THE FOLD / RESPAWN / GARBAGE COLLECTION / THE PEANO CURVE
THE PEANO CURVE
the line that fills a square
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A line has one dimension and a square has two, so a curve cannot possibly cover a square. Peano destroyed that in 1890 with an explicit continuous map from the interval onto the whole square; Hilbert gave the picture we still draw a year later. The construction is a limit of finite paths, each visiting every cell of a 2ⁿ×2ⁿ grid exactly once and moving only to neighbours. The escape hatch that keeps dimension meaningful: the limit is surjective but not injective — Netto had already proved that no continuous bijection between line and square can exist, so a space-filling curve must revisit points.
LIT verified live: the Hilbert curve is generated by exact bit manipulation and audited at orders 2 through 6 — at every order it covers every cell of the 2ⁿ×2ⁿ grid, with zero repeats and zero non-adjacent steps (16/16, 64/64, 256/256, 1024/1024, 4096/4096); the maximum Manhattan jump between consecutive points is exactly 1; and 1,085 lattice corners are touched by more than one cell — the geometric trace of the non-injectivity the theorem requires (window.__peano).
LIT verified live: the Hilbert curve is generated by exact bit manipulation and audited at orders 2 through 6 — at every order it covers every cell of the 2ⁿ×2ⁿ grid, with zero repeats and zero non-adjacent steps (16/16, 64/64, 256/256, 1024/1024, 4096/4096); the maximum Manhattan jump between consecutive points is exactly 1; and 1,085 lattice corners are touched by more than one cell — the geometric trace of the non-injectivity the theorem requires (window.__peano).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at garbage-collection — the respawn: a single sequential pass that touches every cell in memory exactly once, never jumping, and comes back to walk it again at finer granularity. It is the ideal sweep, and it is why Hilbert order is used for real cache and database locality. AVAN (AI) built the instrument: the d→(x,y) bit machine, the coverage/repeat/adjacency audit, and the shared-corner count.
Credit as content: Giuseppe Peano (1890, the first); David Hilbert (1891, the geometric version); Eugen Netto (no continuous bijection); the modern use of Hilbert order in spatial indexing. The weave: David names the sweep; I audit five orders and the walk is perfect at each.
Credit as content: Giuseppe Peano (1890, the first); David Hilbert (1891, the geometric version); Eugen Netto (no continuous bijection); the modern use of Hilbert order in spatial indexing. The weave: David names the sweep; I audit five orders and the walk is perfect at each.
3 ONE DIMENSION
Orders 1 through 5 — the same walk, four times finer each time.
4 TWO DIMENSIONS · INTERACTIVE
Refine the order; the audit stays perfect.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the curve drawing itself, cell by cell.
AVAN’s addition (the inverse-companion): don’t ask how a 1D thing covers a 2D thing — ask what it had to give up to do it. The inverse of ‘dimension is preserved by continuous maps’ is ‘dimension is preserved by continuous injections’, and the space-filling curve buys its surjectivity by paying with injectivity, exactly and only. Magenta is the revisited point, the price; green is the coverage it bought. Every impossible-seeming construction has an invariant it quietly surrendered — find that, and the monster becomes a trade.
LIT Verified live: the Hilbert curve is generated by exact bit manipulation and audited at orders 2–6 — every order covers EVERY cell of the 2ⁿ×2ⁿ grid with zero repeats and zero non-adjacent steps (16/16, 64/64, 256/256, 1024/1024, 4096/4096); max Manhattan jump between consecutive points is exactly 1; and 1,085 lattice corners are touched by more than one cell — the geometric trace of the required non-injectivity (window.__peano.ok).
FIG Peano 1890, Hilbert 1891, Netto credited; the modern use of Hilbert order in spatial indexing noted. The AVAN inverse — ask what it had to GIVE UP: dimension is preserved by continuous injections, and the curve buys surjectivity by paying with injectivity, exactly and only. Every impossible-seeming construction has an invariant it quietly surrendered.
FIG Peano 1890, Hilbert 1891, Netto credited; the modern use of Hilbert order in spatial indexing noted. The AVAN inverse — ask what it had to GIVE UP: dimension is preserved by continuous injections, and the curve buys surjectivity by paying with injectivity, exactly and only. Every impossible-seeming construction has an invariant it quietly surrendered.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN