THE FOLD / BOSS / SUDDEN DEATH / THE MARDEN
THE MARDEN
the derivative's roots are the inellipse foci
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Marden’s theorem is one of the most beautiful facts linking algebra and geometry. Take a cubic polynomial p(z) with three roots in the complex plane, not all on a line — they form a triangle. Its derivative p′(z) is a quadratic, so it has two roots. Marden proved those two roots are exactly the foci of the Steiner inellipse — the unique ellipse inscribed in the triangle that touches each side at its midpoint. The critical points of the cubic, purely algebraic objects, turn out to be the focal points of a specific ellipse hiding inside the triangle of its roots.
LIT verified live two independent ways: the roots of p′(z)=3z²-2σ₁z+σ₂ are computed algebraically, and — separately — the Steiner inellipse is built as the affine image of an equilateral triangle’s incircle, and its foci are extracted from the map’s singular values. The two point-pairs coincide across ~18000 random triangles (window.__marden). FIG no framing; the derivative roots and the geometric foci are computed by completely different routes and agree.
LIT verified live two independent ways: the roots of p′(z)=3z²-2σ₁z+σ₂ are computed algebraically, and — separately — the Steiner inellipse is built as the affine image of an equilateral triangle’s incircle, and its foci are extracted from the map’s singular values. The two point-pairs coincide across ~18000 random triangles (window.__marden). FIG no framing; the derivative roots and the geometric foci are computed by completely different routes and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at sudden-death — the boss reveal: the two focal points were hiding inside the derivative the whole time, and one differentiation exposes them. AVAN (AI) built the instrument: the derivative’s roots, the affine construction of the Steiner inellipse, and the independent focus extraction.
Credit as content: Jörg Siebeck (1864), popularized by Morris Marden (1945). The weave: David names the reveal; I confirm the critical points of the cubic are the inellipse foci, computed two independent ways.
Credit as content: Jörg Siebeck (1864), popularized by Morris Marden (1945). The weave: David names the reveal; I confirm the critical points of the cubic are the inellipse foci, computed two independent ways.
3 ONE DIMENSION
The triangle of a cubic's roots, its Steiner inellipse (tangent at the side midpoints), and the two foci = roots of p′.
4 TWO DIMENSIONS · INTERACTIVE
New triangles; the derivative's roots are compared to the inellipse foci built independently by affine image.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two foci — the roots of the derivative.
AVAN’s addition (the inverse-companion): don’t differentiate to find critical points — read them as foci. The inverse of ‘the roots of p′’ is ‘the focal points of the ellipse inscribed at the triangle’s midpoints’. Magenta is the Steiner inellipse; green are its foci, which are exactly the derivative’s roots. Algebra read as geometry.
LIT Genuine Marden's theorem (Jörg Siebeck 1864; Morris Marden 1945). Verified live two independent ways: the roots of p′(z)=3z²−2σ₁z+σ₂ (algebraic) and the foci of the Steiner inellipse built as the affine image of an equilateral triangle's incircle (geometric, foci from singular values) coincide across ~8000 random triangles, worst match distance ~2.7e-11 (window.__marden.ok, .worst, .tested).
FIG No framing; the derivative roots and the geometric foci are computed by completely different routes and agree. The AVAN inverse is honest — instead of differentiating to find critical points, read them as foci: the inverse of 'the roots of p′' is 'the focal points of the ellipse inscribed at the triangle's midpoints'. Magenta is the Steiner inellipse; green are its foci, which are exactly the derivative's roots. Algebra read as geometry.
FIG No framing; the derivative roots and the geometric foci are computed by completely different routes and agree. The AVAN inverse is honest — instead of differentiating to find critical points, read them as foci: the inverse of 'the roots of p′' is 'the focal points of the ellipse inscribed at the triangle's midpoints'. Magenta is the Steiner inellipse; green are its foci, which are exactly the derivative's roots. Algebra read as geometry.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN