ISEGORIA / MATHEMATICS
Math encyclopedia
From sets to spaces of functions, and on into physics and engineering. 79 explainers and experiments, by subject.
Foundations and algebra
14 entries
Set theorySets, relations, functions, cardinality, and infinity.Discrete mathematicsLogic, counting, proof, and cellular automata.Logic and computability: what can be computedTuring machines, the halting problem, and Gödel numbering.Cellular automata: computation from local rulesThe 256 elementary rules, Conway’s Life, Langton’s ant, and rule 184 as traffic.Category theoryObjects, arrows, composition, functors, and universal properties.Linear algebra: the geometry of transformationsCoordinates, area, eigenvectors, projection, and singular values.Group theory: symmetry as algebraCompose symmetries, inspect group tables, and count orbits.Representation theory: groups acting as matricesDihedral groups as matrices, character tables that decompose anything, and symmetry setting vibration frequencies.Quaternions: rotation as multiplicationRotation as a sandwich q v q*, slerp between orientations, and gimbal lock.Number theory: patterns in the integersCongruences, primes, and public-key ideas.Elliptic curves: geometry that addsThe chord-and-tangent law, curves mod p, and the torus behind the curve.Galois theory: the symmetry of equationsWhich permutations of the roots are allowed, the correspondence between subgroups and subfields, and what ruler and compass can draw.Graph theory: routes, trees, and networksShortest paths, spanning trees, and centrality.Structure Before AlgebraRecognize a problem’s structure and choose a useful first step.
Topology and analysis
14 entries
Analysis AtlasTopology, metric spaces, real analysis, complex analysis, and functional analysis.Lebesgue vs Riemann integrationCompare partitions, null sets, and convergence.Lebesgue integration: visual guideA visual introduction to measure and integration.Lebesgue Integration LabSimple functions, level sets, and convergence theorems.Measure theory: size, approximation, and convergencePartitions, simple functions, and convergence warnings.Functional analysis: norms, projections, and operatorsNorms, Hilbert-space projection, and operator spectra.Fourier analysis: functions made of wavesHarmonics, convolution, and the limits of sampling.Wavelets: zooming in on a signalScalograms that follow a chirp, the Haar and Daubechies ladders, and thresholding that compresses and denoises.Complex analysis: maps, residues, and harmonic fieldsConformal maps, residues, and harmonic functions.Gamma and Beta functionsExplore integral identities, substitutions, and worked examples.Integrating factorsSee how a multiplier simplifies differential equations.Dynamical systems: stability and chaosPhase portraits, changing equilibria, and sensitive dependence.Fractal geometry: dimension between the integersIterated function systems, box counting, Julia and Mandelbrot sets, L-systems, and Newton’s basins.Partial differential equations: fields in motionHeat, waves, and potential fields.
Jump to an Analysis Atlas lesson: Topology · Metric spaces · Real analysis · Complex analysis · Functional analysis
Geometry and transformations
13 entries
Differential geometry: curvature and shapeMetrics, geodesics, and curvature on surfaces.Hyperbolic geometry: where parallels multiplyPoincaré-disc distance, angle defect, and {p, q} tilings.Knot theory: telling tangles apartReidemeister moves, tricolouring and the determinant, and torus knots.Differential forms: circulation, curl, and pullbacksLine integrals, Stokes theorem, and coordinate change.Push-forward and pullback: integrating through a mapPull a function or form back and integrate upstairs, or push a measure forward and integrate downstairs.Lie groups and Lie algebras: continuous symmetryExponentials of rotations, the double cover, and the bracket.Algebraic topology: detecting holesBuild complexes, distinguish cycles from holes, and vary scale.Algebraic varietiesPolynomial equations, curves, and singularities.Calculus of variations: paths and principlesExtremal paths, action, and geodesics.Geometry and mensurationExplore lengths, areas, volumes, and geometric relationships.Cone dimensionsRelate height, slant height, surface area, and volume.Jacobians, Wronskians, and HessiansLocal maps, independence of functions, and curvature.Möbius transformations and the SchwarzianComplex-plane transformations and projective structure.
Probability, computation, and reference
13 entries
Probability: learning from uncertaintyBayesian updating, sums of random variables, and conditional probability.Statistics and inference: signals in dataSampling, confidence, and regression.Markov chains: transition and stationarityTransition matrices, stationary distributions, and absorbing walks.Stochastic processes: Brownian motion and noiseScaling limits, quadratic variation, and Fokker–Planck densities.Information theory: uncertainty and codesEntropy, noisy channels, and information distance.Optimization: the geometry of the best choiceDescent, conditioning, constraints, and shadow prices.Control theory: feedback, poles, and stabilityPole locations, PID tuning, and Bode and Nyquist plots.Complex systems: order from many simple partsKuramoto synchronisation, the self-organised sandpile, percolation, and the Vicsek flock.Numerical analysis: when computation misleadsTime stepping, cancellation, and interpolation error.Counting and probabilityPascal’s triangle, distributions, the binomial theorem, and birthdays.The Scharr gradientEdit image samples and inspect gradient size and direction.Mathematics reference sheetsCompact reference sheets for studying and revision.Daily Mathematics: solutionsFull solutions to each day's hard integral and original Putnam-style problem cards.
Physics
19 entries
Classical mechanics: motion and forcesProjectiles, oscillators, and orbital motion.Hamiltonian mechanics: phase space and its geometryLiouville’s theorem, symplectic integrators, and Noether’s theorem.Rigid-body dynamics: spinning, tumbling, precessingThe tennis-racket flip, gyroscopes, and principal axes.Electromagnetism: fields and inductionElectric fields, magnetic forces, and changing flux.Optics: rays, waves, and colourLenses, interference, and polarization.Laser physics: light that copies itselfThreshold and switch-on spikes, the cavity that picks the frequencies, and mode-locked pulses.Thermodynamics: energy, work, and entropyPV paths, reversible limits, and entropy flow.Statistical mechanics: microstates and temperatureBoltzmann weights, partition functions, and random walks.Magnetism and the Ising model: order from alignmentOnsager’s exact transition on the square lattice, the chain that never orders, and mean field beside the truth.Fluid dynamics: flow, pressure, and vorticityStreamlines, Bernoulli pressure, and rotating flow.Plasma physics: screening, orbits, and wavesDebye screening, cyclotron motion, and dispersion.Special relativity: space, time, and lightLorentz diagrams, time dilation, and invariant intervals.General relativity: curvature, clocks, and lightEmbedding diagrams, gravitational time dilation, and light deflection.Cosmology: the expanding universeThe Friedmann equation, cosmological distances, and horizons on a conformal diagram.Quantum mechanics: amplitudes and spinWavefunctions, uncertainty, and a Bloch sphere.Atomic physics: orbitals and spectraHydrogen orbitals, spectral series, and the Zeeman effect.Quantum information: entanglement and its limitsBell’s inequality, teleportation step by step, and entanglement entropy.Solid-state physics: waves in a crystalTight-binding bands, band gaps, and Brillouin zones.Semiconductor physics: bands, doping, junctionsCarriers and the Fermi level, the p–n junction, and the diode run forwards and as a solar cell.
Engineering: watches and sound
6 entries
Oscillators and escapements: how a watch keeps timeThe balance and hairspring, Airy’s theorem, and positional error.Gears and mechanisms: transmitting motion exactlyInvolute teeth, the going train, and epicyclic gears.Quartz resonators: a crystal that keeps timeWhy 32,768 Hz, the crystal as a circuit, and temperature error.Loudspeakers: the moving-coil driverDriver impedance, sealed and vented boxes, and beaming.Filters and crossovers: splitting sound between driversBode plots, crossovers that sum flat, and offset drivers.Room acoustics: the room is part of the speakerRoom modes, reverberation time, and boundary reflections.
Companion PDFs
English-language companions to the interactive guides.