Mathematical Explorations
16 interactive research labs, 6 formal essays, and 6 browser games running live with no install step — each lab a small piece of in-browser mathematical research, not a textbook visualization. Labs span Pentagonal Lattice Geometry, Geometric Entropy, Spacelike Knots, PID-Regulated Cellular Automata, Constrained Mesh Enclosure, No-Three-in-Line, the nam numeric VM, and an Optimization Mechanics Visualizer. Claims verified computationally via Maxima, BFS sweeps, and KPM estimation. Open the labs at math.cognotik.com → — nothing to install, everything runs in your browser.
Key Features
Original Mathematical Research
Research-grade constructions including Pentagonal Lattice Geometry (fractional dimension, spinor-like holonomy), Geometric Entropy (continuous analogue of the Erdős distinct-distance problem), Spacelike Knots (Minkowski-metric knots), and the Irrational Lattice (provably aperiodic algebraic 'colored noise').
- Pentagonal Lattice Geometry: d ≈ 2.37, spinor-like holonomy, exact ℚ(√5) arithmetic
- Geometric Entropy: continuous Erdős distinct-distance analogue on arbitrary manifolds
- Spacelike Knots: crossings as causal inversions under a Minkowski metric
- Irrational Lattice: deterministic, provably aperiodic, spectrally tunable ℚ(√D) fields
- PID-Regulated Cellular Automata: per-cell PID controllers as an update rule
- Constrained Mesh Enclosure: differentiable shrink-wrap with exact continuous collision detection
- No-Three-in-Line: continuous relaxation with entropic simulated annealing
- Fractal Learning: inverse IFS fitting via gradient optimization under a Chamfer metric
Computationally Verified Claims
Symbolic identities verified via Maxima batch scripts; spectral behavior confirmed via BFS dimension sweeps and KPM estimation.
- Maxima batch scripts for symbolic verification
- BFS dimension sweeps for spectral confirmation
- KPM estimation for independent validation
- Reproducible in-browser experiments
Zero Build Step Deployment
Plain HTML, CSS, and JavaScript with no build step, deployed as a static site to AWS S3 + CloudFront via Terraform.
- No framework, no bundler, no build step
- AWS S3 + CloudFront via Terraform
- Instant load, works offline
- Every lab ships with a README covering motivation, math, and method
Formal Essays & Playable Experiments
Six long-form essays — including the RCC/PI_RCC/NAM trilogy on the cost of computing real numbers and an interactive scroll-driven tutorial on QQN — alongside six browser games where a mathematical idea is the mechanic.
- RCC trilogy: rational certificate complexity, x + sin(x) for π, and numbers as machines
- QQN essay plus an interactive 'Which Direction? → Where on the Curve?' tutorial
- The Extension Ladder: watching number systems grow
- Games: Predictive Markov Wordsearch, Stereo-Tac-Toe, Arithmetic Dynamics, Number Crush
- Sister sites: The Arcade of Life and Vocal Parkour
Getting Started
Nothing to install — every lab, essay, and game runs directly in your browser.
While you're here, the sister sites are one click away: The Arcade of Life and Vocal Parkour.
Quick Example
Featured Labs (16) — browse them all →
- Pentagonal Lattice Geometry — branched covering, d ≈ 2.37, spinor-like holonomy
- Geometric Entropy — continuous analogue of the Erdős distinct-distance problem
- PID-Regulated Cellular Automaton — each cell is a PID controller; tunable damping regimes
- Layered Cellular Automata — Langton's ants gate where Conway's Life may live
- Spacelike Knots — knot theory under a Minkowski metric; crossings as causal inversions
- Space-Color Symmetry — symmetry rewires the diffusion graph, not just the strokes
- Irrational Lattice — deterministic, provably aperiodic algebraic "colored noise"
- Relativistic 2-Body Gravity — retarded interactions, precession, and chaos
- Bidirectional Markov Text Analyzer — forward + backward n-gram surprise heatmaps
- Constrained Mesh Enclosure — differentiable shrink-wrap with exact collision clipping
- Static Secrets — publish encrypted content anywhere, decrypt entirely in-browser
- No-Three-in-Line Explorer — entropic annealing with parabola warm starts and tabu search
- nam — Numbers as Machines — a calculator where every number is a forkable nano-VM
- Fractal Learning — inverse IFS fitting via gradient optimization (Chamfer metric)
- Optimization Mechanics Visualizer — GD, Adam, L-BFGS, and QQN on the same landscape
- Chromatic — OKLab vs. naive HSL palettes with live distortion metrics
Formal Essays (6)
- QQN — Quadratic Quasi-Newton, plus an interactive scroll-driven companion
- RCC — Rational Certificate Complexity (trilogy I: the vocabulary)
- PI_RCC — The simplest increment, x + sin(x) (trilogy II: the boundary)
- NAM — Numbers as Machines (trilogy III: the executable substrate)
- TEL — The Extension Ladder: one way to watch number systems grow
Games (6)
- The Arcade of Life ↗ and Vocal Parkour ↗ (sister sites)
- Predictive Markov Wordsearch — filler letters generated to camouflage hidden words
- Stereo-Tac-Toe — tic-tac-toe played entirely inside an autostereogram
- Arithmetic Dynamics — value-preserving moves that teach the mechanics of arithmetic
- Number Crush — match-and-clear on a hex grid, matching arithmetic rules not colors
Technical Details
Technologies
JavaScriptHTML5 CanvasMathematical VisualizationInteractive WebMaximaAWS S3CloudFrontTerraform
Requirements
- Modern web browser with JavaScript enabled
- No installation or account required
Contribute to the Project
This is an open-source project. Contributions, bug reports, and feature requests are welcome from the community.