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X₃₆₀¶

HOFSTADTER ZERO POINT

The limit of this point as r approaches 0 is X₃₆₀. The two Hofstadter points, X₃₅₉ and X₃₆₀ are examples of transcendental triangle centers, since they have no trilinear or barycentric representation using only algebraic functions of a,b,c (or sin A, sin B, sin C).

X₃₆₀ (HOFSTADTER ZERO POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_360 = HOFSTADTER ZERO POINT."""

        from __future__ import annotations

        from pathlib import Path

        import sympy.geometry as spg
        from geometor.model import Model

        from geometor.etc.ops import (
            central_line,
            centroid_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            isogonal_conjugate,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00360(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_360 = HOFSTADTER ZERO POINT on reference triangle ABC."""
            X_359 = eval_trilinear_point(model, A, B, C, 'a/A', ID="X_359", classes=["center"])
            X_360 = isogonal_conjugate(model, A, B, C, X_359, ID="X_360", classes=["selected", "target"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_1115 = eval_trilinear_point(model, A, B, C, '( pi  - A)/a', ID="X_1115", classes=["center", "incident"])
            central_line(model, X_2, X_1115, 2, 1115, incident=True)
            return promote_point_to_top(model, X_360)


        def build_model(
            sides: tuple[float, float, float] = (5.0, 6.0, 7.0),
            show_guides: bool = False,
        ) -> Model:
            """Build a `geometor.model.Model` for X_360 on the given reference triangle."""
            model = create_model('X_360 = HOFSTADTER ZERO POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00360(model, A, B, C)
            return model


        if __name__ == "__main__":
            here = Path(__file__).resolve().parent
            model_acute = build_model((5.0, 6.0, 7.0))
            model_acute.save(str(here / "x-00360.json"))
            write_center_svg(
                model_acute,
                here / "x-00360.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X360¶

X₃₆₀¶