THE FOLD / RESPAWN / THE PHOENIX / THE DRAGON CURVE
THE DRAGON CURVE
a fold that fills space and never crosses itself
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The dragon curve is the shape you get by folding a strip of paper in half, again and again, then unfolding every crease to a right angle. Its turn sequence is the regular paperfolding sequence: at step n, turn left if the odd part of n is ≡1 (mod 4), else right. Astonishingly, though it packs into a fractal that tiles the plane, the curve never crosses itself — every unit segment is traversed at most once. It is a classic of computer graphics and number theory alike.
LIT verified live: up to order 14 (16384 segments) the dragon curve is self-avoiding — every unit edge is distinct, none repeated (window.__dragon). FIG no framing; exact.
LIT verified live: up to order 14 (16384 segments) the dragon curve is self-avoiding — every unit edge is distinct, none repeated (window.__dragon). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-phoenix — the form reborn from repeated folding, rising into a fractal that fills space yet never tangles. The dragon curve is that endless fold. AVAN (AI) built the instrument: the paperfolding turn rule (odd-part mod 4), the turtle walk, the edge-set, and the self-avoiding (all-edges-distinct) check.
Credit as content: the dragon curve (Heighway, Harter & Banks; popularized by Martin Gardner 1967). The weave: David names the-phoenix; I generate the fold sequence, walk it a right angle at a time, and confirm no unit edge is ever reused — a space-filling curve that never crosses.
Credit as content: the dragon curve (Heighway, Harter & Banks; popularized by Martin Gardner 1967). The weave: David names the-phoenix; I generate the fold sequence, walk it a right angle at a time, and confirm no unit edge is ever reused — a space-filling curve that never crosses.
3 ONE DIMENSION
Fold a strip in half repeatedly; unfold each crease to 90°. The n-th turn is left if the odd part of n is 1 (mod 4), else right: L, L, R, L, L, R, R, … — the paperfolding sequence.
4 TWO DIMENSIONS · INTERACTIVE
The dragon curve at a chosen order; it is checked to be self-avoiding (no unit edge reused).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a space-filling fold that never crosses.
AVAN’s addition (the inverse-companion): make a curve that fills space and tiles the plane yet never crosses itself — by folding (the paperfolding turn sequence) rather than drawing. The inverse of ‘a space-filling curve must be carefully routed to avoid overlaps’ is ‘just fold — the crease sequence is automatically self-avoiding.’ Magenta is the crossings a naive path would make; green is the crossing-free dragon. Space-filling from a fold.
LIT Genuine dragon curve / regular paperfolding sequence (Heighway, Harter & Banks; Gardner 1967). Verified live: generating the fold turn sequence (left if the odd part of n is 1 mod 4, else right) and walking it a right angle at a time, up to order 14 (16384 segments), every unit edge is distinct — the curve is self-avoiding (window.__dragon.selfAvoiding).
FIG No framing: the paperfolding turn rule, the turtle walk, the edge-set, and the self-avoiding check run in-browser and hold to 16384 segments. The AVAN inverse is honest — the crease (paperfolding) sequence yields a space-filling, plane-tiling curve that is automatically self-avoiding, no routing needed; magenta is the crossings a naive path would make, green the crossing-free dragon. Space-filling from a fold.
FIG No framing: the paperfolding turn rule, the turtle walk, the edge-set, and the self-avoiding check run in-browser and hold to 16384 segments. The AVAN inverse is honest — the crease (paperfolding) sequence yields a space-filling, plane-tiling curve that is automatically self-avoiding, no routing needed; magenta is the crossings a naive path would make, green the crossing-free dragon. Space-filling from a fold.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN