Unit VII: Pointer-Free Visual Pseudocode — 3-Way Rosetta Analysis

Side-by-Side Formalism Alignment: Heath (1908) · Byrne (1847) · GEOMETOR DAG

1. Formal Grammar of Byrne’s 1847 Visual Pseudocode

In geometor.byrne.pseudocode, every proposition in Oliver Byrne's 1847 treatise is modeled as an abstract syntax tree (ByrneProgram) over typed pass-by-value geometric literals:

Program     ::= PhaseBlock+
        PhaseBlock  ::= ("GIVEN" | "SOUGHT" | "CONSTRUCTION" | "DEMONSTRATION" | "QED") Statement+
        Statement   ::= (VisualToken | Operator | Connector)+ ("[" Justification+ "]")?
        VisualToken ::= Sort "(" ColorExpr ("," StrokeMode)? ("," FillMode)? ")" ("²")?
        Sort        ::= line | angle | tri | sq | rect | arc | circle | mag
        ColorExpr   ::= red | blue | yellow | black | Color "+" Color | Color "/" Color "/" Color
        StrokeMode  ::= solid | dashed | dotted | solid-dotted
        Operator    ::= "=" | "+" | "-" | "⊏" | "⊐" | "∥" | "⊥" | "∷" | "×" | "∟" | "2∟" | "∴" | "∵"
        

2. Rosetta Case I: I.1 (Equilateral Triangle, G.84)

• Heath / Classical Alphabetic Prose (179 words, 31 vertex-pointer lookups): “With centre A and distance AB let the circle BCD be described; again, with centre B and distance BA let the circle ACE be described; and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined…”

• Byrne 1847 Visual Pseudocode (22 words, 0 pointers, 6 registers): Describe and ; draw and ([I.post.3], [I.post.1]). Then = ([I.def.15]) and = ([I.def.15]), ∴ = = ([C.N.1]).

• GEOMETOR Formal Pseudocode (``elements`` / ``model``):

  c_ab = model.construct_circle(A, B, by="I.post.3")
          c_ba = model.construct_circle(B, A, by="I.post.3")
          C = model.intersect(c_ab, c_ba, side_of=(A, B, +1), ID="C")
          assert quadrance(A, C) == quadrance(A, B) == quadrance(B, C)
          

3. Rosetta Case II: I.5 (Pons Asinorum, G.91)

• Heath (329 words, 62 vertex-pointer lookups): Requires tracking overlapping triangles AFC, AGB, BFC, CGB and three-letter angles ABG, ACF, CBG, BCF, ABC, ACB.

• Byrne 1847 (32 words, 0 pointers, 4 :term:`congruence-twin <congruence-twin>` pairs): + = + around common ([C.N.2]) $\implies$ = and = ([I.4]) $\implies$ remaining base angles = ([C.N.3]).

• GEOMETOR Formal Pseudocode:

  G = model.intersect(line_AC, circle(A, F), beyond=(A, C), ID="G")
          assert spread(B, A, B, C) == spread(C, A, C, B)
          

4. Rosetta Case III: I.47 (Pythagorean Theorem, G.140)

• Heath (314 words, 64 vertex-pointer lookups): Refers repeatedly to four-letter squares BDEC, ABFG, ACKH and parallelograms BLMD, CLME.

• Byrne 1847 (25 words, 0 pointers, 8.0% verbal load): = 2 × windmill triangle = ([I.4], [I.41]), and = 2 × symmetric windmill triangle = , ∴ + = + ([C.N.2]).

• GEOMETOR Formal Pseudocode:

  assert spread(A, B, A, C) == 1
          assert quadrance(B, C) == quadrance(A, B) + quadrance(A, C)
          

5. Rosetta Cases IV–VI: II.11, IV.10–IV.11 & Book V V.16

• Golden Section & Pentagon (``II.11`` → ``IV.10`` → ``IV.11``): Byrne links colored area subtraction ( = in II.11) to tangent-secant power (² in IV.10) and five-chord cyclotomic symmetry (, , , , in IV.11), mirrored in SymPy by the algebraic tower $\mathbb{Q} \subset \mathbb{Q}(\phi) \subset \mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})$.

• Book V Colored Ratio Calculus (``V.16`` *Alternando*, ``G.276``): : ∷ : $\implies$ : ∷ : (7.1% verbal load, connecting directly to geometor/codes).