Unit III — Inverse Reduction, Polygonal Gnomons, Same/Other & Sum of Cubes — Structural Summary

Figurate Arithmetical Geometry & Gnomonic Identities · Book II, Chs. 1–2, 6–20

Nicomachus II.20 — Sum of Consecutive Cubes Equals the Squared Triangle 1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)² = 10² = 100 · Odd-block sums & L-gnomon square dissection 2³ = 8 3³ = 27 4³ = 64 0 T₁=1 T₂=3 T₃=6 T₄=10 Nicomachus's Consecutive Odd-Block Theorem (II.20) k = 1 : 1 = 1 = 1³ k = 2 : 3 + 5 = 8 = 2³ k = 3 : 7 + 9 + 11 = 27 = 3³ k = 4 : 13 + 15 + 17 + 19 = 64 = 4³ k = 5 : 21 + 23 + 25 + 27 + 29 = 125 = 5³ Why Shell k Has Exact Area k³: Tₖ² − Tₖ₋₁² = (Tₖ − Tₖ₋₁) · (Tₖ + Tₖ₋₁) = k · k² = k³ (By II.12, two consecutive triangles Tₖ₋₁ + Tₖ = k²!)
Book II, Ch. 20: Odd blocks summing to k^3 and L-gnomons of width k tiling the square of side T_n.

Architectural Thesis

Unit III completes Nicomachus’s philosophy of Equality by proving the inverse reduction theorem (II.1–2) and then constructs the entire hierarchy of plane and solid figurate numbers from alpha-pebble gnomons (II.6–20):

  1. Inverse Reduction to Equality (II.1–2): Subtracting \((x, y, z) \mapsto (x, y-x, z-x-2(y-x))\) and sorting if needed reduces every geometric progression uniquely back to (1, 1, 1).

  2. Universal Polygonal Gnomon & Triangle Decomposition Laws (II.6–12): Summing arithmetic progressions with difference \(d = s - 2\) builds all \(s\)-gonal numbers \(P(s, n) = n + (s-2)\frac{n(n-1)}{2}\), each decomposing into elementary triangles via P(s, n) = Tn + (s - 3)Tn-1.

  3. Plato’s Same & Other and Nicomachus’s Cube Theorem (II.16–20): Odd gnomons build Squares n2 (Same), even gnomons build Heteromecics n(n+1) (Other), and consecutive odd blocks of length k sum to k3, proving \(\sum_{k=1}^n k^3 = T_n^2\).

Summary of Figurate Sequences

Figurate Class

Gnomon \(g_k\)

Closed Formula

Initial Terms (\(n = 1, \dots, 6\))

Triangular (s = 3)

\(k\) (\(d = 1\))

\(T_n = \frac{n(n+1)}{2}\)

\(1, 3, 6, 10, 15, 21\)

Square / Same (s = 4)

\(2k - 1\) (\(d = 2\))

\(S_n = n^2 = T_n + T_{n-1}\)

\(1, 4, 9, 16, 25, 36\)

Pentagonal (s = 5)

\(3k - 2\) (\(d = 3\))

\(\frac{n(3n - 1)}{2} = T_n + 2T_{n-1}\)

\(1, 5, 12, 22, 35, 51\)

Heteromecic / Other

\(2k\) (evens)

\(H_n = n(n+1) = 2 T_n\)

\(2, 6, 12, 20, 30, 42\)

Cubes from Odd Blocks

\(k\) odds after \(T_{k-1}\)

\(k^3,\quad \sum_{k=1}^n k^3 = T_n^2\)

\(1, 8, 27, 64, 125, 216\)