Companion Reference Pages

Pentagonal Lattice Geometry

A multi-sheeted covering construction in which the regular pentagon develops fractional dimension (d ≈ 2.37) and spinor-like holonomy — a discrete…

Geometric Entropy

A continuous analogue of the Erdős distinct-distance problem. Points on a manifold are arranged to extremize the Shannon entropy of their…

PID-Regulated Cellular Automaton

A cellular automaton whose update rule is a per-cell PID controller: each cell measures its active-neighbor count against a target and computes…

Layered Cellular Automata

Langton's ants write a colored substrate that gates where Conway's Life is allowed to live. Three feedback layers — ants → substrate → life →…

Spacelike Knots

A knot equipped with a Minkowski metric, turning crossings into causal inversions. The distance matrix becomes a causal diagram: timelike,…

Space-Color Symmetry

A pixel canvas where symmetry doesn't just mirror your strokes — it rewires the diffusion graph. Heat flows along symmetry orbits, producing…

Irrational Lattice

A new primitive: deterministic, algebraic “colored noise” for lattices. Coordinates in ℚ(√D) deform ℤᵈ and snap back via a nearest-lattice map —…

Relativistic 2-Body Gravity

A 2-body gravitational system with relativistic adjustments and time-delayed (retarded) interactions. Gravity propagates at finite speed c, breaking…

Bidirectional Markov Text Analyzer

Build a language model from your own corpus, then watch another text light up token-by-token by how surprising it is. Forward and backward order-n…

Constrained Mesh Enclosure Lab

A deformable surface mesh flows downhill on geometric fitness functionals (area, volume, edge-length entropy, angular regularity, Laplacian…

Static Secrets

Publish encrypted content anywhere and decrypt it entirely in your browser. The host serves opaque bytes; only you hold the key.

The Prime Sieve as a Stack of Orthogonal Periodic Fields

An interactive essay on the periodic machinery underneath the primes: each prime emits an orthogonal exclusion field, their stacked product has…

nam — Numbers as Machines

A calculator where every number is a deterministic, forkable virtual machine emitting an infinite digit stream on demand. Comparison is honestly…

Fractal Learning

Inverse iterated-function-system fitting: gradient-optimize a small alphabet of affine maps so the depth-N orbit of the origin matches a target…

Optimization Mechanics Visualizer

Drop GD, Adam, L-BFGS, and QQN onto the same rugged 2D loss landscape and watch, step by step, how each finds its way downhill. Configurable terrain…

Chromatic — OKLab vs Naive HSL

A color engine that thinks in perception, not hue wheels. The same palette — N accent colors evenly spaced in hue — built two ways: a…

No-Three-in-Line: A Visual Theory Explorer

An infinite-canvas explorer for the no-three-in-line rule under deterministic greedy growth: ring-major Chebyshev order, forced empty strips,…

No-Three-in-Line Explorer

An interactive laboratory for the classic no-three-in-line problem: place ~2n points on an n×n grid so no three are collinear along any rational…

Quadratic Quasi-Newton

A new optimization algorithm for smooth functions that interpolates between gradient descent and L-BFGS.

Which Direction? → Where on the Curve?

An interactive, scroll-driven tutorial on the Quadratic Quasi-Newton (QQN) optimizer. Reframes optimization from "which direction should I move?"…

RCC — Rational Certificate Complexity

The foundation of the trilogy. Treats every convergence sequence as a rational certificate engine, measures its bit-length versus demanded tolerance…

PI_RCC — The Simplest Increment: x + sin(x)

A cubic-convergent iteration for π whose outer recurrence is not hypergeometric — it falls outside RCC's classified class, yet still attains optimal…

NAM — Numbers as Machines

A generator-based numerics library where every number is a forkable, deterministic nano-VM emitting an infinite digit stream. RCC's cost classes…

The Extension Ladder: One Way to Watch Number Systems Grow

Mandelbrot Set

Explore the fractal boundary defined by iteration of z ↦ z² + c in the complex plane. Pan and zoom to examine its self-similar structure.

Prime Number Sieve

A real-time visualization of the Sieve of Eratosthenes, illustrating the elimination of composite numbers to isolate the primes.

Fourier Series

Construct periodic waveforms as sums of rotating phasors, illustrating how harmonic components combine to approximate arbitrary signals.

Collatz Conjecture

Trace the trajectory of any positive integer under the 3n + 1 iteration. Whether every sequence reaches 1 remains an open problem.