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Math encyclopedia
86 pages
Acoustics: strings, plates and moving sourcesPlucked and bowed strings, Chladni figures, and the Mach cone.Algebraic topology: detecting holesBuild complexes, distinguish cycles from holes, and vary scale.Analysis AtlasTopology, metric spaces, real analysis, complex analysis, and functional analysis.Atomic physics: orbitals and spectraHydrogen orbitals, spectral series, and the Zeeman effect.Calculus of variations: paths and principlesExtremal paths, action, and geodesics.Category theoryObjects, arrows, composition, functors, and universal properties.Cellular automata: computation from local rulesThe 256 elementary rules, Conway’s Life, Langton’s ant, and rule 184 as traffic.Classical mechanics: motion and forcesProjectiles, oscillators, and orbital motion.Combinatorics: counting by buildingInteger partitions, Catalan objects, and growth read off singularities.Complex analysis: maps, residues, and harmonic fieldsConformal maps, residues, and harmonic functions.Complex systems: order from many simple partsKuramoto synchronisation, the self-organised sandpile, percolation, and the Vicsek flock.Control theory: feedback, poles, and stabilityPole locations, PID tuning, and Bode and Nyquist plots.Cosmology: the expanding universeThe Friedmann equation, cosmological distances, and horizons on a conformal diagram.Filters and crossovers: splitting sound between driversBode plots, crossovers that sum flat, and offset drivers.Daily Mathematics: solutionsFull solutions to each day's hard integral and original Putnam-style problem cards.Differential forms: circulation, curl, and pullbacksLine integrals, Stokes theorem, and coordinate change.Differential geometry: curvature and shapeMetrics, geodesics, and curvature on surfaces.Discrete mathematicsLogic, counting, proof, and cellular automata.Dynamical systems: stability and chaosPhase portraits, changing equilibria, and sensitive dependence.Electromagnetism: fields and inductionElectric fields, magnetic forces, and changing flux.Elliptic curves: geometry that addsThe chord-and-tangent law, curves mod p, and the torus behind the curve.Cone dimensionsRelate height, slant height, surface area, and volume.Counting and probabilityPascal’s triangle, distributions, the binomial theorem, and birthdays.Geometry and mensurationExplore lengths, areas, volumes, and geometric relationships.Integrating factorsSee how a multiplier simplifies differential equations.Jacobians, Wronskians, and HessiansLocal maps, independence of functions, and curvature.Möbius transformations and the SchwarzianComplex-plane transformations and projective structure.The Scharr gradientEdit image samples and inspect gradient size and direction.Structure Before AlgebraRecognize a problem’s structure and choose a useful first step.Fluid dynamics: flow, pressure, and vorticityStreamlines, Bernoulli pressure, and rotating flow.Fourier analysis: functions made of wavesHarmonics, convolution, and the limits of sampling.Fractal geometry: dimension between the integersIterated function systems, box counting, Julia and Mandelbrot sets, L-systems, and Newton’s basins.Functional analysis: norms, projections, and operatorsNorms, Hilbert-space projection, and operator spectra.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.Game theory: strategy when others choose tooNash equilibria, replicator dynamics, and the minimax theorem.Gamma and Beta functionsExplore integral identities, substitutions, and worked examples.Gears and mechanisms: transmitting motion exactlyInvolute teeth, the going train, and epicyclic gears.General relativity: curvature, clocks, and lightEmbedding diagrams, gravitational time dilation, and light deflection.Graph theory: routes, trees, and networksShortest paths, spanning trees, and centrality.Group theory: symmetry as algebraCompose symmetries, inspect group tables, and count orbits.Hamiltonian mechanics: phase space and its geometryLiouville’s theorem, symplectic integrators, and Noether’s theorem.Hyperbolic geometry: where parallels multiplyPoincaré-disc distance, angle defect, and {p, q} tilings.Information theory: uncertainty and codesEntropy, noisy channels, and information distance.Knot theory: telling tangles apartReidemeister moves, tricolouring and the determinant, and torus knots.Laser physics: light that copies itselfThreshold and switch-on spikes, the cavity that picks the frequencies, and mode-locked pulses.Lebesgue integration: visual guideA visual introduction to measure and integration.Lebesgue Integration LabSimple functions, level sets, and convergence theorems.Lebesgue vs Riemann integrationCompare partitions, null sets, and convergence.Lie groups and Lie algebras: continuous symmetryExponentials of rotations, the double cover, and the bracket.Linear algebra: the geometry of transformationsCoordinates, area, eigenvectors, projection, and singular values.Logic and computability: what can be computedTuring machines, the halting problem, and Gödel numbering.Loudspeakers: the moving-coil driverDriver impedance, sealed and vented boxes, and beaming.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.Markov chains: transition and stationarityTransition matrices, stationary distributions, and absorbing walks.Math encyclopedia knowledge graphFollow conceptual connections and jump into an experiment.Measure theory: size, approximation, and convergencePartitions, simple functions, and convergence warnings.Nuclear physics: binding, decay and tunnellingThe liquid-drop model, decay chains, and alpha decay by tunnelling.Number theory: patterns in the integersCongruences, primes, and public-key ideas.Numerical analysis: when computation misleadsTime stepping, cancellation, and interpolation error.Optics: rays, waves, and colourLenses, interference, and polarization.Optimization: the geometry of the best choiceDescent, conditioning, constraints, and shadow prices.Partial differential equations: fields in motionHeat, waves, and potential fields.Piston engines: from crank to cycleCrank-slider kinematics, the Otto and Diesel cycles, and engine balance.Plasma physics: screening, orbits, and wavesDebye screening, cyclotron motion, and dispersion.Probability: learning from uncertaintyBayesian updating, sums of random variables, and conditional probability.Push-forward and pullback: integrating through a mapPull a function or form back and integrate upstairs, or push a measure forward and integrate downstairs.Quantum information: entanglement and its limitsBell’s inequality, teleportation step by step, and entanglement entropy.Quantum mechanics: amplitudes and spinWavefunctions, uncertainty, and a Bloch sphere.Quartz resonators: a crystal that keeps timeWhy 32,768 Hz, the crystal as a circuit, and temperature error.Quaternions: rotation as multiplicationRotation as a sandwich q v q*, slerp between orientations, and gimbal lock.Representation theory: groups acting as matricesDihedral groups as matrices, character tables that decompose anything, and symmetry setting vibration frequencies.Rigid-body dynamics: spinning, tumbling, precessingThe tennis-racket flip, gyroscopes, and principal axes.Room acoustics: the room is part of the speakerRoom modes, reverberation time, and boundary reflections.Semiconductor physics: bands, doping, junctionsCarriers and the Fermi level, the p–n junction, and the diode run forwards and as a solar cell.Set theorySets, relations, functions, cardinality, and infinity.Mathematics reference sheetsCompact reference sheets for studying and revision.Solid-state physics: waves in a crystalTight-binding bands, band gaps, and Brillouin zones.Special relativity: space, time, and lightLorentz diagrams, time dilation, and invariant intervals.Statistical mechanics: microstates and temperatureBoltzmann weights, partition functions, and random walks.Statistics and inference: signals in dataSampling, confidence, and regression.Stochastic processes: Brownian motion and noiseScaling limits, quadratic variation, and Fokker–Planck densities.Switching power converters: chopping to convertBuck and boost converters, conversion limits, and discontinuous conduction.Thermodynamics: energy, work, and entropyPV paths, reversible limits, and entropy flow.Oscillators and escapements: how a watch keeps timeThe balance and hairspring, Airy’s theorem, and positional error.Wavelets: zooming in on a signalScalograms that follow a chirp, the Haar and Daubechies ladders, and thresholding that compresses and denoises.Algebraic varietiesPolynomial equations, curves, and singularities.
Interactive explainers
27 pages
The Bladeless FanHow a ring with a one-millimetre slot and no visible blades moves fifteen times more air than its motor pushes, and where that multiplication actually comes from.Apple silicon · The M-series atlasExplore every announced Apple M-series chip, with top-down architecture, comparisons, memory experiments and downloadable guides.The Optical Pulse · Apple Watch Heart RateExplore how Apple Watch turns reflected light into a heart rate with an interactive 3D wrist cutaway and pulse laboratory.Inside an Automatic Chronograph: the Whole Movement, TurningA complete automatic chronograph calibre built part by part and set running in the browser: rotor and winding, barrel and going train, lever escapement and balance, keyless…How my headphones measure my earsWhat Bose CustomTune actually does in the half second after you put the headphones on: a multitone chime, a microphone inside the cup, and a retuned noise-cancelling loop and…Carrera Panda: An Interactive StudyDiscover how an automatic watch works through nine animated, exploded mechanisms and 36 guided stages. Explore the TAG Heuer Carrera Panda, TH20-00 chronograph, escapement,…DualSense · Inside the feelExplore the PS5 DualSense in 3D. Experiment with left and right haptics, voice-coil motion and adaptive trigger resistance, then inspect every controller system.KEF Coda W · Inside Uni-QAn interactive 3D companion to the Bose Lifestyle Ultra and HomePod studies. Explore KEF Coda W, Uni-Q and the geometry of coherent sound.Inside the Kuru TogaA mechanical pencil that turns its own lead nine degrees every time you lift it off the paper. The engine that does it, half a tooth at a time, and a simulation of what the…Laurent series × Bessel functions: a visual math labAn interactive visual lab for Laurent expansions on a domain-coloured plane, Bessel partial sums and the vibration modes of a circular drum.Apple M-series · Die floorplansEvery Apple M-series chip from M1 to M6 as an interactive top-down floorplan: CPU clusters, GPU, Neural Engine, SLC and memory, multi-die Ultras and the M5 Fusion dies, with…Skin Deep · Oral versus Topical MelatoninTopical melatonin makes a gradient through the dermis and oral melatonin makes a plateau. An interactive model of where they cross, and what oral dose it would take to match a…Model Benchmark AtlasAstra, Fable 5.1 and Opus 5: how reasoning effort changes measured quality, cost and response time, and which tests matter for game development.Membrane: Action PotentialAn interactive 3D neuron with Hodgkin–Huxley membrane dynamics. Explore travelling action potentials, sodium and potassium ion flow, and voltage-gated channels.Nicotine Dose CurveAn illustrative 24-hour nicotine exposure model comparing cigarette boluses with vape micro-doses.OLED: A panel made of lightExplore OLED panel layers in 3D, follow electron–hole recombination, and mix RGB subpixels. An interactive English and Japanese explainer.Inside a Perpetual CalendarA classic 48-month-cam perpetual calendar made to run in the browser: the grand lever, the cam that knows the length of every month for four years, the stars, the moon disc,…Inside a Quartz WatchInteractive visual guide to the quartz watch: piezoelectric tuning fork, Pierce oscillator, 15-stage divider chain, Lavet stepper motor, and the gear train that drives the hands.Electricity → Quartz → MotorWhere energy actually changes form between the battery and the hands of a quartz watch: two transductions, one gap that nothing physical crosses, and the Lavet stepper at the end.Cold, explained · Refrigerator labExplore a refrigerator in 3D, follow the refrigerant cycle, and experiment with heat flow, energy use and thermostat control. English and Japanese.SodaStream ART: Inside the fizzExplore the SodaStream ART in 3D. Inspect a reference-based ART model and explore carbonation, Henry’s law, gas release, and carbonic acid with interactive chemistry controls.Heat, moved: Split System LabExplore a reverse-cycle split system in 3D. See how heat moves and compare useful heating or cooling with the electricity and cost it takes.Studio Display · Sound in a thin spaceExplore how Apple's Studio Display produces bass, controls vibration, and creates a spacious soundstage. An interactive English and Japanese guide.Inside a Flying TourbillonA one-minute flying tourbillon taken apart and made to run in the browser: the cage, the fixed wheel, the escapement turning with the balance, and why the rotation averages out…The Writing Lab · uniExplore uni JETSTREAM Lite touch ink and KURU TOGA DIVE: interactive 3D inspection and bilingual mechanism experiments.Inside an Automatic WatchInteractive visual guide to the movement, gear train, escapement, balance wheel, and hairspring inside an automatic watch.The Mechanical Watch: Interactive AtlasExplore automatic winding, escapements and watch complications through interactive diagrams in English and Japanese.
Research
34 pages
The Dislike of AI Art Is Not AestheticLooking at art is inference about a mind. Generated images do not fail on quality; they collapse the posterior over the maker, and because the cost of production stops being…Analytics preferenceTurn analytics off for this browser on theisegoria.github.io, check the preference now in force, and see how Do Not Track and Global Privacy Control are honoured.Computational Acoustics: HomePod 2 vs Bose Lifestyle UltraA patent-led visual investigation of how Apple HomePod 2 and Bose Lifestyle Ultra create bass, spatial sound, and room-aware playback.Homepod Vs Bose Acoustic ExplainerHomePod 2's sealed woofer against a five-driver circular beamforming array, compared on excursion, room sensing, and where each design puts its energy.Bose Lifestyle Ultra Acoustic ExplainerHow a small cabinet makes bass: CleanBass excursion, QuietPort damping, the DSP chain, and the ceiling reflections TrueSpatial steers to widen the image.Coding reasoning: Astra, Sol and FableCoding benchmark performance, token use, cost and latency across reasoning levels. Expanded research report with interactive comparisons and a seven-page PDF.Astra Fable Effort CurvesEffort curves for GPT-6 Astra and Claude Fable 5.1 across five reasoning settings: Terminal-Bench and SciCode pass rates, tokens, API cost and time to first answer.Astra Medium Versus SolGPT-6 Astra, GPT-5.6 Sol and Claude Fable 5.1 compared at selected reasoning levels on coding pass rates, output and reasoning tokens, API cost and latency.Cold Water and the MindWhat a cold shower does to mood, anxiety and attention, traced from skin receptor to randomised trial: what the evidence establishes, what it only suggests, and a protocol that…Concerta 54-108 mg - extracellular NE and DA modelAn evidence-bounded interactive model of plasma methylphenidate, transporter occupancy, and relative extracellular NE and DA across 54-108 mg Concerta.Creatine, the Brain, and Cognition - Research ReportA critical research report on creatine, brain energetics, cognition, sleep loss, aging, mental health, neurological disease, and safety.Creatine, Amphetamine and the Shape of the LoadAmphetamine adds an ATP cost that no action potential switches off, and lowers dopamine-neuron firing rate. A kinetic model asks what the phosphocreatine buffer is worth…Creatine, longevity and the central nervous systemPlausible mechanisms, human evidence, and the limits of translation.Inside an Automatic Watch - Interactive mechanismFollow energy from rotor and mainspring through the gear train, escapement, balance oscillator, and hands of an automatic watch.The effort dial runs out before the money doesClaude Opus 5, Claude Fable 5.1 and GPT-6 Astra swept across all five reasoning-effort settings: where the marginal return on deliberation reaches zero, which model is cheapest…A model that abstains is worth more than one that guessesA thirty-item closed-book probe for measuring how much domain knowledge a language model actually holds, with a forced-abstention protocol, a confidence rating, an answer key…Frontier Coding Atlas - IsegoriaA source-backed comparison of frontier models for software engineering, reasoning effort, and Ultra orchestration.The Gamepad Froze in 2005. The Hardware Did Not.Game input did not stop being invented after 2005. It stopped converging: no controller feature added since then has reached all three console makers, so multiplatform games…Can Concerta and Intuniv ease heartbreak?The neurobiology of romantic loss and the case that methylphenidate (Concerta) and guanfacine ER (Intuniv) could help people function through it: the circuit, the pharmacology,…The Longevity StackCreatine, cold exposure, fasting, aerobic and resistance training, and melatonin: mechanisms, the human evidence behind each, and hypotheses about what they do together.The Menstrual Cycle and ADHDA critical narrative review of menstrual-cycle variation in catecholamine signalling and CNS function in relation to ADHD. Eight competing hypotheses, a staged research…Monster Beverage, Tech Stocks & the Geography of DemandA sourced 25-year return comparison and country-level sales test of Monster Energy versus tech-industry concentration.Nanite and Lumen: the algorithms behind Unreal Engine 5How Unreal Engine 5 draws film-density geometry at one triangle per pixel (Nanite) and lights a fully dynamic world with multi-bounce global illumination (Lumen): the cluster…What Fable 5.1 still buys you over Opus 5.5Claude Opus 5.5, Claude Fable 5.1 and GPT-6 Astra compared on the Artificial Analysis index, Anthropic’s launch benchmarks and three independent testers: where Opus 5.5 matches…Real-Time Natural Worlds - Research BriefA source-backed 2026 research brief on procedural terrain, procedural flora, and real-time water rendering for videogames.Soulblighter InternalsA visual research pass through the internal architecture of Bungie's Myth II: Soulblighter.The Best Argument for T-RexVortioxetine plus brexpiprazole, audited for cognition, mood and anxiety. The dendrite claim rests on one assay in a tumour cell line; the real argument is receptor…Threads: agreement, disagreement, praise and criticismAn anonymized observational study of reply stance, account patterns, praise and criticism, with context from four behavioral studies.{{TITLE}}{{DESCRIPTION}}What the Diamond in uni NanoDia DoesWhat nanodiamond physically does inside uni NanoDia mechanical pencil lead: void filling, crack pinning and ball-bearing friction, set against the one public head-to-head with…Violin Études de perfectionnementA series of original violin études for the first level, built on Michel Sogny's model: one technical target per study, a bilingual instruction page with notated practice…Two Routes to a Cheaper BrainVortioxetine and creatine both lower task-evoked BOLD. I argue they do it by two mechanistically independent routes, and that only one of them has any claim on general…Weekly PutnamOne original problem a week at the hard end of the Putnam, with a full worked solution as a PDF: hints, the key idea, a complete proof and commentary.World Model MCP: evidence-aware research memoryLocal research memory for Codex and Claude: findings, evidence, assumptions, human review, and revision history. Explore a clearly labeled synthetic demo.
Books
3 pages
C++23, Metal and ARM64 field guidesThree typeset field guides for writing a Metal C++ engine on Apple silicon: modern C++23 with pointers explained slowly, graphics with C++ and Metal 4, and arm64 assembly. Free…Game Design as the Dynamics of LearningMy book arguing that games are systems regulating the rate at which players reduce uncertainty, that fun is that reduction at an optimal rate, and that the learning gradient is…Integrated Medical FoundationsA bilingual English and Japanese study resource running from MCAT foundations to pre-clinical medicine: 105 expanded chapters, EPUB editions and Anki card exports.
Projects
3 pages
Material LabMaterial Lab is a local macOS app with an iOS companion and a CLI that turns one photograph into editable PBR texture maps: albedo, 16-bit height, normal, roughness, GTAO…STEM Genius: Daily STEM PracticeShort, exact daily practice across calculus, physics, chemistry and logic. Runs entirely on your device, with no account and no internet connection.STEM Genius · Privacy PolicyPrivacy policy for STEM Genius. The app collects no data, has no accounts, and works entirely offline.