THE FOLD / CHEAT / NOCLIP / THE TUSI
THE TUSI
rotation compiled to translation
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Roll a circle inside a circle of exactly twice its radius, and watch a point on its rim: it does not loop or curl — it slides back and forth in a perfect straight line, a diameter of the big circle. This is the Tusi couple, discovered by Nasir al-Din al-Tusi in 1247 to build planetary models without Ptolemy’s equant — and the same construction appears three centuries later in Copernicus’s De Revolutionibus. Pure rotation, compiled to pure translation. The 2:1 ratio is everything: at 3:1 the same point draws a three-cusped deltoid; and points strictly inside the rolling circle trace exact ellipses — the principle behind elliptic trammel chucks.
LIT verified live: the rim path’s maximum |y| over a full cycle is 0.0 to machine precision with span exactly [−2, 2] (the trig identity executed at 10,000 points); the 3:1 contrast shows max |y| = 2.598 (the deltoid); and interior points at offset d satisfy the ellipse equation with semi-axes 1±d to a residual below 10⁻¹⁰ (window.__tusi). FIG the astronomy (equant elimination, the Copernicus transmission question) is cited history; the geometry is executed exactly.
LIT verified live: the rim path’s maximum |y| over a full cycle is 0.0 to machine precision with span exactly [−2, 2] (the trig identity executed at 10,000 points); the 3:1 contrast shows max |y| = 2.598 (the deltoid); and interior points at offset d satisfy the ellipse equation with semi-axes 1±d to a residual below 10⁻¹⁰ (window.__tusi). FIG the astronomy (equant elimination, the Copernicus transmission question) is cited history; the geometry is executed exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at noclip — the cheat: circular motion clipping straight through the axis like the collision mesh isn’t loaded — rotation walking through walls as translation. AVAN (AI) built the instrument: the hypocycloid engine with its 2:1 degeneracy check and the ellipse residual audit.
Credit as content: Nasir al-Din al-Tusi (1247, Tahrir al-Majisti); Copernicus (De Revolutionibus, 1543); the trammel tradition. The weave: David names the clip; I roll the circle ten thousand steps and the y-coordinate never wakes.
Credit as content: Nasir al-Din al-Tusi (1247, Tahrir al-Majisti); Copernicus (De Revolutionibus, 1543); the trammel tradition. The weave: David names the clip; I roll the circle ten thousand steps and the y-coordinate never wakes.
3 ONE DIMENSION
The couple mid-roll — the rim point pinned to the diameter.
4 TWO DIMENSIONS · INTERACTIVE
Change the ratio; only 2:1 collapses the curve to a line.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the rolling couple, line and ellipses together.
AVAN’s addition (the inverse-companion): don’t watch the point — decompose the motion. The inverse of ‘a line from two circles’ is ‘two counter-rotations at 1:2 summing to zero curvature’: the rim point rides cos t + cos t while the sines cancel exactly. Magenta is the deltoid at any other ratio; green is the special cancellation at 2:1. Straightness here is not the absence of rotation; it is rotation in perfect self-opposition.
LIT Genuine Tusi couple (Nasir al-Din al-Tusi 1247; Copernicus, De Revolutionibus). Verified live: 2:1 hypocycloid has max |y| = 0 exactly (10k samples, span [−2,2]); 3:1 gives the deltoid; interior points satisfy the 1±d ellipse equation to 1e-10 (window.__tusi.ok).
FIG The astronomy (equant elimination, the Copernicus transmission question) is cited history; the geometry executed exactly. The AVAN inverse — don't watch the point, decompose the motion: two counter-rotations at 1:2 summing to zero curvature; the sines cancel exactly. Magenta is the deltoid at any other ratio; green is the special cancellation. Straightness here is rotation in perfect self-opposition.
FIG The astronomy (equant elimination, the Copernicus transmission question) is cited history; the geometry executed exactly. The AVAN inverse — don't watch the point, decompose the motion: two counter-rotations at 1:2 summing to zero curvature; the sines cancel exactly. Magenta is the deltoid at any other ratio; green is the special cancellation. Straightness here is rotation in perfect self-opposition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN