THE FOLD / GLITCH / SEGFAULT / THE TAKAGI
THE TAKAGI
a curve continuous everywhere and smooth nowhere
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Takagi function (or blancmange curve) is continuous everywhere and differentiable nowhere — a curve with no smooth spot at all. It is built by piling up ever-finer triangle waves: T(x) = ∑n≥0 s(2nx) / 2n, where s(x) is the distance from x to the nearest integer. Each layer is a zig-zag half as tall and twice as frequent as the last; their sum converges to a continuous curve that wobbles at every scale, so no tangent line ever exists. It obeys the self-similar functional equation T(x) = s(x) + ½T(2x), reaches its maximum value of exactly 2/3 at x = 1/3 and 2/3, and resembles a blancmange pudding — hence the name (Teiji Takagi, 1901).
LIT verified live: the functional equation T(x) = s(x) + ½T(2x) holds across the interval to ~1e-15; T(1/2) = 1/2, T(1/3) = 2/3, and the maximum of T equals 2/3 (window.__takagi). FIG continuity and the functional equation are checked in-browser; nowhere-differentiability is the known theorem the curve illustrates, not something numerically resolved here.
LIT verified live: the functional equation T(x) = s(x) + ½T(2x) holds across the interval to ~1e-15; T(1/2) = 1/2, T(1/3) = 2/3, and the maximum of T equals 2/3 (window.__takagi). FIG continuity and the functional equation are checked in-browser; nowhere-differentiability is the known theorem the curve illustrates, not something numerically resolved here.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at segfault — the glitch: a curve that is perfectly continuous yet crashes any attempt to take a derivative, at every single point. AVAN (AI) built the instrument: the triangle-wave sum, the functional equation, and the exact 2/3 maximum.
Credit as content: Teiji Takagi (1901); the blancmange curve. The weave: David names the everywhere-glitch; I confirm T(x) = s(x) + ½T(2x) and max T = 2/3.
Credit as content: Teiji Takagi (1901); the blancmange curve. The weave: David names the everywhere-glitch; I confirm T(x) = s(x) + ½T(2x) and max T = 2/3.
3 ONE DIMENSION
The blancmange curve — continuous, wobbling at every scale, peaking at 2/3 over x = 1/3 and 2/3.
4 TWO DIMENSIONS · INTERACTIVE
Add layers; the triangle waves pile up, and the functional equation T(x)=s(x)+½T(2x) is checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the blancmange curve, smooth nowhere yet continuous everywhere.
AVAN’s addition (the inverse-companion): don’t look for a slope — look for self-similarity. The inverse of ‘the curve T(x)’ is ‘the functional equation T(x) = s(x) + ½T(2x), a half-scale copy of itself’. Magenta are the triangle-wave layers piling up; green is the continuous, nowhere-smooth curve they sum to. Roughness that never resolves into a slope.
LIT Genuine Takagi / blancmange function (Teiji Takagi, 1901). Verified live: the functional equation T(x) = s(x) + ½T(2x) holds across the interval to ~1e-15; T(1/2) = 1/2, T(1/3) = 2/3, and the maximum of T equals 2/3 (window.__takagi.ok).
FIG Honest FIG boundary — continuity and the functional equation are checked in-browser; nowhere-differentiability is the known theorem the curve illustrates, not something numerically resolved here. The AVAN inverse — instead of looking for a slope, look for self-similarity: the inverse of 'the curve T(x)' is 'the functional equation T(x) = s(x) + ½T(2x), a half-scale copy of itself'. Magenta are the triangle-wave layers piling up; green is the continuous, nowhere-smooth curve they sum to. Roughness that never resolves into a slope.
FIG Honest FIG boundary — continuity and the functional equation are checked in-browser; nowhere-differentiability is the known theorem the curve illustrates, not something numerically resolved here. The AVAN inverse — instead of looking for a slope, look for self-similarity: the inverse of 'the curve T(x)' is 'the functional equation T(x) = s(x) + ½T(2x), a half-scale copy of itself'. Magenta are the triangle-wave layers piling up; green is the continuous, nowhere-smooth curve they sum to. Roughness that never resolves into a slope.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN