashing with virtual nodes for distributed caching and load balancing.
Visualize how contrastive learning pulls similar examples together and pushes dissimilar ones apart in embedding space.
Convex Hull
Visualize how a kernel slides over an input image to produce a feature map via the convolution operation.
Explore how copulas link marginal distributions to form a joint probability distribution.
Visualize the apparent deflection of moving objects in a rotating reference frame.
Learn how covariance and correlation measure relationships between random variables and how they differ.
Simulate and visualize the electrostatic force between two point charges based on distance and charge magnitude.
Visualize the expected number of random draws required to collect all unique coupons in a set.
Visualize Cramer's Rule for solving a 2x2 system of linear equations.
Cross Product
Visualize K-Fold cross-validation, model fitting, and training versus validation error.
Visualize curl and divergence fields in vector calculus, exploring fluid flow and vector field operations.
Visualize how image transformations like rotation, scaling, and noise artificially expand training datasets.
Visualize how model complexity and regularization shape the separation between classes.
A non-parametric supervised learning method for classification.
Visualize how a 2x2 matrix transforms the area of a unit square.
Visualize matrix diagonalization as changing to an eigenbasis to scale coordinates.
Visualizing slope fields and numerical solutions to first-order ordinary differential equations.
Observe the interference pattern created by light passing through single and double slits.
Finding the shortest path in a weighted graph by greedily exploring the nearest unvisited node.
Visualize the relationship between the surface integral of a vector field over a closed surface and the volume integral of its divergence.
Observe how the frequency of a wave changes as the source moves relative to the medium.
Visualizing vector projection and orthogonality.
Simulate the chaotic motion of a double pendulum system.
Visualize multi-variable integration by partitioning domains into sub-regions.
Visualize how randomly disabling neurons during training prevents neural network overfitting.
Dynamic Programming
Discovering the special directions that remain unchanged under linear transformations.
Simulate 1D elastic collisions between two objects.
Visualizing the electric field created by two point charges.
Electromagnetic Waves
Combine multiple weak learners to create a single strong model with lower error and greater generalization.
Visualize Shannon entropy, probability distributions, and information content for binary and ternary systems.
Visualize the formal definition of a limit using epsilon and delta bounds.
Visualizing e^(iθ) on the complex plane and the world's most beautiful equation.
Visualize the exponential distribution PDF and CDF with an adjustable rate parameter.
Visualize the Generalized Extreme Value distribution including Gumbel, Fréchet, and Weibull families.
Visualize electromagnetic induction, magnetic flux, and the resulting electromotive force from a rotating coil.
Visualize SHAP values to explain machine learning model predictions based on feature contributions.
Exploring the Fibonacci sequence and its connection to the golden ratio.
Finite State Machine.
Visualize Fisher Information and the Cramér-Rao bound for normal distribution parameter estimation.
Visualize fixed-point iteration and cobweb plots for finding roots of functions.
Visualize the dynamic programming approach to finding shortest paths between all pairs of vertices in a weighted graph.
Interactive fluid dynamics simulation.
Decompose a complex wave into simple sine waves.
Visualizes the classic gambler's ruin random walk and explores ruin probabilities over time.
Visualize the generator mapping latent noise to a target distribution while the discriminator learns to separate real from fake.
Visualize how a Givens rotation zeroes out specific elements in a vector while preserving its norm.
Visualize the relationship between scalar fields and their gradient vector fields.
Visualize how exploding gradients occur during backpropagation and how gradient clipping mitigates them.
Finding the minimum of a cost function.
Visualize gradient vector fields, equipotential lines, and vector calculus operations.
Gram-Schmidt Process
Visualize how Breadth-First and Depth-First Search algorithms navigate through graph data structures.
Gravity Simulation.
Visualize the relationship between a line integral around a simple closed curve and a double integral over the plane region it encloses.
Simulating spring-mass systems and damping.
Heap & Priority Queue.
Simulate the transfer of heat through a 2D material using the heat equation and finite difference method.
Visualize the relationship between force, spring constant, and displacement in elastic materials.
Visualize how a Householder matrix reflects a vector across a hyperplane.
Visualize character frequencies, tree building, and prefix-free encoding.
Visualize the relationship between significance level, effect size, and statistical power in a two-tailed Z-test.
Implicit Differentiation
Visualize the convergence and divergence of improper integrals with infinite intervals or infinite discontinuities.
Visualize how different inner products change the concepts of length, angle, and orthogonality.
Visualize the convergence of the Jacobi iterative method for solving systems of linear equations.
Visualize how 2D coordinate transformations scale local areas using the Jacobian determinant.
Visualize continuous phase portraits and canonical forms of 2x2 Jordan blocks.
A complex fractal exploring the boundary of bounded orbits for a fixed parameter c.
Visualize the construction of a K-D tree and nearest neighbor search in 2D space.
Partitioning data into clusters by iteratively assigning points and updating centroids.
Estimate time-to-event outcomes and survival probabilities using the Kaplan-Meier estimator.
Visualize non-parametric density estimation using different kernel functions and bandwidths.
Optimize resource allocation by maximizing total value within a given weight capacity constraint.
Transfer knowledge from a large teacher model to a smaller student model using temperature scaling.
Visualize how the Kronecker product combines two matrices to form a larger block matrix.
Evaluate limits of indeterminate forms using derivatives of the numerator and denominator.
Visualize how the gradient of the objective function aligns with the gradient of the constraint function at extreme points.
Laplace Transform
The sample average converges to the expected value as the number of trials increases.
Simulate the continuous exchange of energy between an inductor and a capacitor in an ideal LC circuit.
Visualize how learning rate decay schedules impact optimization convergence and stability.
Find the minimum energy solution to an underdetermined system of linear equations.
Explore how least squares approximation fits a line to data points by minimizing the sum of squared residuals.
Visualize how converging and diverging lenses and mirrors form images using ray tracing and the thin lens equation.
Limits & Continuity
Visualize the calculation of work done by a vector field along various parametric paths.
Fitting a line to data by minimizing the sum of squared residuals.
Visualizing 2D transformations via matrix operations.
Visualize dynamic memory allocation through node insertion, deletion, and traversal.
Visualize complex harmonic motion through parametric equations.
Exploring bifurcation and deterministic chaos.
Visualize the Least Recently Used (LRU) cache eviction policy.
Visualize the internal workings of an LSTM cell and its gating mechanisms.
LU Decomposition
Visualize the magnetic field lines generated by straight current-carrying wires.
A famous fractal generated from a simple equation in the complex plane.
Visualize the Metropolis-Hastings algorithm sampling from complex distributions.
Visualize state transitions, probability distributions, and steady states in stochastic systems.
Visualize the continuous evolution of systems using the matrix exponential.
Visualize how 2x2 matrices transform space and combine through multiplication.
Visualize how matrix norms measure size and condition number affects numerical stability.
Find the parameters of a probability distribution that best explain the observed data.
Visualizing how secant lines relate to tangent lines for continuous and differentiable functions.
Optimize parameters efficiently by estimating gradients on small data batches.
Visualize Kruskal's algorithm finding the minimum spanning tree of an edge-weighted graph.
Visualize how a gating network routes inputs to specialized expert networks to form a combined prediction.
Visualize moment generating functions for different probability distributions and see how they derive moments.
Explore how mass distribution affects rotational inertia and angular acceleration.
Estimating Pi using random points in a circle.
Visualize how the Moore-Penrose pseudoinverse finds the least squares solution for linear systems.
Visualize the multivariable chain rule showing how changes in independent variables propagate to the dependent variable.
Visualize 2D normal distributions with varying correlation and standard deviations.
Interactively explore how a Naive Bayes classifier learns probabilities and makes predictions.
Visualize how the maximum flow in a network is calculated using augmenting paths.
Understanding forward propagation, backpropagation, and weight updates in artificial neural networks.
Compare and visualize various data normalization and scaling techniques used in machine learning.
Visualize the null space, column space, and rank-nullity theorem of a 2x2 transformation matrix.
Visualizing Riemann sums, Trapezoidal, and Simpson's rules to approximate definite integrals.
Explore how light refracts and reflects at the boundary between different media.
Visualize gradient descent path finding minimums on 2D contour loss surfaces.
Simulate planetary orbits using Newtonian gravity.
Visualize the distribution of the k-th smallest value in a sample of size n.
Visualize orthogonal projections of vectors onto subspaces and compute projection matrices.
Parametric Curves.
Visualize 3D surfaces, partial derivatives as slopes, and the resulting tangent planes.
Find the principal axes that maximize variance and project data into a lower-dimensional space.
Simulating nonlinear pendulum dynamics.
Visualize the Perceptron algorithm learning a linear decision boundary.
Experiment with different light wavelengths and target metals to observe the ejection of photoelectrons.
Poisson Process.
Visualize and explore polar and spherical coordinate systems and their transformations to Cartesian coordinates.
Visualize the polarization of electromagnetic waves including linear, circular, and elliptical states.
Visualize positive definite matrices and their quadratic forms.
Visualize statistical power, Type II error (β), and their relationship with effect size, sample size, and significance level.
Visualize the power iteration algorithm converging to the dominant eigenvector.
Visualize how the sequence of partial sums behaves inside and outside the radius of convergence.
Visualize the distribution and underlying order of prime numbers using the Ulam spiral.
Exploring continuous probability density functions and their parameters.
Simulating parabolic trajectories under gravity.
Visualize the decomposition of a matrix into orthogonal and upper triangular components.
Visualize a quantum wave packet tunneling through a potential barrier.
Visualize an M/M/1 queue with adjustable arrival and service rates.
Visualize the rolling hash and matching process of the Rabin-Karp algorithm.
Visualize how integer arrays are sorted digit by digit using counting sort as a subroutine.
Visualize a 1D random walk over time.
Visualize how 3D transformations split space into image and kernel dimensions.
Simulate resistor-capacitor and resistor-inductor circuit responses over time.
Recurrent Neural Network.
Visualize self-balancing binary search trees through nodes and recoloring.
Regression to the Mean
Visualize how regular expressions are parsed and evaluated against text strings.
Visualize an agent learning to navigate a grid world using the Q-Learning algorithm.
Generate samples from a complex target distribution by using a scaled, simpler proposal distribution.
Visualize time dilation and length contraction as a spaceship approaches the speed of light.
Visualize how skip connections help mitigate the vanishing gradient problem in deep networks.
Evaluate complex contour integrals by summing enclosed pole residues.
Visualizing the path of the Riemann Zeta Function in the complex plane.
Visualize True Positives, False Positives, and trade-offs in binary classification.
Visualize finding points where the derivative is zero for a function with equal endpoints.
Visualizes RSA key generation, modular exponentiation, and the flow between public and private key encryption/decryption.
Visualize alpha particle scattering by a heavy nucleus to demonstrate the nuclear model of the atom.
Visualize Schur decomposition as the orthogonal transformation of a matrix into upper triangular form.
Visualize how segment trees enable efficient range queries and point updates in logarithmic time.
Visualize the convergence of sequences and series, exploring conditions for a limit to exist.
Visualize simple harmonic motion of a spring-mass system with adjustable parameters.
Visualize multi-level linked list operations with probabilistic balancing.
Visualizing how data is ordered.
Visualize how symmetric matrices are guaranteed to have real eigenvalues and orthogonal eigenvectors.
Fundamental linear data structures enforcing LIFO and FIFO access patterns.
Standing Waves & Resonance
Visualize the relationship between the surface integral of the curl of a vector field over a surface and the line integral of the field around its boundary.
Visualize suffix sorting, lexicographical ordering, and efficient substring searching using an array of suffix indices.
Visualize how a linear SVM learns a maximum-margin decision boundary using gradient descent on hinge loss.
Visualize the computation of flux for vector fields across parametrized 3D surfaces.
Estimate time-to-event outcomes using the Kaplan-Meier estimator.
Visualize Singular Value Decomposition as a sequence of rotation, scaling, and rotation transformations.
Visualizing the intersection and solution of multiple linear equations.
Reduce high-dimensional data into a low-dimensional map by matching pairwise probability distributions.
Approximating functions with polynomials.
Visualize tensor contractions and transformations on multi-dimensional matrices.
Visualize the relationship between pressure, volume, temperature, and amount of an ideal gas.
Visualize Kahn's algorithm and DFS for finding linear orderings of directed acyclic graphs.
Visualize how applied force, lever arm radius, and application angle create torque and rotational acceleration.
Visualize the core components and connectivity of a Transformer encoder block.
Tree Traversal
Visualize string insertion, exact matching, and prefix search operations within a Trie data structure.
Visualize the difference between pointwise and uniform convergence of function sequences.
Visualize how the Union-Find data structure efficiently tracks and merges disjoint sets.
Visualize the encoding of data into a probabilistic latent space and its reconstruction.
Vector Spaces
Visualizing constructive and destructive interference from two point sources.
Simulate training 2D word vectors using the Skip-gram architecture with Negative Sampling.
Determine linear independence of functions using the Wronskian determinant.