Unit III Notes: Exact $\mathbb{Q}(\phi)$ Coordinates & Half-Filled Woodblocks

Typographic & Symbolic Commentary on Book II

Why Byrne Invented Half-Filled Squares in II.4

Inspection of Pickering pp. 56–57 (Leaves 90–91) shows a striking Chiswick Press graphic device: instead of printing the two diagonal sub-squares in solid ink, Byrne fills only the upper-left triangle of each square while outlining the boundary (, ). Why? Because in II.4 Euclid first proves that the two sub-triangles cut by the diagonal are isosceles ([I.6]) before concluding that the four-sided figures are squares! Half-filling makes both the constituent triangle and the enclosing square simultaneously visible in a single woodblock impression.

Symbolic Verification in $\mathbb{Q}(\phi)$

In geometor.byrne.constructions.build_ii11(), setting $|AB| = 1$ yields exact coordinates in $\mathbb{Q}(\sqrt{5})$:

\[|EB| = \frac{\sqrt{5}}{2}, \qquad |AH| = |AF| = \frac{\sqrt{5} - 1}{2} = \phi^{-1}, \qquad |HB| = 1 - \phi^{-1} = \frac{3 - \sqrt{5}}{2} = \phi^{-2}.\]

Both the blue golden square < A F G H > and the yellow complement rectangle < H B D K > simplify in SymPy to the exact irrational area $\frac{3 - \sqrt{5}}{2} = 2 - \phi$, with difference identically 0.