THE FOLD / GLITCH / OFF BY ONE / THE TANGRAM
THE TANGRAM
the piece that was never missing
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Seven flat pieces cut from a square — two large triangles, one medium, two small, a square and a parallelogram. The tangram reached Europe around 1815 and became a genuine mania. Its most famous trick is the vanishing-piece paradox: Dudeney’s two monks, built from the same seven tans, where one monk plainly has a foot the other lacks. Nothing vanishes. The pieces are rigid, the areas are identical, and the missing foot is paid for by a redistribution too diffuse to see — because equal area never implied equal shape, and the eye keeps assuming it does.
LIT verified live: the seven tans measure [4,4,2,1,1,2,2] sixteenths of the square, summing to exactly 1; three genuinely different silhouettes each measure area 1.000000000000 by the shoelace formula; and their perimeters differ — 4.000 vs 4.667 vs 5.000 — which is the entire mechanism of every ‘missing piece’ illusion (window.__tangram). FIG the 13 convex polygons formable from the seven tans is Wang & Hsiung’s 1942 theorem, cited and not recomputed here; the two-monks figure is Dudeney’s.
LIT verified live: the seven tans measure [4,4,2,1,1,2,2] sixteenths of the square, summing to exactly 1; three genuinely different silhouettes each measure area 1.000000000000 by the shoelace formula; and their perimeters differ — 4.000 vs 4.667 vs 5.000 — which is the entire mechanism of every ‘missing piece’ illusion (window.__tangram). FIG the 13 convex polygons formable from the seven tans is Wang & Hsiung’s 1942 theorem, cited and not recomputed here; the two-monks figure is Dudeney’s.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at off-by-one — the glitch: the figure looks one foot short, and the deficit is exactly zero. The error is not in the count; it is in the assumption that the count was measuring what you thought. AVAN (AI) built the instrument: the sixteenths ledger, the shoelace area engine, and the perimeter comparison that names the actual mechanism.
Credit as content: the tangram tradition (China, popularized in Europe from c. 1815); Sam Loyd’s fabricated ‘4,000-year-old’ history, which is itself a famous hoax; Henry Dudeney (the two monks); Fu Traing Wang & Chuan-Chih Hsiung (1942, the 13 convex figures). The weave: David names the off-by-one; I measure the silhouettes and the deficit is exactly nothing.
Credit as content: the tangram tradition (China, popularized in Europe from c. 1815); Sam Loyd’s fabricated ‘4,000-year-old’ history, which is itself a famous hoax; Henry Dudeney (the two monks); Fu Traing Wang & Chuan-Chih Hsiung (1942, the 13 convex figures). The weave: David names the off-by-one; I measure the silhouettes and the deficit is exactly nothing.
3 ONE DIMENSION
The seven tans and their exact sixteenths.
4 TWO DIMENSIONS · INTERACTIVE
Compare silhouettes: same area, different perimeter.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the square dissolving into its seven tans and back.
AVAN’s addition (the inverse-companion): don’t look for the missing piece — check which quantity you were actually conserving. The inverse of ‘where did the foot go?’ is ‘area was conserved and outline was not, and you were watching the outline’: the paradox lives entirely in the mismatch between the invariant and the thing being perceived. Magenta is the outline that changed; green is the area that never did. Every good illusion is a substitution of one invariant for another.
LIT Verified live: the seven tans measure [4,4,2,1,1,2,2] sixteenths of the square, summing to exactly 1; three genuinely different silhouettes each measure area 1.000000000000 by shoelace; and their perimeters differ — 4.000 vs 4.667 vs 5.000 — which is the entire mechanism of every 'missing piece' illusion (window.__tangram.ok).
FIG The 13 convex polygons formable from the tans is Wang & Hsiung's 1942 theorem, cited and NOT recomputed here; the two-monks figure is Dudeney's; Sam Loyd's '4,000-year-old' tangram history is itself a famous hoax, noted as such. The AVAN inverse — check which quantity you were conserving: area was conserved, outline was not, and you were watching the outline. Every good illusion substitutes one invariant for another.
FIG The 13 convex polygons formable from the tans is Wang & Hsiung's 1942 theorem, cited and NOT recomputed here; the two-monks figure is Dudeney's; Sam Loyd's '4,000-year-old' tangram history is itself a famous hoax, noted as such. The AVAN inverse — check which quantity you were conserving: area was conserved, outline was not, and you were watching the outline. Every good illusion substitutes one invariant for another.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN