Unit III · Executive Geometric Summary¶
Core Epistemological & Algebraic Theses (Harmonices Mundi Book I, 1619)¶
Purpose of Euclid Book X Recovered: Against Petrus Ramus, Kepler identifies the architectural telos of Euclid’s classification of quadratic irrationals: establishing a rigorous hierarchy of degrees of geometric knowability (scibilitas) that separates constructible regular polygons from non-constructible ones.
Cyclotomic Minimal Polynomials of $2\cos(2\pi/n)$: In
geometor.kepler.harmonices_geometry.classify_polygon_scibilitas, every regular $n$-gon studied by Kepler is classified by its exact minimal polynomial over $\mathbb{Z}$:
$n$ |
Polygon |
Minimal Polynomial of $x = 2\cos(2\pi/n)$ |
Deg |
Kepler’s Book I Classification |
|---|---|---|---|---|
$3$ |
Equilateral Triangle |
$x + 1 = 0$ ($s_3^2 = 3R^2$) |
$1$ |
III. Effabilis potentiâ tantùm |
$4$ |
Square |
$x = 0$ ($s_4^2 = 2R^2$) |
$1$ |
III. Effabilis potentiâ tantùm |
$5$ |
Regular Pentagon |
$x^2 + x - 1 = 0$ |
$2$ |
VIII. Elasson / Mizon over $\phi$ |
$6$ |
Regular Hexagon |
$x - 1 = 0$ ($s_6 = R$) |
$1$ |
II. Effabilis longitudine |
$7$ |
Regular Heptagon |
$x^3 + x^2 - 2x - 1 = 0$ |
$3$ |
*Inscibilis* (Prop. XLV cubic) |
$8$ |
Regular Octagon |
$x^2 - 2 = 0$ |
$2$ |
VI–VIII. Bisected square |
$9$ |
Regular Nonagon |
$x^3 - 3x + 1 = 0$ |
$3$ |
*Inscibilis* (Trisected trigon) |
$10$ |
Regular Decagon |
$x^2 - x - 1 = 0$ |
$2$ |
VI. Apotome ($s_{10} = R/\phi$) |
$11$ |
Regular Hendecagon |
$x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1 = 0$ |
$5$ |
*Inscibilis* (Irreducible quintic) |
$12$ |
Regular Dodecagon |
$x^2 - 3 = 0$ |
$2$ |
VI–VIII. Bisected hexagon |
$15$ |
Regular Pentadecagon |
$x^4 - x^3 - 4x^2 + 4x + 1 = 0$ |
$4$ |
VIII. Compounded $3$- & $5$-gon |
$16$ |
Regular Hexadecagon |
$x^4 - 4x^2 + 2 = 0$ |
$4$ |
VIII. Iterated quadratic tower |
Kepler’s Anticipation of Galois Multiplicity: In Proposition XLV, Kepler points out not only that Jost Bürgi's chord polynomial $y^6 - 7y^4 + 14y^2 - 7 = 0$ has cubic degree in $y^2$, but also that its three positive roots $2\sin(\pi/7), 2\sin(2\pi/7), 2\sin(3\pi/7)$ are algebraically indistinguishable conjugates over $\mathbb{Q}$—remarkably foreshadowing Galois-group symmetry two centuries before Galois and Wantzel (1837).