Unit II — Five Species of Inequality & the Equality Unfolding Tree — Structural Map

Ratio Taxonomy & Ternary Tree Generation · Book I, Chs. 17–23

Nicomachus I.23 & II.1–2 — Unfolding All Inequality from Equality (1, 1, 1) Direct rule N₁: (a, a+b, a+2b+c) · Anastrophe rule N₂ = N₁R · Inverse reduction N₁⁻¹ (gold dashed) N₁ N₁ (direct) N₂ (anastrophe) N₁ Book II.1–2 Inverse Subtraction N₁⁻¹ returns every ratio to (1, 1, 1) (1, 1, 1) · α, α, α Equality · 1:1 (1, 2, 4) · α, β, δ 1. Multiplex · 2:1 (1, 3, 9) · α, γ, ϑ Multiplex · 3:1 (4, 6, 9) · δ, ϛ, ϑ 2. Superparticularis · 3:2 (1, 4, 16) Multiplex · 4:1 (9, 12, 16) Epitritos · 4:3 (4, 10, 25) · δ, ι, κε 4. Mult. superpartic. · 5:2 (9, 15, 25) · ϑ, ιε, κε 3. Superpartiens · 5:3 (9, 24, 64) 5. Mult. superpartiens · 8:3
Every coprime rational ratio q : p > 1 appears uniquely as a three-term geometric progression (p^2, pq, q^2) unfolded from Equality (1, 1, 1).

Architectural Thesis

Unit II transitions from absolute to relative quantity (pros ti poson), proving that every rational inequality is classified into five exact species and generated deterministically from Equality:

  1. Exhaustive Fivefold Classification of Ratios (I.17–22): Every reduced rational ratio \(a : b\) (\(a > b\), \(\gcd(a, b) = 1\)) written via Euclidean division \(a = m \cdot b + r\) (\(m \ge 1, 0 \le r < b\)) falls into one and only one of five species depending on whether \(m = 1\) vs. \(m \ge 2\) and \(r = 0\), \(r = 1\), or \(r \ge 2\).

  2. Nicomachus’s Rule of Three Terms (I.23): Applying the linear operator \(L(a, b, c) = (a, a+b, a+2b+c)\) directly (\(D = L\)) and after reversal (\(R = L \circ \text{rev}\)) to Equality (1, 1, 1) unfolds a complete binary tree of three-term geometric progressions \((p^2, pq, q^2)\) isomorphic to the Calkin–Wilf / Stern–Brocot tree of positive rationals!

Taxonomy of the Five Species

Species (Greek / Latin)

Reduced Form

Canonical Examples

Birthplace in the Equality Tree

Multiple (pollaplasios / multiplex)

\(m : 1\) (\(m \ge 2\))

\(2:1, 3:1, 4:1\)

Direct spine from (1,1,1): (1,2,4), (1,3,9), (1,4,16)

Superparticular (epimorios / superparticularis)

\((n + 1) : n\) (\(n \ge 2\))

\(3:2, 4:3, 5:4, 9:8\)

Reversed child of Multiple: (4,6,9), (9,12,16)

Superpartient (epimerēs / superpartiens)

\((n + r) : n\) (\(1 < r < n\))

\(5:3, 7:4, 9:5\)

Reversed child of Superparticular / higher node: (9,15,25)

Multiple Superparticular (pollaplasiepimorios)

\((mn + 1) : n\) (\(m, n \ge 2\))

\(5:2, 7:3, 7:2\)

Direct child of Superparticular: (4,10,25)

Multiple Superpartient (pollaplasiepimerēs)

\((mn + r) : n\) (\(m \ge 2, r > 1\))

\(8:3, 11:4\)

Direct child of Superpartient: (9,24,64)