Explorations

Visualize the relationship between a function and its derivative through synchronized graphs and tangent lines

Explore functions, derivatives, and tangent lines interactively with 8 common functions

Find maximum and minimum values using derivatives

Visualize how changing one quantity affects another through derivatives

Visualize and calculate the area enclosed between two functions

Visualize definite integrals and compare Riemann sum approximation methods

Use random sampling to estimate definite integrals - watch probability converge to calculus

Compare left, right, midpoint, and trapezoidal methods side-by-side on a single function to see which converges fastest

Visualize left, right, midpoint, and trapezoidal approximations for any function on any interval

Interactively explore the formal epsilon-delta definition of limits and understand how epsilon and delta relate

Visualize limits, one-sided limits, and continuity for functions with removable, jump, and infinite discontinuities

Explore how functions behave as they approach specific values and what makes a function continuous

Visualize gradient vectors on a loss surface to understand how gradient descent finds minima

Explore gradient vector fields, contour plots, and gradient descent optimization for multivariable functions

Explore an energy landscape with critical points, showing minima, maxima, and saddle points

Explore how any periodic function can be approximated by sines and cosines

Explore how polynomial approximations converge to common functions

Step through common algorithms with interactive flowcharts showing decision structures, loops, and execution paths

Compare algorithm time complexities and see how different Big-O classes scale with input size

Master algorithm complexity analysis by exploring time complexity, growth rates, and classifying algorithms into Big-O categories

Compare time complexities across different algorithm classes and understand asymptotic behavior

Step through the execution of the Fibonacci function to visualize recursion, understand exponential complexity, and see how memoization optimizes performance

Visualize recursive call patterns, stack depth, and memoization benefits

Step through bubble sort, selection sort, and insertion sort to understand how different sorting algorithms work and compare their efficiency

Compare time complexity and operations for classic sorting algorithms

Explore Boolean algebra, logic gates, and digital circuit design with interactive truth tables and circuit diagrams

Explore finite state machines (FSMs) - fundamental models of computation used in software design, hardware, and digital systems

Visualize how regular expressions are processed by finite automata - step through string matching character by character

Explore regular expression pattern matching, character classes, quantifiers, and the state machines that power regex engines

Explore the classic substitution cipher and understand why simple encryption is vulnerable

Understand how digital signatures provide authentication, integrity, and non-repudiation

Compare encryption methods and understand their tradeoffs in security and performance

Explore hash function properties including collision resistance and preimage resistance

Analyze password entropy, crack time, and security against brute force attacks

Explore BST operations and understand how tree balance affects search performance

Explore binary search trees and understand tree traversal algorithms through interactive visualization

Explore Floyd's cycle detection algorithm (tortoise and hare) to detect cycles in linked lists

Understand load factors, collision probability, and hash table efficiency

Explore linked list data structures and understand common operations like traversal, insertion, and deletion

Analyze SQL query performance, understand table scans vs. index scans, join costs, and optimize database queries

Explore BFS and DFS traversal strategies and their time/space complexity on graphs

Explore Kruskal's and Prim's algorithms for finding minimum spanning trees in weighted graphs

Explore maximum flow problems and the Ford-Fulkerson algorithm in flow networks

Compare Dijkstra's and Bellman-Ford algorithms for finding shortest paths in weighted graphs

Explore topological ordering of directed acyclic graphs (DAGs) with Kahn's algorithm

Explore the TSP, exact vs heuristic solutions, and the complexity of combinatorial optimization

Understand how histogram equalization enhances image contrast by remapping intensity distributions using the cumulative distribution function

Explore how JPEG-style compression works, understand the tradeoff between quality and file size, and learn about DCT quantization

Learn how convolution kernels transform images through blur, sharpen, and edge detection operations

Explore erosion, dilation, opening, and closing operations on binary images using structuring elements

Calculate the maximum data rate of a communication channel using Shannon's channel capacity theorem

Analyze compression ratios, space savings, and theoretical entropy bounds for data compression

Calculate information entropy and understand uncertainty quantification in probability distributions

Analyze Huffman coding efficiency for optimal prefix-free compression codes

Explore how text can be decomposed into n-grams (unigrams, bigrams, trigrams) and analyze their frequency distributions

Explore lexicon-based sentiment analysis and understand how word scores combine to determine the emotional tone of text

Explore text tokenization, the Porter Stemming algorithm, and word frequency statistics used in natural language processing

Explore Term Frequency-Inverse Document Frequency, the foundational text weighting scheme used in information retrieval and text mining

Explore word vector representations, cosine similarity, and vector arithmetic in word embedding spaces

Explore CSPs, backtracking search, and constraint propagation for solving puzzles and scheduling problems

Explore the 0/1 knapsack problem with dynamic programming and greedy approximation

Explore linear optimization with two decision variables, constraints, and the simplex method

Compare pseudo-random number generators (LCG, Mersenne Twister, xorshift) by their period, state space, and statistical quality

Apply statistical tests to evaluate random number generator quality using frequency tests, runs tests, and chi-square analysis

Visualize and compare multiple datasets using box plots and five-number summaries

Explore and visualize data distributions using dot plots and stem-and-leaf plots

Learn to explore data by creating, moving, and analyzing points in real-time

Understand data position with percentiles, quartiles, and the interquartile range

Compare two distributions side-by-side using back-to-back stem-and-leaf plots

Explore how variance and standard deviation measure the spread of data

Learn k-fold, stratified, and leave-one-out cross-validation for robust model evaluation

Understand standardization, normalization, and why feature scaling matters for ML algorithms

Explore how gradient descent finds function minima by iteratively following the steepest descent direction

Learn IQR, z-score, MAD, and isolation forest methods for detecting outliers in data

Visualize how PCA finds directions of maximum variance and projects data onto principal components

Understand how to split data for model training and evaluation, including stratification and random state

Explore the surprising probability that two people share a birthday

Explore probability concepts including union, intersection, complement, conditional probability, and independence using visual representations of events within a sample space

Use probability trees to understand conditional probability and medical test accuracy

Explore the relationship between set theory and probability using Venn diagrams and the Addition Rule

Visualize and understand correlation strength and direction

Understand how least squares regression finds the best-fit line

Visualize binary classification with the sigmoid function, decision boundary, and probability predictions

Compare volumes of 3D shapes - prisms, cylinders, cones, and spheres

Calculate arc length and sector area with real-world examples like pizza slices and clock hands

Discover how radius, diameter, circumference, and area relate to each other

Explore circles, ellipses, parabolas, and hyperbolas through eccentricity, focus-directrix properties, and real-world applications

Discover the relationship between inscribed and central angles in circles

Explore the divide-and-conquer algorithm for finding the closest pair of points in a plane

Explore Delaunay triangulation properties, the empty circumcircle condition, and its dual relationship with Voronoi diagrams

Explore how two line segments intersect using parametric equations and cross products from computational geometry

Explore Voronoi tessellations, nearest-neighbor partitions, and Fortune's sweep line algorithm

Explore how translations, rotations, reflections, and dilations transform shapes in the coordinate plane

Explore the beautiful golden spiral through Fibonacci rectangles and discover how ratios converge to φ

Convert between rectangular and polar coordinates, visualizing the relationship between (x, y) and (r, θ)

Explore vector operations including addition, scalar multiplication, and dot product

Explore complementary, supplementary, vertical, and corresponding angles

See how all polygon areas relate to rectangles

Calculate distance and midpoint between two points using coordinate geometry

Discover why the interior angle sum of any polygon equals (n-2)×180°

Visualize a² + b² = c² with interactive area squares

Explore triangle similarity through AA, SAS, and SSS criteria with interactive ratio relationships

Discover the centroid, incenter, circumcenter, and orthocenter of a triangle

Visualize causal relationships and test structural models

Explore how decision trees make splits using Gini impurity and entropy, visualize decision boundaries, and understand the tradeoff between tree depth and overfitting

Classify observations and reduce dimensions using LDA

Classify points using majority voting from k nearest neighbors

Learn how Naive Bayes uses Bayes theorem with conditional independence to classify data

Explore the perceptron - a single artificial neuron as a linear classifier

Understand how SVMs find optimal decision boundaries using hyperplanes, margins, and support vectors

Build nested cluster hierarchy using agglomerative or divisive methods

Partition data into k clusters using centroid-based algorithm

Learn how gradient boosting builds sequential ensemble models by iteratively fitting to residuals

Understand how random forests combine multiple decision trees through bagging and feature randomization to create robust ensemble predictions

Learn how to evaluate classification models using confusion matrices and key metrics like precision, recall, and F1 score

Learn how ROC curves visualize classifier performance and how AUC measures the probability that a classifier ranks positive examples higher than negative ones

Step through the forward and backward passes of a 2-layer neural network to see how gradients flow and weights update

Understand how model complexity affects underfitting, overfitting, and generalization error

Understand binary and categorical cross-entropy loss functions and their relationship to maximum likelihood estimation

Compare StandardScaler, MinMaxScaler, and RobustScaler to understand how each method transforms data for machine learning

Compare gradient descent variants including SGD, momentum, Adam, and RMSProp for function optimization

Visualize how the learning rate affects gradient descent convergence, oscillation, and divergence on different loss landscapes

Understand the bias-variance tradeoff and how model complexity affects generalization

Understand L1 (Lasso), L2 (Ridge), and Elastic Net regularization to prevent overfitting

Visualize how splitting a dataset into training and test sets affects model evaluation, and explore the tradeoffs between split ratios

Compare different neural network activation functions and understand their properties

Visualize a simple multi-layer neural network with forward propagation

Explore electric fields using field lines to visualize the invisible force around charges

Explore hydrostatic pressure, flow rates, and tank drainage using Torricelli's law and Bernoulli's equation

Explore how a mass-spring system loses energy to friction over time

Explore how potential and kinetic energy transform while total mechanical energy remains constant

Explore forces on an inclined plane and understand equilibrium conditions

See how position, velocity, and acceleration relate through derivatives

Explore how a planet orbits a star following Kepler's laws

Explore synchronized views of a simple harmonic oscillator showing position, velocity, and phase portrait

Explore projectile motion with initial height, velocity components, and optimal launch angles

Explore the fundamentals of simple harmonic motion - amplitude, period, frequency, displacement, and the connection to springs and pendulums

Explore the physics of a simple pendulum, including period, frequency, energy conservation, and the small angle approximation

Explore Young's famous double-slit experiment and discover how light creates interference patterns

Explore how lenses and mirrors form images using ray diagrams

Understand the difference between real and virtual images formed by lenses and mirrors

Explore the wave nature of light through interference and diffraction patterns, from single slits to diffraction gratings

Explore the relationship between frequency, amplitude, and waveform characteristics in audio signals

Explore the relationship between time and frequency domains in signal analysis

Explore how two wave sources create 2D interference patterns through constructive and destructive interference

Explore how two sinusoidal waves combine through constructive and destructive interference, including beat frequency analysis

Explore Analysis of Covariance — comparing group means while controlling for a covariate

Test whether observed frequencies match expected frequencies using the chi-square statistic

Calculate posterior probabilities using Bayes' theorem with prior, likelihood, and evidence

Understand Bayes' theorem through medical testing scenarios and the base rate fallacy

Compare Bayesian and frequentist approaches to statistical inference using Beta-Binomial conjugacy, prior sensitivity, and credible vs confidence intervals

Explore the Box-Cox power transformation to normalize skewed data and understand how lambda affects distribution shape

Compare imputation methods (listwise deletion, mean, median, regression) and understand how missingness patterns affect statistical bias

Assess whether data follows a normal distribution using skewness, kurtosis, and statistical tests

Understand Q-Q plots for assessing normality through quantile comparison

Compute descriptive statistics including mean, median, standard deviation, and quartiles

Test your understanding of core statistical concepts including mean, median, standard deviation, confidence intervals, and hypothesis testing

Explore univariate statistics including measures of central tendency, spread, distribution shape, and outlier detection using IQR, z-score, and modified z-score methods

Calculate and interpret Cohen's d, Pearson's r, and eta-squared effect sizes

Design crossover trials where subjects receive multiple treatments in sequence, controlling for individual differences and carryover effects

Design factorial experiments to study multiple factors simultaneously, estimate main effects and interactions, and optimize experimental efficiency

Explore simple random, systematic, and stratified sampling methods, and see how sample size and method affect coverage and representativeness

Compare randomization strategies (CRD, RCBD, Latin Square) for controlling experimental error and improving precision

Evaluate A/B test results with z-tests for proportions, p-value calculations, minimum detectable effect, and sample size estimation

Perform one-sample and two-sample z-tests and t-tests with step-by-step calculations

Explore nonparametric hypothesis tests including Mann-Whitney U and Wilcoxon signed-rank

Understand what p-values really mean, common misconceptions, and the relationship between statistical and practical significance

Explore permutation (randomization) tests as a non-parametric alternative to t-tests, with no distributional assumptions required

Visualize statistical power through overlapping distributions, understand Type I and II errors, and determine required sample sizes for hypothesis tests

Understand what p-values really mean and explore common statistical misconceptions

Determine required sample sizes for hypothesis tests using power analysis

Visualize the tradeoff between false positives (alpha) and false negatives (beta), understand statistical power, and see how effect size and sample size influence error rates

Explore the Law of Large Numbers through coin flipping, seeing how observed proportions converge to the true probability as sample size grows

Explore dice probability, expected values, and how the Central Limit Theorem makes the sum of multiple dice approach a normal distribution

Understand why past outcomes don't affect future probabilities in independent events, and why the belief that they do is one of the most persistent cognitive biases

Explore Markov chains - stochastic processes where the future depends only on the present, not the past

Explore how random sampling estimates values through the Law of Large Numbers, with applications from estimating pi to option pricing

Explore the famous Monty Hall paradox through simulation, understanding why switching doors doubles your chance of winning

Given data characteristics like mean, standard deviation, skewness, and kurtosis, find which probability distribution best fits your data

Compare and explore common probability distributions, their parameters, properties, and relationships

Explore 1D random walks - how simple coin-flip steps lead to diffusion, Brownian motion, and the gambler's ruin problem

Explore patterns in randomness, test for uniformity, and understand the chi-square test for random distributions

Discover how aggregated data can show the opposite trend from subgroup data, and learn to identify confounding variables

Explore how different probability distributions generate data, compare uniform, normal, exponential, and Poisson distributions, and understand their parameters and properties

Explore Pearson correlation coefficient, understand r values from -1 to 1, and see how outliers affect correlation

Explore Pearson correlation between two variables with interpretation and r² analysis

Understand why extreme values tend toward average on remeasurement, and why this statistical phenomenon is often mistaken for a causal effect

Visualize resampling with replacement and learn how bootstrap builds confidence intervals without distributional assumptions

Detect when statistical properties change using CUSUM — explore the tradeoff between sensitivity and false alarms.

Two non-stationary series can share a stationary linear combination — the basis of pairs trading and error correction.

Measure cross-correlation between two series at different lags to detect lead-lag relationships and coupling strength.

When two series are cointegrated, the ECM describes how each adjusts to restore equilibrium. Explore adjustment speeds, half-lives, and weak exogeneity.

Optimal state estimation from noisy observations — see how the Kalman gain balances model trust vs data trust.

Between stationary and unit root: fractional d creates long memory with hyperbolic ACF decay, Hurst exponent, and spectral poles.

View time series in the frequency domain — how AR and MA processes concentrate energy at different frequencies.

How an input X affects output Y through delay, response shape, and decay. Impulse and step responses for dynamic systems.

Understand the Dickey-Fuller unit root test: non-standard distributions, critical values, test power, and why normal tables don't apply.

Explore how multiple time series interact through a VAR(1) model with impulse response functions, Granger causality, and stationarity analysis

Explore how AR(1) processes revert to the mean, with impulse responses, variance growth, and shock half-lives.

Explore oscillatory behavior in AR(2) processes: stability triangle, complex vs real roots, damped oscillations, and characteristic equations.

Understand ARIMA(p,d,q): how autoregressive, differencing, and moving average components combine for time series modeling.

Identify AR and MA processes from their autocorrelation and partial autocorrelation signatures.

Understand MA(1) processes — finite memory, ACF cutoff, invertibility, and the maximum autocorrelation bound.

Add seasonal components to ARIMA — see how ACF spikes at seasonal lags and how SARIMA captures periodic patterns.

Explore the bias-variance decomposition for time series models: how model complexity affects forecast error through bias, variance, and irreducible noise.

Combining forecasts often beats any individual one. Explore optimal weights, the diversification benefit, and parallels with portfolio theory.

Watch forecast intervals widen with horizon — bounded for AR processes, unbounded for random walks.