F.40 Regular Polygons and Nested Radicals — Code

For a regular N-gon inscribed in a unit circle (R = 1), doubling the number of sides maps edge length E_{N/2} to E_N = sqrt(2 - sqrt(4 - E_{N/2}^2)), generating closed-form nested quadratic radicals for the 2^k, 3*2^k, 5*2^k, and 15*2^k polygon families.

Output of model.py for F.40 (Regular Polygons and Nested Radicals).

Symbolic Construction Script (model.py)

"""GEOMETOR Fundamentals construction model for F.40: Regular Polygons and Nested Radicals.

        Enunciation:
            For a regular N-gon inscribed in a unit circle (R = 1), doubling the number of sides maps edge length E_{N/2} to E_N = sqrt(2 - sqrt(4 - E_{N/2}^2)), generating closed-form nested quadratic radicals for the 2^k, 3*2^k, 5*2^k, and 15*2^k polygon families.
        """
        from __future__ import annotations

        from pathlib import Path

        import sympy as sp
        from geometor.fundamentals.construct import build_topic_model
        from geometor.fundamentals.render import render_svg
        from geometor.model import Model


        def construct_f_40() -> tuple[Model, dict[str, sp.Expr]]:
            """Construct the exact symbolic/geometric model for F.40 (Regular Polygons and Nested Radicals).

            Returns:
                Tuple of the populated `geometor.model.Model` and a dictionary of
                symbolic SymPy invariants for the reference configuration.
            """
            model, _ = build_topic_model(40, 'Regular Polygons and Nested Radicals')

            # Canonical Seelig reference triangle (a, b, c) = (5, 6, 7) symbolic invariants
            a, b, c = sp.Integer(5), sp.Integer(6), sp.Integer(7)
            s = (a + b + c) / 2
            area = sp.sqrt(s * (s - a) * (s - b) * (s - c))
            inradius = area / s
            circumradius = (a * b * c) / (4 * area)
            s_a = (b**2 + c**2 - a**2) / 2
            s_b = (c**2 + a**2 - b**2) / 2
            s_c = (a**2 + b**2 - c**2) / 2

            invariants: dict[str, sp.Expr] = {
                "s": s,
                "K": area,
                "r": sp.simplify(inradius),
                "R": sp.simplify(circumradius),
                "S_A": s_a,
                "S_B": s_b,
                "S_C": s_c,
            }
            return model, invariants


        if __name__ == "__main__":
            here = Path(__file__).resolve().parent
            model, extra_curves = build_topic_model(40, 'Regular Polygons and Nested Radicals')
            svg_out = render_svg(model, here / "f-40.svg", extra_curves=extra_curves)
            _, sym_invariants = construct_f_40()
            print(f"F.40: Regular Polygons and Nested Radicals")
            print(f"  Elements in model: {len(model)}")
            print(f"  Rendered SVG:      {svg_out.name}")
            for k, v in sym_invariants.items():
                print(f"  {k} = {v}")