Wolfram Mathematica Masterclass: from Beginner to Expert

Embark on a four-hour crash course in Mathematica for beginners, covering functions, lists, plotting, and advanced techniques to support project work.

Explore defining basic functions such as f(x) equals sine of x, generalizing the concept, and building multi-argument, code-driven functions that perform tasks like image changes, with exercises and a quiz.

Define functions in Mathematica with F[x], plot and clear them, and use solve and find root to locate real and complex roots, even restricting to real values.

Explore creating and visualizing multi-argument functions in the Wolfram Mathematica Masterclass, defining a two-variable function, and plotting it in 3D with x and y axes and interactive rotation via dragging.

Explore defining multi-line functions in the Wolfram Language, including end-based sums, function notation vs decompile, and optional compile for performance.

Define built-in and user-defined functions in Mathematica with proper syntax, plot simple functions, and solve roots using solve or the Find Route command, including real and complex options.

Explore how lists in Wolfram Mathematica store different elements like numbers, words, and characters, how to address and modify them, and how lists form vectors and matrices.

Master list operations in Mathematica: define lists with curly braces and Table, select elements by index, and modify lists with join, append to, and drop.

Explore how three-component lists act as vectors and matrices in Wolfram Mathematica, performing vector calculus, dot and cross products, and matrix operations such as determinant, inverse, and multiplication.

Learn how to import and export data in Mathematica, manipulate and flatten data, modify elements, and save updated data. Explore basic statistics and visualize with list plots and bar charts.

Learn nest and nest list to apply a function repeatedly, view intermediate steps, and handle numeric evaluation with slots; then build polynomials efficiently using power lists.

Explore working with lists in Wolfram Mathematica by creating, indexing, joining, and dropping lists and arrays, manipulating vectors and matrices, and applying flatten, import/export, statistics, and plotting.

Master plotting and visualization in Mathematica by modifying plots, adding options, and visualizing listed data; create three-dimensional plots with the graphics command and use animate and manipulate for interactive plots.

Explore how to create and customize 2d plots in Wolfram Mathematica, including setting plot ranges, styling with frames and labels, applying color functions, and exporting figures as pdf.

Learn to visualize data points with list plots in Wolfram Mathematica, using frame, aspect ratio, and plot style, then connect points with list line plots and handle two-dimensional coordinates.

Explore creating and refining three-dimensional plots in Mathematica, using plot3D, contour and density plots, with axis labels, viewpoints, colors, opacities, and performance-aware point options.

Explore graphics options in Wolfram Mathematica by creating shapes such as rectangles, disks, circles, polygons, and arrows, and learn to combine graphics with plots using show.

Explore parametric plots in Wolfram Mathematica, drawing circles with x=cos(phi)*r and y=sin(phi)*r, extending to 3D with a z component, and enhancing visuals with color, opacity, and the Manipulate function.

Master the manipulate function to create interactive plots and gif animations, exploring a sine-based function, animating with sliders, fixing plot ranges, and exporting frame-packed gifs.

Explore advanced plotting techniques in Wolfram Mathematica, including 2D and 3D plots, color functions, filling between curves, graphics objects, parametric plots, and animated visualizations with manipulate and export GIFs.

Build on core concepts by adding interactive dynamic plots, advanced list techniques, and personalization, then briefly cover riveters and integrals in this masterclass section.

Explore dynamic module concepts to build interactive graphics that respond to mouse clicks, updating a points list and lines or polygons in real time.

Master map commands in Wolfram Mathematica by applying scalar and vector functions to lists and images, using map, table, and the slash at notation.

Master pattern matching in Wolfram Mathematica by using blank arguments, underscores, and patterns in cases and replace list. Learn to extract and transform expressions with named blanks.

Explore associations and their relation to lists by mapping keys to values with the arrow syntax. Apply functions to numerical values while preserving keys and convert between lists and associations.

Explore how Mathematica uses entities to store and retrieve data, featuring planets and cities, interactive entity lists, and properties like diameter, population, and images.

Learn to speed up independent operations in Mathematica by parallelization across multiple kernels, using parallel map, parallel table, and parallel array operations.

Learn to use Mathematica to compute derivatives, gradients, curls, and integrals for single- and multi-variable functions, including areas under curves and parameterized expressions.

Explore limits and series in Mathematica, learning how to compute limits of singular functions like sgn(x)/x, test convergence of p-series, harmonic numbers, and the zeta function.

Simulate random walks from one-dimensional to three-dimensional steps and analyze mean, variance, and the binomial distribution in Mathematica; explore stock market data with moving averages, candlestick charts, and daily returns.

Analyze a one-dimensional random walk with steps of length one, using mean, variance, and standard deviation; relate end positions to a binomial distribution and extend to higher dimensions.

Simulate a one-dimensional random walk with unit steps, moving left or right at random from zero, and visualize 1000 steps to characterize fluctuations and zero crossings.

Simulate random walks by generating random integers, using accumulate to sum steps, and visualize many simulations with plots, grids, and tables to quantify randomness.

Simulate ten thousand random walk end positions, then analyze their distribution with mean, standard deviation, and the binomial distribution's pdf to reveal zero-centered variability.

Develop and analyze two- and three-dimensional random walks in Wolfram Mathematica, visualize paths with color coding, compare multiple runs, and compute the expectation value for the x and y coordinates.

Learn to work with financial data in Mathematica, load stock prices (S&P 500), compare stocks with plots and moving averages, use candlestick diagrams, statistics such as mean and variance.

Learn to access stock data with the financial data command, plot time series and date list plots for multiple stocks, and compare performance using cumulative fractional change and logarithmic scales.

Analyze stock data in Mathematica by computing moving averages, building candlestick charts, calculating daily returns, statistics like mean and standard deviation, and comparing Tesla with the S&P 500.

Learn how to import Bitcoin price data from CoinMarketCap into Mathematica, using web sessions and table extraction to build a time series and plot historical closing prices.

Explore geography, chemistry, and biology projects in Mathematica by working with entities, maps, and data visualizations, including shortest paths, museum and park data, and interactive DNA plots.

Explore geographical data as entities in Mathematica, building interactive maps that show day and night, compute shortest paths, and list 20 largest U.S. national parks with their areas and locations.

Color each European country by its national flag on a geographic map, load country boundaries and flag images, and apply geo styling to create a satellite-backed, flag-colored map.

Learn to map Paris and the 20 most important museums, plot them on street or satellite maps, and plan a shortest-route trip using geospatial data in Mathematica.

Explore how to use Wolfram Mathematica to discover Paris museums within a 10 km radius, access museum entities and geo data, and run Find Shortest Tour to plot optimized route.

Identify the 50 largest United States cities, map them, and compute the shortest possible road trip using city entities and each city's GOP position.

Apply Wolfram Mathematica to determine the shortest tour across U.S. cities and national parks, plot locations, order the route, and visualize the connecting path.

Learn to load India’s age distribution from Wolfram Alpha, extract three age-group totals, and generate a color-coded pie chart for India, China, and Japan with flag imagery and a legend.

Create a world day and night map by shading day and night on a background world map, animated with sliders for June 21 to show sun position and twilight.

Learn to plot a global map with country borders in Mathematica, compute the sun altitude at each location, and overlay a day–night contour with twilight boundaries using dynamic manipulation.

Explore chemistry data in Mathematica by analyzing chemical entities, generating 3D plots of structures, and comparing properties like molar volume and boiling point using tables and charts.

Explore chemical data in Mathematica, accessing properties, formulas, and alternate names for more than 43000 compounds. Filter by class, search by strings, and visualize molecules with three dimensional plots.

Access chemical data to retrieve the first 28 alcohols and plot their molecular representations in a grid. Explore map and molecule plot approaches, and add names or formulas for presentation.

Search the chemical database for strings by listing all chemical names, then filter with select and string match to find names that contain a target term, handling case sensitivity.

Explore chemical data in Wolfram Mathematica by accessing properties, filtering nitrogen compounds, mapping properties into a table of formula, molar mass, melting and boiling points, and plotting data.

Plot chemical data in Wolfram Mathematica by extracting boiling points and vaporization heat for liquids, then analyze linear relationships and molar volumes across gases using list plots and linear fits.

Analyze the molar volume of gases using a chemical data table, bean plots, and binning in Mathematica to reveal surprising variation among gases.

Explore biology through interactive Mathematica plots using entities, from ranking animals by speed with hover data to anatomy visuals and a DNA double-helix visualization colored by nucleotides.

Compare animal maximum speeds by retrieving species data with entity lists, extracting maximum speed, and visualizing velocities in kilometers per hour as interactive plots with images.

Explore anatomy plots in Wolfram Mathematica to visualize heart anatomy, view regional structures from multiple viewpoints, and create interactive 3D plots with X-ray comparisons for medical learners.

Explore visualizing DNA in Mathematica by loading a DNA sequence, computing codons, and building a 3D DNA visualization with two spirals and colored nucleotides.

Learn how interpolation and neural networks solve problems by minimizing error, then train a neural network in Mathematica to recognize handwritten digits, from scratch to a pre-trained 98 percent accuracy.

Explore linear interpolation by fitting five data points, minimize the squared error over all points with gradient descent, and see how Mathematica speeds it with a single line of code.

Explore interpolation through fitting data with a linear model and gradient descent, comparing manual fits to automated optimization using error minimization and contour plots.

Explore easy interpolation with mathematica by comparing manual gradient descent to built-in gradient and fit functions, and learn how to quickly obtain accurate polynomial fits.

Discover how neural networks extend linear fits to non-linear problems by training weights and biases with hidden layers and sigmoid activations to minimize error, using Mathematica.

Train neural networks in Mathematica to recognize handwritten digits, compare a self-built model with a pre-trained 11-layer network, and deploy a life number recognition tool.

Load a standard handwritten digits dataset from the Mathematica database for a neural network, split into 60000 training and 10000 test samples, then flatten 28x28 images into 784-length input vectors.

Learn to use Mathematica's classify command to train a number recognizer on a large sample, test accuracy, and then contrast with programming a neural network from scratch.

Explore building a neural network from scratch with an input layer, a hidden layer, and an output layer for digits, using random weights, biases, and a sigmoid activation.

Compute the neural network error by squaring the difference between outputs and true labels for all 10 digits, then apply gradient descent with a learning rate to update the weights.

Explore gradient descent and backpropagation in neural networks, derive and verify weight updates, and learn how learning rate and derivatives shape training and error reduction.

Reinitialize weights across a wider range, adjust the learning rate, and use a results function with input to predict digits, then measure accuracy on 10,000 cases using cronica delta function.

Learn how to train a neural network in Wolfram Mathematica from scratch, tune steps and weights, track error and accuracy, and improve with larger hidden layers.

Use the classify function in Mathematica to generate a neural network from training data, apply it to the test set for handwritten digit recognition, and compare accuracy to improve results.

Build an interactive dynamic module in Wolfram Mathematica to recognize handwritten digits from a 28×28 pixel image using a trained neural network and real-time predictions.

Apply existing pre-trained neural networks to new problems by retraining only the final layers and adjusting output dimensions, such as distinguishing normal versus mirrored digits.

Evaluate a modified neural network's ability to distinguish normal and mirrored handwritten digits using a 14,000-image test set. Explore excluding certain digits and axis flips to test robustness.

Explore supervised and unsupervised machine learning by using interpolation-based neural networks on projects like predicting Boston housing prices from 13 criteria, wine quality, anomaly detection, and image feature maps.

Explore supervised learning to predict house prices from multiple criteria and compare to predicted values, then examine neural networks on image data with training sets and pre-trained networks for insights.

Explore how supervised learning uses training and test data to train a predictive function in Mathematica, illustrated by predicting Boston house prices.

Train a predictive function on the wine quality dataset in Mathematica, using 11 input features to predict wine quality on test data and analyze fit quality and possible reasons.

Train a predictive function on wine data to predict quality, compare predicted versus actual values, and conclude that data quality and input features like acidity and sugar shape poor fit.

Compare predictive functions for house prices using gradient boosted trees, linear regression, neural networks, random forests, and Gorshin process; gradient boosted trees show the best fit.

Train a neural network on the 1000 most common female given names to generate new, plausible names. Observe the decreasing loss function and the generated names like Adrianne.

Explore unsupervised learning in a Wolfram Mathematica masterclass project by generating a feature space that reveals image relationships and detects anomalies without labels.

Apply a feature extract function to all images and visualize similarity with feature space plots to reveal how images cluster by inside versus outside without labels.

Apply a pre-trained Wolfram neural network to map images to the most prominent feature, demonstrating supervised learning and how outputs like lake, airport terminal, or giraffe label pictures.

In unsupervised learning, apply anomaly detection to the amnesty data, tune the acceptance threshold to filter anomalies, then train and test the network on the filtered dataset to improve accuracy.

Explore image processing in Mathematica through art manipulation, colorizing grayscale images with neural networks stored in Mathematica, face detection, and crystallography imaging, fractals, and an optical illusion animation.

Merge portraits with paintings to explore art styles and create style-blended images, like Mona Lisa in Starry Night, and recolor grayscale images with a trained neural network in Mathematica.

Explore mixing painting styles by applying Kandinsky and van Gogh to the Mona Lisa, adjust style weights, view a grid of results, and upload an image to create blended artworks.

Color black-white image: learn how a pre-trained neural network colorizes black-and-white photos by predicting the a and b channels of the lab color space, producing vivid yet subtle results.

Apply an alternative colorization network trained on places data to images and compare results to a network trained on image net data, revealing how training data shapes colorization.

Use Mathematica neural networks to detect faces and coordinates of eyes and mouth, align positions, and compare twin faces. Train a network to estimate age and classify gender.

Tweak an existing neural network to detect age from faces, using a 24,000-image dataset with age labels in Mathematica, with random 1,000-sample training and a train-test split.

adapt a pre-trained neural network to estimate ages from images by replacing the last layers with an 111-element output and a softmax layer, then train on the age dataset.

Train the network to determine age from facial images by evaluating output probabilities, using a weighted sum to predict age and discussing loss function improvements.

Modify the detection network to predict gender instead of age. Shrink the output to a binary gender value, assess accuracy, and try the exercise.

Modify the network to determine gender by switching targets to one and two. Train on eight thousand examples, and evaluate predictions with confidence probabilities.

Explore Mathematica's face detection with the find faces command, adjust thresholds, and visualize faces; then use facial features to estimate eye and mouth positions and infer age and gender.

Discover how to extract eye and mouth coordinates in Mathematica, overlay them on faces, compute landmarks for each feature using the mean, and align two faces for blending.

Blend two aligned face images by mapping landmarks for the eyes and mouth to a common reference, apply a geometric transformation, and interpolate between faces to assess similarity.

Learn to apply Mathematica to image processing in scientific contexts, extracting pixel coordinates of atom dots in a perovskite crystal, distinguishing two species, and computing unit cell parameters.

Learn to determine atom positions in crystal images with Mathematica by applying max detect, component measurements, and centroid to extract, sort, and visualize coordinates over the image.

Distinguish two atom types by adjusting tolerance to split coordinates, use select and norm to filter coordinates, determine unit cell and angles to analyze crystal structure in Wolfram Mathematica.

Explore fractals by generating a snowflake using Mathematica and the manipulate command with a complexity slider, learning rule-based growth and the built-in approach.

Generate a self-similar Koch curve by iteratively replacing line segments with an angled path in Mathematica. Apply insert, nest, and manipulate to control order and visualize the evolving snowflake geometry.

Build a snowflake fractal in Mathematica by rotating three copies of a base curve around the triangle center, and analyze its area convergence as order increases.

Explore how the snowflake fractal's circumference grows with order while its area converges, illustrating the coastline paradox and natural fractal patterns, with Mathematica-based curve generation.

Explore fractals and space-filling curves through Mathematica, generating variations of snowflake-inspired curves and visualizing Hilbert and Hébert curves with adjustable order.

Explore how zooming animations create fractal loops by repeating a pizza with salami slices, producing a perfect self-similar illusion you can reproduce.

Generate a 3d pizza with salami slices using graphics 3d in Mathematica, defining cylinder coordinates, polar placement, and color/opacity for realistic rendering.

Demonstrate adding fractal orders by constructing multiple scaled layers of salami slices in Mathematica, enabling deeper zoom and a looping fractal animation.

Generate a 30-frame fractal pizza zoom animation in Mathematica using the graphics 3d option, adjust the camera and view vector, and export a looping gif.

Explore text analysis in Mathematica by comparing word usage, studying first and last letters, translating phrases, and visualizing data with word clouds, including political speeches.

Analyze words and languages with case-sensitive frequency history, translations, and dictionary comparisons, from English to German, and explore word length trends and word clouds in Mathematica.

Learn to manipulate text in Wolfram Mathematica, generate random words, analyze word frequency and historic data with time series plots, and examine text structure.

Identify languages, translate between German and English, perform dictionary lookups and word translations, and explore Greek, Russian, Polish, and Danish alphabets with the complement function.

Explore how to compare average word lengths across languages by plotting word length versus frequency, using English and German as examples, and learn to solve the task in your notebook.

Compare word lengths in English and German using word lists, string length, and histograms in Wolfram Mathematica, normalize by total words, and visualize with legends.

Analyze the first character of a 39,000-word English list, lowercase, count occurrences, compute relative frequencies, and visualize with a word cloud showing s as the top starter.

Analyze all characters in the word list to compute frequencies for every letter, not just the initial ones, and attempt a self-generated solution. Post a video.

Learn to analyze all characters by joining words into a single string, splitting into individual characters, lowering case, and computing character frequencies to create a word cloud.

Learn to compare character frequency across word clouds in Mathematica by transforming associations to lists, sorting alphabetically, and creating bar charts and grids to visualize first and last letters.

Analyze political speeches from a database of US presidents, examining word frequency, ages, and word clouds. Compare recent presidents and explore total word counts and word prominence across ranking.

Analyze the U.S. president speeches to reveal word frequency and historical themes across state of the union addresses, loading data from the Mathematica server and plotting year vs. word count.

Practice plotting the president's age by year in Wolfram Mathematica, accessing the age property from the data table, and comparing to the upcoming solution.

Explore how to analyze presidential ages over time using data plots, compare age trends at the start of presidencies, and generate word clouds from speeches to identify buzzwords.

Learn to compare main topics in speeches by generating word clouds from selected speeches, extracting nouns, counting occurrences, and filtering common words to reveal changing themes.

Explore Zipf's law through word-frequency analysis, lowering case, counting words, and visualizing with a list plot and log-log plot to reveal ranking versus frequency in speeches, books, and cities.

Generate and layer sounds in Mathematica, explore midi and real instruments, analyze spectra to distinguish timbre, speech synthesis, and an upcoming interactive piano.

Explore how different instruments in Mathematica alter a 440 Hz note, using sine sums and play to study overtones and the spectrum.

Analyze the spectrum with spectrograms using the spectrogram function, comparing flute, violin, guitar, and baritone sax, and observe time-frequency axes, harmonics, and the decay of intensity in higher overtones.

Explore speech synthesis in Mathematica by generating spoken words, experimenting with different speakers and pitch shifts, and analyzing audio with spectrograms to understand voice characteristics and noise.

Build an interactive piano in Mathematica by generating sounds, applying different instruments, and creating a graphic keyboard with clickable keys and notes to play songs.

Explore physics through a rolling ball described by Newton's equation, Monte Carlo pi estimation, and magnet energy minimization, then introduce quantum computing with superposition, gates, and teleportation.

Solve a Newtonian differential equation with gravity and friction to model a point mass in a bowl using Mathematica, visualize its trajectory, and animate the rolling ball.

Define a two-dimensional paraboloid potential from the center distance, visualize with plot3d and region function, then set up Newton's equations to simulate a ball sliding in the bowl.

Define the motion equations from Newton's law for a unit mass, solve the ODEs with NDSolve, and visualize the ball’s trajectory and the potential landscape.

Apply friction by adding a velocity-proportional term, giving a damped trajectory that settles at the bottom. Build a Mathematica animation by exporting frames at 30 fps to visualize the motion.

Explore how starting conditions and a periodic external force shape a ball’s motion: initial velocity alters trajectory, chaos emerges, then damped oscillations align along a 50-degree line.

Explore the Monte Carlo algorithm to use randomness for approximating pi and simulating magnet interactions, including parallel alignment, temperature fluctuations, and color-based visualization.

Learn how to estimate pi with a Monte Carlo method by comparing the circle area to the rectangle area, using the circle-to-rectangle area ratio pi/4, then multiply by four.

Estimate pi with a Monte Carlo method by generating random points in the square and counting those inside the circle, then multiply by four, illustrated with loop and list approaches.

Explore how to simulate nanoscale magnets with a Monte Carlo algorithm, using random numbers to model how magnetic moments align parallel or antiparallel under interactions.

Set up a three-dimensional magnet lattice in Mathematica, generate random unit-vector orientations with spherical coordinates, visualize arrows, and compute nearest-neighbor energy with a Monte Carlo approach under periodic boundaries.

Program the Monte-Carlo steps by selecting a random magnet, flipping its orientation, computing the new energy, and storing energy for plotting convergence, accepting only if the energy decreases.

Apply a weak magnetic field to bias the magnetic moments along the z direction, stabilizing the ground state in a nearest-neighbor exchange Monte Carlo simulation.

Explore a Monte Carlo approach to magnetism by using a Boltzmann acceptance criterion at finite temperature to gradually lower energy and reach a ferromagnetic configuration of 100 moments.

Explore simulating antiferromagnets with a Monte Carlo method by flipping the interaction sign to favor opposite neighboring moments, yielding near-zero magnetization and nonlinear configurations.

Solve the Schrödinger equation for a particle in a box in Mathematica, then explore spin qubits, superposition, gates, entanglement, and basic quantum teleportation.

Explore wave-particle duality, the Schrödinger equation, and the stationary Schrödinger problem with a potential V(x); solve in Mathematica to reveal energy levels and wave functions inside and outside a box.

Solve the stationary Schrödinger equation for a particle in a box using shooting methods in Mathematica, obtaining eigenvalues and eigenfunctions, and visualizing normalization and probability density.

Explore electron spin as a two-component spinor, use Pauli matrices to compute orientation along x, y, z, and visualize states on the block sphere for quantum computing basics.

Explore how Hadamard and other gates transform single- and two-qubit states in Mathematica, generating superpositions and entanglement with the CNOT gate, and illustrating quantum teleportation.