phunction.sh

a wizard's lab on the open web — math, code, simulation, art

palette/preview · chalkboard · dark

incipit

Description is revelation. The world is the description, & the description is the world. The fragment shader runs once per pixel, in parallel, on a screen that is already glowing. You have ~200 lines and a clock. What makes it feel like a poem rather than a function: it has a subject, a constraint, and a rhythm.

Inline code: const φ = (1 + Math.sqrt(5)) / 2;. Inline math: $\oint_{C} f (z) d z = 2 π i \sum_{k} Res (f, z_{k})$ . Inline link: an anchor. Inline strike: ~~Mercury retrograde~~. Inline emphasis: panta rhei.

display math

\nabla \cdot E \nabla \times E = \frac{ρ}{ε _{0}}, = - \frac{\partial B}{\partial t}, \nabla \cdot B \nabla \times B = 0, = μ_{0} J + μ_{0} ε_{0} \frac{\partial E}{\partial t}

The Mandelbrot escape iterates $z_{n + 1} = z_{n}^{2} + c$ starting at $z_{0} = 0$ ; the smooth-coloured iteration count is $μ = n - lo g_{2} lo g_{2} ∣ z_{n} ∣^{2} + C$ .

The Riemann hypothesis: every non-trivial zero of $ζ (s)$ lies on the critical line $ℜ (s) = \frac{1}{2}$ .

ζ (s) = n = 1 \sum \infty \frac{1}{n ^{s}} = p prime \prod \frac{1}{1 - p ^{- s}}

A matrix and a continued fraction:

(a c b d)^{- 1} = \frac{1}{a d - b c} (d - c - b a), φ = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \dots}}}

lists

unordered

Mandelbrot — $z_{n + 1} = z_{n}^{2} + c$
Julia — same recurrence, fixed $c$
Reaction–diffusion — $\partial U / \partial t = D \nabla^{2} U - U V^{2} + F (1 - U)$
Helmholtz — $\nabla^{2} ψ + k^{2} ψ = 0$

ordered

open a lab URL on a laptop or phone
plug into a projector if you've got one
leave it running
let the room change

nested

shaders
- vertex
- fragment
- compute
analysis
- real
- complex
- functional

table

symbol	lab	domain	tech
`∂`	fractal	ℂ	WebGL2
`∮`	flow	ℝ²	GPU PDE
`ζ`	scry	webcam	getUserMedia
`∇²`	lattice	ℝ³	vertex shader

code block

// generate evenly-distributed hues by the golden angle
const phi = (1 + Math.sqrt(5)) / 2;
const goldenStep = 360 / phi;            // ≈ 222.4922°

const palette = (n: number, h0: number, l = 76, c = 0.18) =>
  Array.from({ length: n }, (_, i) =>
    `oklch(${l}% ${c} ${(h0 + i * goldenStep) % 360})`
  );

palette(8, 170);
//  → ['oklch(76% 0.18 170)', 'oklch(76% 0.18 32.5)', ...]

blockquote

"the universe is not only queerer than we suppose, but queerer than we can suppose."

— J.B.S. Haldane, Possible Worlds, 1927

definition list

math: the language of structure
code: the executable form of math
art: math made visible

buttons

⌬ enter the lab § read the prose ! destructive

card row

∂ /lab/fractal Mandelbrot ↔ Julia morpher M = { c ∈ ℂ : orbit of 0 stays bounded } ∮ /lab/flow Gray–Scott reaction–diffusion ∂U/∂t = D∇²U − UV² + F(1−U) ζ /lab/scry webcam through shader oracles I(x,y,t) = (𝓚 ∗ V)(x,y,t) ∇² /lab/lattice 3D parametric wave field ∇²ψ + k²ψ = 0

palette generated by golden ratio over dark.