The pattern seam · continuity without calm
A Curve With No Quiet Place
Tune a Weierstrass function, zoom into its recurring roughness, test difference quotients and finite box counts, then turn any drawn polygon into a certified finite shadow of the Baire category theorem.
A continuous curve usually arrives in the imagination already smooth. In 1872 Karl Weierstrass put a formula on the table that broke that habit. The formula below is not the historical normalization, but an equivalent classical one:
W(x) = Σn≥0 an cos(2πbnx).
Each added wave is smaller by a and faster by b. Uniform convergence makes the infinite sum continuous. When ab ≥ 1, Hardy proved that it has no finite derivative at any point. The instrument renders a finite sum, and it keeps that distinction visible.
Move the cursor. Then zoom.
Drag the gold marker across the curve, or use the cursor slider. Use the mouse wheel over the graph to zoom horizontally. The quotient ladder asks the same local slope question at eighteen geometrically shrinking steps.
Wheel over the graph to zoom. Double-click to return to x from 0 to 1. Vertical scale stays fixed, so a screen pixel keeps the same meaning.
Hardy condition
No finite derivative theorem for the infinite series.
finite render
exact dimension theorem
Shen 2018, only while integer b and 1/b < a < 1.
symmetric quotient ladder
finite box census
At the default values, the theorem gives computing. The finite box fit should be read beside it, never promoted into it. Exact whole-graph repetition is absent too: frequencies and amplitudes recur geometrically, but lower-frequency terms do not disappear when the view is rescaled.
One monster proves too little. Build the dense-open engine.
The sophisticated dismissal is right: one hand-built counterexample does not say what most continuous functions do. In the complete metric space C[0,1] with the uniform norm, define Um to contain each function for which every x has a nearby y, with 0 < |x-y| < 1/m, whose secant slope has magnitude greater than m. Each Um is open and dense. Their countable intersection contains only functions with no finite derivative anywhere.
Draw any 257-knot polygon below. The button adds a triangular wave small enough to remain inside your chosen uniform tube, yet fast enough that every linear piece has absolute slope greater than M.
Press Make Generic to build and inspect the certificate.
η = ε/2
not built
Exact uniform perturbation bound.
polygon slope L
not built
Maximum absolute knot-to-knot slope.
chosen frequency q
not built
Ceiling of (M + L + 1)/(4η).
certified minimum slope
not built
4ηq - L, algebraic lower bound.
The construction is f(x)=p(x)+η tri(qx), where the unit triangular wave ranges from minus one to one and has segment slopes ±4q. Thus ||f-p||∞ ≤ η = ε/2 < ε, while each linear piece of f has slope magnitude at least 4ηq-L ≥ M+1. The visible witness picks a nearby point in the same piece and evaluates the actual quotient.
Residual does not mean a percentage, random draw, or probability. Smooth functions are dense in the same uniform norm, even though they form a meagre set. Category-small does not mean hard to find. The construction above makes the other side visible: inside every uniform tube around every polygon lies a function passing as many steep-slope tests as you choose here.
The check
recomputing the shipped claims
| quantity | recomputed live | what can vary |
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Every free choice and uncertainty: cosine normalization 2π; 4097 horizontal samples; fixed vertical range ±1/(1-a); a 320 CSS-pixel tail target; at most 96 terms; quotient steps h=b-k for k=1,…,18; box widths 2-3,…,2-9; equal x and raw-y units; standard grid traversal with diagonal entry at an exact corner; ordinary least squares over all seven scales; 257 polygon knots; η=ε/2; the triangular wave ranges from minus one to one; M≤12 and q≤8192. Floating-point cosine reduction, canvas rasterization, cursor position, chosen parameters, drawn knots, and regression window all affect the experiments. None affects the cited theorems.
The historical values are transcriptions, not browser discoveries: Weierstrass presented the original result on 18 July 1872 with odd positive integer b and the sufficient bound ab > 1 + 3π/2 = computing. Hardy's 1916 theorem uses 0<a<1, b>1, ab≥1 and says no finite differential coefficient. Mazurkiewicz's 1931 theorem gives the residual statement in the uniform norm. These are source claims checked against the cited record, not facts a finite script can prove.
The global dimension is known. The worst local scaling is not.
Shen closed the classical Hausdorff-dimension problem in 2018: for integer b≥2 and 1/b<a<1, the graph has DH=2+log(a)/log(b). The nearest live edge asks for the Assouad dimension, which tests the most difficult local covering behavior across two scales rather than the global average scaling measured here.
As of 1 September 2025, Efstathios-K. Chrontsios-Garitsis and Jeremy T. Tyson wrote that the exact Assouad dimension of Weierstrass graphs remains open, with no partial result known specifically for the Weierstrass function. For the default curve, the honest current bracket is therefore computing ≤ DA ≤ 2. Their general Hölder-spectrum bound is progress around the question, not its answer. This page's one-scale box census cannot adjudicate it.