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…

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…

Geometric Entropy

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

Spacelike Knots

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

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 →…

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…

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…

Quadratic Quasi-Newton

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

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.