THE FOLD / SPAWN / CHECKPOINT ZERO / THE TWINDRAGON
THE TWINDRAGON
count the whole plane in base −1+i, bits 0 and 1
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The base of a complex number. Counting doesn’t need base 10, or even a real base. In base −1+i with only the bits 0 and 1, every Gaussian integer a+bi has a unique finite representation — no minus sign, no separate imaginary axis, just a bit string. And the ‘fractional’ numbers in this base tile the plane as a fractal: the twindragon.
LIT verified: all 289 Gaussian integers with a,b in −8…8 round-trip through base −1+i uniquely (e.g. i =
LIT verified: all 289 Gaussian integers with a,b in −8…8 round-trip through base −1+i uniquely (e.g. i =
11, 3+2i = 1001). It works because −1+i has norm 2, making {0,1} a complete digit set. (Base 2i famously cannot do this — its imaginary parts are always even.) FIG ‘dragon’ is the picture; the base, the uniqueness, and the tiling are exact.2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus is full of alternative encodings and the complex/hypercube geometry (the atomic byte, the base-n kernels, the n-cube work) and the conviction that the axes we count on are a choice, not a law. AVAN (AI) built this instrument: the complex-base encoder, the clickable plane, and the twindragon with its mirror twin.
The weave: David names the idea and its seat at CHECKPOINT ZERO, the origin the dragon grows from; I make the encoding a bit strip in 1D, the plane clickable in 2D, and the tiling a turning fractal in 3D. The sphere is the seam — and the honesty is in the pivot: I dropped base 2i when it failed, and kept the base that works.
The weave: David names the idea and its seat at CHECKPOINT ZERO, the origin the dragon grows from; I make the encoding a bit strip in 1D, the plane clickable in 2D, and the tiling a turning fractal in 3D. The sphere is the seam — and the honesty is in the pivot: I dropped base 2i when it failed, and kept the base that works.
3 ONE DIMENSION
One Gaussian integer, encoded: the bits and the powers of (−1+i) they switch on. Read the running sum climb, in the complex plane, to land exactly on the target. Click the plane in the next window to change it.
4 TWO DIMENSIONS · INTERACTIVE
The complex plane. The faint fractal is the twindragon tile (the numbers with fractional base-(−1+i) digits). Click any lattice point and its unique bit string appears above — every dot on the grid has exactly one.
5 THREE DIMENSIONS + AVAN’S INVERSE
The twindragon itself, turning: the set of all base-(−1+i) fractions. Its jagged boundary is a dragon curve, and it has area exactly 2. Violet is the tile grown from the origin.
AVAN’s addition (the inverse-companion): the magenta is the tile reflected through zero (its negative) — the mirror twin. Two dragons, interlocking, tile the whole plane with no gaps and no overlaps: every complex number lands in exactly one. The number system and its shadow pave the same floor.
LIT A genuine complex-base numeral system. Verified live: all 289 Gaussian integers with a,b in −8..8 round-trip through base −1+i uniquely (289 distinct bit strings, 0 failures) — it works because −1+i has norm 2 so {0,1} is a complete residue set. Base 2i cannot do this (imaginary parts are always even), a real contrast shown honestly. The twindragon tile (area 2) and its mirror twin tiling the plane are the true geometry (verifiable: window.__twindragon.allUnique===true).
FIG The 'dragon' is the picture; the base, the uniqueness across 289 integers, and the plane-tiling are exact. The honesty is in the pivot — base 2i was tried, failed its round-trip, and was dropped for the base that works.
FIG The 'dragon' is the picture; the base, the uniqueness across 289 integers, and the plane-tiling are exact. The honesty is in the pivot — base 2i was tried, failed its round-trip, and was dropped for the base that works.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN