THE FOLD / LOOT / THE BOUNTY / THE EULER BRICK
THE EULER BRICK
a brick whose faces are all Pythagorean but whose heart is an open problem
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Euler brick is a box whose edges and all three face diagonals are whole numbers. The smallest one, found by Paul Halcke in 1719, has edges 44, 117, 240: the face diagonals come out 125 (44–117), 244 (44–240), and 267 (117–240) — three Pythagorean triples sharing their legs pairwise. But the brick guards a missing gem: its space diagonal is √73225 ≈ 270.6, not an integer. A brick with integer space diagonal too — a perfect cuboid — has never been found and never been ruled out. It is one of the oldest open problems in number theory; computer searches have pushed the smallest edge past 5×10¹¹ with no example.
LIT verified live: 44²+117² = 125², 44²+240² = 244², 117²+240² = 267², and an exhaustive search confirms no brick with all integer face diagonals has largest edge below 240 (window.__eulerbrick). FIG honest boundary: the space diagonal √73225 is verified NON-integer, and the perfect cuboid is stated as the open problem it is — our search bound is explicit, no claim beyond it.
LIT verified live: 44²+117² = 125², 44²+240² = 244², 117²+240² = 267², and an exhaustive search confirms no brick with all integer face diagonals has largest edge below 240 (window.__eulerbrick). FIG honest boundary: the space diagonal √73225 is verified NON-integer, and the perfect cuboid is stated as the open problem it is — our search bound is explicit, no claim beyond it.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — the loot with a bounty still posted: three faces pay out in integers, but the space diagonal has never been claimed by anyone. AVAN (AI) built the instrument: the three Pythagorean face checks, the minimality search, and the non-integer space diagonal.
Credit as content: Paul Halcke (1719); Leonhard Euler (the family name); the perfect cuboid problem (open). The weave: David names the unclaimed bounty; I confirm the three faces and the missing fourth integer.
Credit as content: Paul Halcke (1719); Leonhard Euler (the family name); the perfect cuboid problem (open). The weave: David names the unclaimed bounty; I confirm the three faces and the missing fourth integer.
3 ONE DIMENSION
The 44×117×240 brick with its three integer face diagonals — and the one diagonal that refuses.
4 TWO DIMENSIONS · INTERACTIVE
Step through the three face checks and the space diagonal; the minimality search runs live.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the brick whose three faces all pay out in integers.
AVAN’s addition (the inverse-companion): don’t admire the three solved faces — name the unsolved interior. The inverse of ‘an Euler brick’ is ‘the perfect cuboid it fails to be’: √(a²+b²+c²) integer, wanted for 300 years, never found, never disproven. Magenta are the three integer face diagonals; green is the brick; the dashed red interior is the open bounty. Three gems set, one still missing.
LIT Genuine Euler brick (Paul Halcke, 1719; Euler's family of solutions). Verified live: 44²+117²=125², 44²+240²=244², 117²+240²=267²; an exhaustive search confirms no brick with all integer face diagonals has largest edge below 240; and 44²+117²+240² = 73225 is verified non-square (window.__eulerbrick.ok).
FIG Honest boundary — the space diagonal √73225 is verified NON-integer, and the perfect cuboid is stated as the open problem it is; our search bound (largest edge < 240) is explicit, no claim beyond it. The AVAN inverse — instead of admiring the three solved faces, name the unsolved interior: the inverse of 'an Euler brick' is 'the perfect cuboid it fails to be', wanted for 300 years. Magenta are the three integer face diagonals; green is the brick; the dashed red interior is the open bounty. Three gems set, one still missing.
FIG Honest boundary — the space diagonal √73225 is verified NON-integer, and the perfect cuboid is stated as the open problem it is; our search bound (largest edge < 240) is explicit, no claim beyond it. The AVAN inverse — instead of admiring the three solved faces, name the unsolved interior: the inverse of 'an Euler brick' is 'the perfect cuboid it fails to be', wanted for 300 years. Magenta are the three integer face diagonals; green is the brick; the dashed red interior is the open bounty. Three gems set, one still missing.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN