Weierstrass Function

DEFINITION

W(x) = Σ aⁿ cos(bⁿ π x)
0 < a < 1, b odd integer, ab > 1 + 3π/2  (Weierstrass, 1872)
0 < a < 1, b real, ab ≥ 1  (Hardy, 1916)
continuous everywhere · differentiable nowhere

NOTES

The function that ended the belief that a continuous curve must have a tangent somewhere — Weierstrass read it to the Berlin Academy and the century’s geometers called it a scandal. Every term is a smooth cosine; the sum has a corner at every point. Weierstrass needed b to be an odd integer and ab past 1 + 3π/2; Hardy pulled both away, leaving ab ≥ 1 for real b, and that is the condition on the plate. What the canvas shows is necessarily a partial sum, because any finite sum is smooth and the pathology lives only in the limit — terms whose frequency passes the pixel grid are dropped, since past that they alias into noise the mathematics never contained.

PROVENANCE

Origin

K. Weierstrass, read to the Königlich Preussische Akademie der Wissenschaften, Berlin, 18 July 1872 — recorded in the Monatsberichte for 1872, p. 560; G. H. Hardy, "Weierstrass’s non-differentiable function", Trans. AMS 17 (1916), 301–325, dropped the odd-integer requirement on b and weakened ab > 1 + 3π/2 to ab ≥ 1

Standing

Public domain — 19th-century analysis

Constants

The term count is set by resolution, not by a dial: a finite sum is smooth, so more terms than the pixels resolve would claim a depth the plate cannot show. Both sliders stay inside Hardy’s condition across their whole range — the weakest corner, a = 0.34 against b = 3, still gives ab = 1.02

HOUDINI · VEX

The same published mathematics as a Detail Wrangle body. Paste it into a Wrangle with Run Over set to Detail; every constant is the published value plus a tweak channel, so Create Spare Parameters gives a slider that starts where the paper does.

// FORMA — PL. 69 · WEIERSTRASS FUNCTION — Karl Weierstrass, 1872 · G. H. Hardy, 1916
//   W(x) = Σ aⁿ cos(bⁿ π x)
//   0 < a < 1, b odd integer, ab > 1 + 3π/2  (Weierstrass, 1872)
//   0 < a < 1, b real, ab ≥ 1  (Hardy, 1916)
//   continuous everywhere · differentiable nowhere
// Paste into a Detail Wrangle (Run Over: Detail), no inputs needed.
// Written from the published mathematics, not adapted from any code.
// Constants arrive at their published values. Press the node's Create
// Spare Parameters button and every tweak becomes a slider — starting
// at 0, the published figure, and moving in the constant's own units.
// https://forma-gen.com/#plate=weierstrass

float p_a = 0.55 + chf('a_tweak');    // a — amplitude ratio · live 0.34 .. 0.68
float p_b = 5 + chf('b_tweak');       // b — frequency ratio · live 3 .. 9

// The plate's own colour: FORMA's CURVES accent as a cosine ramp,
// brightest near t = 0 and t = 1, near-black around t = 0.5.
vector forma_ramp(float t){
  return set(
    0.11 + 0.1196 * cos(6.28318530718 * (t + 0)),
    0.46 + 0.5 * cos(6.28318530718 * (t + 0.05)),
    0.2453 + 0.2667 * cos(6.28318530718 * (t + 0.1)));
}

// Continuous everywhere, differentiable nowhere. Depth is bounded by what
// the samples can hold: bⁿ stays under a quarter of the sample count so the
// top term still renders as oscillation rather than aliasing into noise the
// mathematics never contained — the plate's own ceiling.
int forma_n = 1600;

int terms = max(2, int(floor(1.0 + log(float(forma_n) / 4.0) / log(p_b))));
// Σ|aⁿ| — normalising by it keeps every (a, b) on the same amplitude
float norm = (1.0 - pow(p_a, terms)) / (1.0 - p_a);
int prim = addprim(0, "polyline");
for (int i = 0; i <= forma_n; i++){
    float x = float(i) / float(forma_n) * 2.0 - 1.0;
    float y = 0.0;
    for (int k = 0; k < terms; k++)
        y += pow(p_a, k) * cos(pow(p_b, k) * 3.14159265359 * x);
    // the graph is drawn y-up already; no flip needed
    int pt = addpoint(0, set(x, y / norm, 0.0));
    // colour sweeps the bright lobe of the ramp along the graph
    float u = float(i) / float(forma_n);
    setpointattrib(0, "Cd", pt, forma_ramp(0.8 + 0.3 * u));
    addvertex(0, prim, pt);
}

AFTER EFFECTS · EXPRESSION

The same published mathematics as a Shape Layer path expression. Paste it onto a Path property; every constant is the published value plus a Slider Control named _tweak, so a bare paste already draws the figure and each slider moves one constant in its own units. Trim Paths is the comet.

// FORMA — PL. 69 · WEIERSTRASS FUNCTION — Karl Weierstrass, 1872 · G. H. Hardy, 1916
//   W(x) = Σ aⁿ cos(bⁿ π x)
//   0 < a < 1, b odd integer, ab > 1 + 3π/2  (Weierstrass, 1872)
//   0 < a < 1, b real, ab ≥ 1  (Hardy, 1916)
//   continuous everywhere · differentiable nowhere
// After Effects port — paste onto a Shape Layer's Path property
// (Contents › Shape › Path). Written from the published mathematics, not
// adapted from any code. Constants arrive at their published values; add a
// Slider Control (Effect › Expression Controls) named <k>_tweak and that
// constant moves in its own units, starting at 0 — the published figure.
// The plate's comet and its reveal are Trim Paths; the stroke colour is
// FORMA's CURVES accent, #3DFF88. Animation runs on time.
// https://forma-gen.com/#plate=weierstrass

// A missing slider reads 0, so a bare paste already draws the figure.
function forma_tweak(n){ try { return effect(n)("Slider"); } catch (e){ return 0; } }
var p_a = 0.55 + forma_tweak("a_tweak");  // a — amplitude ratio · live 0.34 .. 0.68
var p_b = 5 + forma_tweak("b_tweak");     // b — frequency ratio · live 3 .. 9

// The frame: the plate's W × H canvas is this comp, with the origin at the
// layer's anchor; canvas y already runs down, as After Effects' does.
var forma_W = thisComp.width, forma_H = thisComp.height, forma_t = time;
var forma_phase = 0.22420714446343482;   // this plate's own fixed phase, as the page has it
function forma_pt(x, y){ return [x - forma_W / 2, y - forma_H / 2]; }

// W(x) = Σ aᵏ cos(bᵏ π x) on x ∈ [−1, 1], as many terms as 1,600 samples can
// resolve (k up to log_b(N/4)), normalised by Σ|aᵏ| so the graph always fits
// 0.4 of the height. The page's pointer walks the graph back and forth over
// 13 s: Trim Paths again, Start and End a short window apart.
var N = 1600;
var terms = Math.max(2, Math.floor(1 + Math.log(N / 4) / Math.log(p_b)));
var norm = (1 - Math.pow(p_a, terms)) / (1 - p_a);
var pts = [];
for (var i = 0; i <= N; i++){
  var x = i / N * 2 - 1, y = 0;
  for (var k = 0; k < terms; k++) y += Math.pow(p_a, k) * Math.cos(Math.pow(p_b, k) * Math.PI * x);
  pts.push(forma_pt(forma_W * (0.05 + 0.9 * i / N), forma_H * 0.5 - (y / norm) * forma_H * 0.4));
}
createPath(pts, [], [], false);