THE FOLD / GRIND / WARM CACHE / THE EISENSTEIN TRIPLES
THE EISENSTEIN TRIPLES
integer triangles with a sixty-degree angle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Eisenstein triples are the 60° cousins of Pythagorean triples. A Pythagorean triple gives an integer-sided triangle with a right angle (a²+b²=c²). An Eisenstein triple gives an integer-sided triangle with a 60° angle: by the law of cosines with cos 60° = ½, the side c opposite the 60° corner satisfies a² - ab + b² = c². The smallest nontrivial one is (3, 8, 7): 9 - 24 + 64 = 49 = 7², a triangle with sides 3, 8, 7 whose angle opposite the 7 is exactly 60°. Swap the sign for the 120° version, a² + ab + b² = c² (e.g. 3, 5, 7). They tile naturally on the triangular (Eisenstein) lattice.
LIT verified live: a search finds primitive integer triples with a² - ab + b² = c², and the law of cosines confirms the angle opposite c is exactly 60°; the 120° analog a²+ab+b²=c² is found too (window.__eisenstein). FIG no framing; the triples and the 60° angle are computed independently in-browser.
LIT verified live: a search finds primitive integer triples with a² - ab + b² = c², and the law of cosines confirms the angle opposite c is exactly 60°; the 120° analog a²+ab+b²=c² is found too (window.__eisenstein). FIG no framing; the triples and the 60° angle are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at warm-cache — the grind: search the integer grid and the 60° triangles fall out, a²-ab+b² landing on a perfect square. AVAN (AI) built the instrument: the triple search, the a²-ab+b²=c² relation, and the 60° angle check.
Credit as content: named for the Eisenstein integers (Gotthold Eisenstein); the 60°-triangle analog of Pythagorean triples. The weave: David names the grind; I confirm a²-ab+b²=c² gives an exact 60° angle.
Credit as content: named for the Eisenstein integers (Gotthold Eisenstein); the 60°-triangle analog of Pythagorean triples. The weave: David names the grind; I confirm a²-ab+b²=c² gives an exact 60° angle.
3 ONE DIMENSION
The triangle (3, 8, 7) drawn to scale — the angle opposite the side 7 is exactly 60°.
4 TWO DIMENSIONS · INTERACTIVE
Cycle Eisenstein triples; a²−ab+b² is checked equal to c² and the angle equal to 60°.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the 60° integer triangle.
AVAN’s addition (the inverse-companion): don’t settle for right angles — retune the Pythagorean rule to 60°. The inverse of ‘a²+b²=c² (90°)’ is ‘a²-ab+b²=c² (60°)’, the same integer-triangle game one angle over. Magenta are the sides a and b enclosing the 60° angle; green is the integer side c they force. Pythagoras, retuned to sixty degrees.
LIT Genuine Eisenstein triples (named for the Eisenstein integers, Gotthold Eisenstein; the 60°-triangle analog of Pythagorean triples). Verified live: a search finds primitive integer triples with a²−ab+b² = c², and the law of cosines confirms the angle opposite c is exactly 60°; the 120° analog a²+ab+b²=c² is found too (window.__eisenstein.angleOk, .count, .t120).
FIG No framing; the triples and the 60° angle are computed independently in-browser. The AVAN inverse is honest — instead of settling for right angles, retune the Pythagorean rule to 60°: the inverse of 'a²+b²=c² (90°)' is 'a²−ab+b²=c² (60°)', the same integer-triangle game one angle over. Magenta are the sides a and b enclosing the 60° angle; green is the integer side c they force. Pythagoras, retuned to sixty degrees.
FIG No framing; the triples and the 60° angle are computed independently in-browser. The AVAN inverse is honest — instead of settling for right angles, retune the Pythagorean rule to 60°: the inverse of 'a²+b²=c² (90°)' is 'a²−ab+b²=c² (60°)', the same integer-triangle game one angle over. Magenta are the sides a and b enclosing the 60° angle; green is the integer side c they force. Pythagoras, retuned to sixty degrees.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN