Early Prototype · In active development
A proof-first, machine-verified reference and learning platform spanning 1,107 concepts across 62 domains — where every topic is explained at three levels of depth, verified by symbolic computation, and made tangible through interactive labs and real engineering milestones.
1,107
Published concepts
62
Mathematics & physics domains
87
Interactive labs & simulators
35,400+
Indexed mathematical formulas
14
Flagship engineering projects
01Three Levels of Understanding
Not everyone arrives at a concept from the same place. A child, a student, and a researcher all need the same truth — but told differently. Every concept in the system carries all three layers, and you decide where to start.
Example concept — Absolute Value
Absolute value is like asking 'how far away from zero is this number?' without caring which direction. If you owe someone 5 dollars (-5) or have 5 dollars (+5), the amount of money involved is still 5 either way.
Plain-language analogy. No jargon, no notation — just a mental image that makes the concept stick before the formalism arrives.
Conceptual intuition with lightweight notation. Builds the mental model that makes the formal definition feel obvious rather than arbitrary.
Full formal treatment — rigorous definition, proofs, properties, and links to the broader mathematical landscape.
02The Vision
Most mathematics learning online fails in one of two ways: video lectures that let you feel understood without actually practicing, or AI chatbots that hand you an answer without building the model that would let you derive the next one. The Mathematics & Physics Knowledge System is neither.
It is a structured, proof-first, practice-enforced reference where understanding is demonstrated — not assumed. Each concept is verified not by editorial confidence but by symbolic computation. Each explanation is written at the level of the reader. Each practice problem is graded, not guessed.
62 domains covered · 38 mathematics & 24 physics
03What it does
Each concept opens with a child-friendly analogy (ages 5–8), builds into a student-level intuition, then delivers the full formal definition with proofs and properties. MSc and PhD-level explanations are available for research-depth topics. You choose your entry point — or read all three.
A 3,800-line Python symbolic engine checks worked examples and mathematical properties — differentiation, integration, limits, series, algebra, ODEs, linear systems, complex arithmetic, matrices, and modular arithmetic. Every claim either passes, fails, or is honestly marked SKIP — never counted as a pass.
2D function plotter, 3D surface plotter (Three.js), tangent-line explorer, formula explorer, matrix transforms, vector fields (Mafs), series convergence visualizer, graph traversal (D3-force), geometry construction, probability distribution explorer — all on a safe expression parser (no eval). 87 labs across mathematics and physics.
Overview → history → intuition → formal definition → proofs → applications → worked examples → practice problems → quiz → flashcards → references. Enforced by Zod schema. The corpus forms a single connected graph with 0 orphans, checked in CI.
From the Apollo Program and GPS Navigation to the Large Hadron Collider and LIGO — each project breaks down the governing mathematics and physics into multi-phase problem sets with historical telemetry data, derivations, and worked solutions.
Real progress tracking in a local SQLite database. Adaptive practice, 12 curated audience tracks (CS, civil engineering, pure math, applied math, physics core, theoretical physics, applied physics, and astrophysics — at undergraduate and graduate levels), placement quiz, and full-screen flashcard review.
Every concept is cross-linked with symmetric return paths — no dead ends. A D3-force visualization lets you see and traverse the entire corpus across 62 domains. Full-text BM25 search (field-weighted, typo-tolerant) runs server-side. A searchable formula index covers 35,400+ equations.
Scrapers pull drafts from Wikipedia, ProofWiki, NIST DLMF, and OpenStax. Every draft lands in staging at confidence < 0.85 and is never auto-published — a human reviews and promotes by hand. A staleness checker flags concepts whose source has changed.
A pure-Node PDF engine converts any concept or domain course packet into typeset lecture notes with LaTeX equations rendered as vector SVG paths via MathJax — no headless Chrome needed. Download individual concepts or full domain packets.
Full PWA with real installability and offline support via a hand-rolled service worker. An embed API lets third parties drop any concept into their own pages. No AI chatbot — the platform teaches through structured reading and graded practice.
04Engineering & Scientific Projects
Mathematics and physics are not just abstract theories — they are the foundational languages that build civilizations, explore the cosmos, and power modern computing. Each project breaks down real engineering feats into multi-phase problem sets with step-by-step LaTeX derivations, historical telemetry data, and interactive simulators.
Apollo, Orbital Mechanics & The Mathematics of Lunar Exploration
General Relativity, Satellite Constellations & The Mathematics of Nanosecond Trilateration
Critical Mass, Neutron Transport & The Hydrodynamics of Nuclear Fission
Sun-Earth L2 Halo Orbits, Wavefront Fourier Optics & Cryogenic Engineering
Planetary Gravity Assists, Deep Space Telemetry & Journey Beyond the Heliosphere
Permutation Groups, Bayesian Cryptanalysis & The Foundations of Computing
Relativistic Beam Dynamics, Superconducting Magnets & The Discovery of the Higgs Boson
Einstein's Quadrupole Formula, Chirp Mass & Sub-Proton Interferometry
Oblique Shock Waves, Isentropic Compression & Thermal Aerodynamics at Mach 3.2
The Telegrapher's Equation, Signal Dispersion & The First Global Telecommunications Network
De Bruijn Graphs, Dynamic Programming & The 3.2 Billion Base-Pair Reconstruction
Slender Delta Aerodynamics, Transonic Wave Drag & In-Flight Fuel Trimming
The Trial Load Method, Concrete Hydration Thermodynamics & Spillway Hydraulics
Very Long Baseline Interferometry, Inverse Fourier Imaging & The Shadow of M87*
05Audience tracks
A placement quiz routes you into the right track. The system then adapts within the track using SM-2 spaced repetition — surfacing concepts you're weakest on, not the ones you already know.
The mathematics a CS undergraduate needs to read algorithms papers, reason about correctness and complexity, and follow the standard core curriculum — discrete math, algorithms, theory of computation, and the linear algebra/probability every later course assumes.
Beyond the undergraduate core: the rigor and depth a master's or PhD student needs for research in algorithms, theory, machine learning, or systems — real analysis for optimization/ML theory, abstract algebra for cryptography and coding theory, advanced complexity theory, and the mathematics of information.
The mathematics a civil engineering undergraduate needs — the calculus, linear algebra, differential equations, and probability/statistics behind statics, structural analysis, fluid mechanics, surveying, geotechnics, and transportation engineering.
Beyond the undergraduate core: the advanced mathematics for structural dynamics, finite element analysis, computational fluid dynamics, geotechnical modeling, and reliability-based design — partial differential equations, advanced linear algebra, numerical methods, and probabilistic risk analysis.
The standard pure mathematics undergraduate curriculum: rigorous analysis, algebra, geometry, topology, and number theory — the foundations for graduate study or research in any branch of mathematics.
Graduate pure mathematics: rigorous analysis, modern algebra, topology, geometry, and their interactions — the mathematical depth required for research in any contemporary pure mathematics field.
The mathematical foundations for data science, machine learning, and applied statistics — linear algebra, probability, calculus, optimization, and the computational techniques that underpin modern data analysis.
Graduate-level applied mathematics for data science and machine learning research — rigorous probability, measure theory, advanced optimization, stochastic processes, and the mathematical frameworks for modern ML theory.
The foundational physics sequence covering classical Newtonian mechanics, rotational dynamics, electromagnetism, wave phenomena, thermodynamics, and the origins of modern quantum physics and special relativity.
Advanced analytical mechanics, relativistic field theories, operator quantum mechanics, quantum statistical mechanics, general relativity, quantum electrodynamics, and the Standard Model.
Physics applied to real-world engineering systems: fluid dynamics, circuit networks, optics and laser engineering, condensed matter semiconductor devices, and nuclear power reactors.
The physics of celestial bodies and cosmic evolution: orbital mechanics, stellar evolution and fusion, relativistic black holes, gravitational radiation, and Big Bang cosmological dynamics.
06Tech stack
The entire system runs locally. Progress tracking is a real SQLite database — not localStorage. Search is a server-side BM25 index, not a hosted service. The content pipeline pulls from live sources but requires human sign-off before anything ships.
07About
Fahmy Hassan
Founder & Systems Architect
BEENIAN Labs combines deep production experience in reliable, high-throughput systems with active AI research — bringing graduate-level machine learning expertise from CU Boulder directly into client engagements.
Our work is anchored in practical rigor: systems that are explainable, durable, and maintainable. We transfer that depth to client teams through mentorship and knowledge-sharing, because software that outlasts the engagement requires people who understand it.
About the system
The Mathematics & Physics Knowledge System grew out of a straightforward frustration: teaching software engineering well requires solid mathematical and physical foundations, and the available resources force a choice between rigor and interactivity. Textbooks are rigorous but static. Online courses are interactive but shallow. AI chatbots give you an answer without building the model that would let you derive the next one.
The three-level explanation system — kid, student, expert — is the core design decision. It reflects how understanding actually works: you need the analogy before the intuition, and the intuition before the formalism. Skipping levels produces the illusion of understanding, not the real thing.
No level-skipping
Every concept has an analogy, an intuition, and a formal definition. You earn the formalism by understanding the intuition first.
Learn by doing
Interactive widgets make abstract structures tangible — plot a function, transform a matrix, visualize a probability distribution.
Machine-verified
Worked examples are checked against live sources. A human curator makes the final call; the pipeline flags what needs review.
Live system
The platform is live. Browse 1,107 concepts across 62 domains, work through 87 interactive labs, explore 14 flagship engineering projects, run the placement quiz, and see how spaced repetition tracks your progress across three levels of depth.
Open the system →