Six ways into one equation
Fractals, domain coloring, Julia sets, Mandelbrot operators, bifurcation diagrams. See what happens when you iterate one equation in the complex plane.
Hear the math. Click a fractal and listen to its orbit. Mix operators like a DJ. Play the depth hierarchy as a keyboard. Mathematics as sound.
Type any equation. See its cost. Optimize it. Annotate your code. The expression optimizer finds cheaper routes through exp and ln.
From zero to the infinite barrier. Interactive lessons, quizzes, and games that teach EML without prerequisites. Start with addition. End with silence.
Browse 265 equations by domain and cost. Visualize the four boundaries of elementary functions. Explore the FMA staircase. The data behind the claims.
Which algorithm stays inside ELC? How many nodes does your FEA cost? Optimize engineering expressions with domain-specific patterns.
Research-derived audio + visual demos: Bessel FM synth (the math behind FM synthesis), math-as-melody chord stacks, Pfaffian phase portraits. Self-contained exhibits from the latest exploration sessions.
Five doors into the mathematics
An entire video game — physics, terrain, controls, rendering — written as eight EML files. Hold UP to cancel gravity, DOWN to drop fast, SPACE to dash. Every equation in the game is a function in `public/play/eml/`. Same compiler that targets FPGAs targets the playable web build.
Drag nodes on the complex plane. Watch the fractal respond. Switch between 16 operators. Turn on audio reactive mode and let music shape the mathematics. This is meditation through computation.
Tap to start the mic. Whistle, hum, or play any instrument — the analyzer snaps every detected tone to an EML equation (sin(2π·440·t) — chain order 1, A4) and logs it so you can read what just came out of you. The math IS the sound.
Climb from arithmetic to the infinite barrier. Each level adds one capability — exp, then ln, then composition, then complex numbers. At the top: a function no finite tree can reach. Interactive, hands-on, no prerequisites.
Six carrier tiers — electronic, photonic, spintronic, phononic, ferronic, quantum — share one Pythagorean axiom. 115 closed Lean proofs. Drag-the-phase Mach-Zehnder, ring resonator, photonic attention head: the same EML proves them all.
Senses
See and hear how different species sense the world: vision, hearing, smell, touch, electroreception, magnetoreception, and infrared.
Pre-alpha interactive models. Not biological proof, not hardware validation, not medical or veterinary advice.
What one equation can and cannot reach
Addition, multiplication, division. Polynomials. Everything you learned before calculus. Simple, complete, and cheap.
Add exp(x). Now you have growth, decay, compound interest, radioactive half-lives. One node. The cheapest transcendental.
Combine exp and ln. Sigmoid, softplus, logarithmic spirals. The compositions are richer than the parts. 68% of science lives here.
Enter the complex plane. exp(ix) = cos(x) + i·sin(x). Euler's formula. Oscillation emerges from exponential. All of wave physics. One identity.
sin(x) over the reals. Infinitely many zeros. No finite tree of exp and ln can produce it exactly. Four classical theorems guard this wall. You can approach it. You cannot cross it.