Unit III Summary: Geometric Algebra Without Symbolic Variables¶
How Byrne’s Colored Areas Realize Book II Polynomial Identities
Book II of Euclid’s Elements establishes quadratic identities—distributivity (II.1), $(a+b)^2$ (II.4), difference of squares and completing the square (II.5–II.6), the Law of Cosines (II.12–II.13), and quadrature of any rectilinear figure (II.14)—entirely through planar dissection without numerical coordinates or ratio theory:
In
II.4, Byrne’s diagonal half-filled squares (, ) visually separate the quadratic terms $a^2, b^2$ from the solid yellow cross-terms $2ab$ ().In
II.11, Euclid solves $x^2 + a x = a^2$ (cutting $a$ in extreme and mean ratio, $x = a / \phi$) by subtracting a shared rectangular strip from two equal composite regions—a proof that feeds directly into the $36^\circ\text{–}72^\circ\text{–}72^\circ$ Golden Triangle (IV.10) and regular pentagon (IV.11).