UNIT III — MATHEMATICAL & PROPORTIONAL SUMMARY¶
Exact Symbolic Classification, Means & Golden Section Identities
1. Boethian Fivefold Classification (classify_boethian_ratio)¶
Given reduced coprime positive integers $p \neq q$, let $M = \max(p, q)$, $m = \min(p, q)$, and write $M = q_{\text{div}} m + r$ with $0 \le r < m$:
Quotient / Remainder |
Greater Genus ($p > q$) |
Lesser Genus ($p < q$) |
Canonical Example |
|---|---|---|---|
$r = 0, \; q_{\text{div}} \ge 2$ |
submultiplex |
$12:4 = 3:1$ (tripla) |
|
$q_{\text{div}} = 1, \; r = 1$ |
subsuperparticularis |
$9:6 = 3:2$ (sesquialtera) |
|
$q_{\text{div}} = 1, \; r > 1$ |
subsuperpartiens |
$5:3$ (superbipartiens tertias) |
|
$q_{\text{div}} \ge 2, \; r = 1$ |
multiplex superparticularis |
submultiplex superparticularis |
$5:2$ (dupla sesquialtera) |
$q_{\text{div}} \ge 2, \; r > 1$ |
multiplex superpartiens |
submultiplex superpartiens |
$11:4$ (dupla supertripartiens quartas) |
2. Semicircle Three Means & Cross-Ratio (construct_three_means)¶
For distinct positive lengths $a, b > 0$:
$$A = \frac{a + b}{2}, \qquad G = \sqrt{ab}, \qquad H = \frac{2ab}{a + b}, \qquad G^2 = A \cdot H, \qquad A - G = \frac{(\sqrt{a} - \sqrt{b})^2}{2} > 0$$
Collinear points $O=(0,0)$, $P_b=(b,0)$, $P_H=(H,0)$, $P_a=(a,0)$ form an exact harmonic range with projective cross-ratio $(P_b, P_a; P_H, O) = -1$.
3. Extreme and Mean Ratio of $10$ (construct_extreme_and_mean_ratio)¶
Continuous Proportion: $10 : (\sqrt{125} - 5) = (\sqrt{125} - 5) : (15 - \sqrt{125}) = \phi = \frac{1 + \sqrt{5}}{2}$.
Euclidean Rectangle Identity (:just:`II.11`): $10 \cdot (15 - \sqrt{125}) = (\sqrt{125} - 5)^2 = 150 - 50\sqrt{5}$.
First Marvelous Effect (:just:`XIII.1`): $(\text{Major} + 5)^2 = 125 = 5 \cdot 5^2$.