Unit IV Notes: Five-Color Overflow in IV.11 & Heronian Integer Incircle¶
Chromatic Register Analysis & Exact Cyclotomic Algebra
Solving the Five-Edge Pentagon Paradox on a Four-Ink Press¶
Book IV, Proposition 11 (Pickering p. 137, Leaf 171) poses a crisp combinatorial test for Oliver Byrne: a regular pentagon has five distinct perimeter chords, yet the Chiswick Press palette has only four inks (red, blue, yellow, black), and the inscribed Golden Triangle already occupies solid yellow ()!
Byrne solves this register pressure cleanly:
Four sides receive solid inks (, , , ), and
The fifth side is printed as red dotted (), keeping all five chords individually addressable in the
Q.E.F.conclusion without adding a fifth press run.
Exact Symbolic Coordinates in geometor.byrne¶
Heronian ``13-14-15`` Triangle in ``IV.4``:
geometor.byrne.constructions.build_iv4()places $B = (0, 0)$, $C = (14, 0)$, $A = (5, 12)$, yielding exact integer side lengths $(13, 14, 15)$, area $84$, semiperimeter $21$, integer incenter $I = (6, 4)$, and integer inradius $r = 4$ over $\mathbb{Q}$.Pentagonal Radical Tower in ``IV.10`` & ``IV.11``:
geometor.byrne.constructions.build_iv11()constructs all five vertices on the unit circle using exact nested radicals $\cos 72^\circ = \frac{\sqrt{5}-1}{4}$ and $\sin 72^\circ = \frac{\sqrt{10 + 2\sqrt{5}}}{4}$, verifying symbolically that all five side quadrances equal $\frac{5 - \sqrt{5}}{2}$ and diagonal-to-side quadrance equals $\phi^2 = \frac{3 + \sqrt{5}}{2}$.