← durer / GEOMETOR • durer

modules¶

geometor.durer¶

GEOMETOR Dürer: Exact symbolic constructions & bilingual codex for Underweysung der Messung (1525).

Provides 100% exact SymPy + geometor.model constructions across all Four Books (Vier Bücher) of Albrecht Dürer’s Underweysung der Messung, mit dem Zirckel un[d] Richtscheyt, in Linien Ebnen unnd gantzen Corporen (Nuremberg, 1525; 2nd ed. 1538; Camerarius Latin ed. 1532):

Book I (geometor.durer.conics_perspective): 12-sector Archimedean spirals (Schneckenlinien) and two-view descriptive geometry of conic sections (Elipsis / Eierlini, Parabola / Brennspiegellini, Hiperbole / Gabellini).
Book II (geometor.durer.polygons): Exact comparison of Dürer’s Single-Aperture (“Rusty Compass”) Pentagon with Euclid’s Q(phi) regular pentagon, plus cubic-impossibility and quadrature audits for workshop 7-, 11-, and 13-gons and pi = 25/8.
Book III (geometor.durer.typography): Compass-and-straightedge Roman Antiqua capitals (A, O, Q) in the 10 x 10 square and modular Gothic Textura minuscules.
Book IV (geometor.durer.polyhedra, geometor.durer.conics_perspective): Planar fold-out polyhedral nets for Platonic & Archimedean solids, Delian cube duplication, and central perspective projection onto the glass picture plane.

Submodules¶

Attributes¶

`PHI`
`DURER_BOOK4_SOLIDS_CATALOG`
`EMBEDDED_SVG_STYLE`
`__version__`

Classes¶

`ConcordanceEntry`	Single cross-corpus concordance link from Dürer's Underweysung to GEOMETOR.
`UnderweysungApparatus`	Loaded JSON apparatus registries for `resources/underweysung/`.
`ConicSectionsResult`	Exact symbolic audit of Dürer's Book I two-view orthographic conic sections (Figs. 34–37).
`DelianCubeCertificate`	Symbolic verification of Plato's / Sporus's two-mean-proportionals cube duplication.
`PerspectiveApparatusResult`	Exact rational perspective projection of a 3D cube onto Dürer's Book IV glass frame.
`SpiralVoluteResult`	Exact 12-sector Archimedean spiral (Schneckenlini, Book I, Figs. 7 & 17) over `Q(sqrt(3))`.
`HeptagonWorkshopReport`	Exact symbolic construction & cubic-impossibility audit for Dürer's Book II heptagon.
`PentagonComparisonResult`	Exact symbolic result comparing Dürer's single-aperture pentagon with Euclid's pentagon.
`WorkshopPolygonEntry`	Exact algebraic vs. trigonometric audit for one of Dürer's Book II workshop rules.
`Book4SolidCatalogEntry`	Historical & combinatorial catalog entry for Dürer's Book IV solids (Figs. 29–43).
`PolyhedralFace`	Single regular polygon face in an unfolded 2D planar polyhedral net.
`PolyhedralNet`	Exact 2D planar fold-out net for a Platonic or Archimedean polyhedron.
`RomanLetterAResult`	Exact symbolic construction of Dürer's Roman Antiqua capital `A` in a `10 x 10` square.
`RomanLetterOQResult`	Exact symbolic construction of Dürer's Roman Antiqua `O` or `Q` in a `10 x 10` square.
`TexturaMinusculesResult`	Exact modular construction of Dürer's Gothic Textura minuscules `i`, `n`, `m`.

Functions¶

`default_resources_dir`() → pathlib.Path	Locate `resources/underweysung/` relative to the repository root.
`load_apparatus`([resources_dir]) → UnderweysungApparatus	Load all five JSON apparatus files from `resources/underweysung/`.
`construct_durer_conic_sections`([model]) → ConicSectionsResult	Construct Dürer's Book I elevation + ground-plan + true-section diagram for the ellipse.
`construct_durer_perspective_apparatus`([model]) → PerspectiveApparatusResult	Project a 3D cube in `Q^3` onto Dürer's picture plane `x = 0` from eye `E(-12, 3, 5)`.
`construct_durer_spiral_volute`([turns, model]) → SpiralVoluteResult	Construct Dürer's Book I Archimedean spiral over `Q(sqrt(3))` using 12 angular sectors.
`construct_durer_heptagon_workshop`([model]) → HeptagonWorkshopReport	Construct Dürer's workshop regular heptagon in the unit circle and audit its cubic residual.
`construct_durer_vs_euclid_pentagon`([model]) → PentagonComparisonResult	Construct Dürer's single-aperture pentagon and Euclid's exact pentagon on base `AB`.
`construct_durer_polyhedral_nets`() → dict[str, PolyhedralNet]	Construct all 6 exact symbolic planar polyhedral nets from Dürer's Book IV.
`polyhedral_net_to_model`(net[, model]) → geometor.model.Model	Populate a `geometor.model.Model` with exact vertices, face polygons, and fold/cut seams.
`export_svg`(svg_text, output_path) → pathlib.Path	Write `svg_text` to `output_path` creating parent directories if needed.
`generate_all_codex_figures`(output_dir) → list[pathlib.Path]	Generate all 10 interactive `.euclid-graphic` SVG figures for `docsrc/_static/figures/`.
`render_model_svg`(model, *[, width, height, padding, ...]) → str	Serialize a `Model` into a crisp `.euclid-graphic` SVG string.
`construct_durer_roman_letter_A`([model]) → RomanLetterAResult	Construct Dürer's Roman capital `A` inside the `10 x 10` square `ABCD`.
`construct_durer_roman_letter_O`([model]) → RomanLetterOQResult	Construct Dürer's Roman capital `O` inside the `10 x 10` square.
`construct_durer_roman_letter_Q`([model]) → RomanLetterOQResult	Construct Dürer's Roman capital `Q` inside the `10 x 10` square.
`construct_durer_textura_minuscules`([model]) → TexturaMinusculesResult	Construct Dürer's modular Textura minuscules `i`, `n`, and `m` from unit squares.

Package Contents¶

class geometor.durer.ConcordanceEntry¶

Single cross-corpus concordance link from Dürer’s Underweysung to GEOMETOR.

id: str¶

book: int¶

unit: str¶

durer_locus: str¶

german_term: str¶

latin_1532_term: str¶

english_title: str¶

classification: str¶

sister_repos: tuple[str, ...]¶

algebraic_field: str¶

summary: str¶

class geometor.durer.UnderweysungApparatus¶

Loaded JSON apparatus registries for resources/underweysung/.

sources_manifest: dict[str, Any]¶

concordance: tuple[ConcordanceEntry, ...]¶

persons: dict[str, dict[str, Any]]¶

citations: dict[str, dict[str, Any]]¶

glossary: dict[str, dict[str, Any]]¶

geometor.durer.default_resources_dir() → pathlib.Path¶: Locate resources/underweysung/ relative to the repository root.

geometor.durer.load_apparatus(resources_dir: str | pathlib.Path | None = None) → UnderweysungApparatus¶: Load all five JSON apparatus files from resources/underweysung/.

class geometor.durer.ConicSectionsResult¶

Exact symbolic audit of Dürer’s Book I two-view orthographic conic sections (Figs. 34–37).

model: geometor.model.Model¶

cone_radius: sympy.Integer¶

cone_height: sympy.Integer¶

ellipse_stations: tuple[tuple[int, sympy.Expr, sympy.Expr, sympy.Expr, sympy.Expr], ...]¶

ellipse_semi_major: sympy.Expr¶

ellipse_semi_minor: sympy.Expr¶

ellipse_focal_distance: sympy.Expr¶

ellipse_eccentricity: sympy.Expr¶

ellipse_bilateral_symmetry_verified: bool¶

parabola_focal_length: sympy.Expr¶

hyperbola_semi_transverse: sympy.Integer¶

hyperbola_semi_conjugate: sympy.Integer¶

class geometor.durer.DelianCubeCertificate¶

Symbolic verification of Plato’s / Sporus’s two-mean-proportionals cube duplication.

given_g: sympy.Expr¶

given_e: sympy.Expr¶

first_mean_c: sympy.Expr¶

second_mean_d: sympy.Expr¶

volume_ratio: sympy.Expr¶

proportion_verified: bool¶

class geometor.durer.PerspectiveApparatusResult¶

Exact rational perspective projection of a 3D cube onto Dürer’s Book IV glass frame.

model: geometor.model.Model¶

eye_3d: tuple[sympy.Integer, sympy.Integer, sympy.Integer]¶

picture_plane_x: sympy.Integer¶

vanishing_point: sympy.geometry.Point¶

cube_vertices_3d: dict[str, tuple[sympy.Integer, sympy.Integer, sympy.Integer]]¶

projected_vertices_2d: dict[str, sympy.geometry.Point]¶

harmonic_cross_ratio_3d: sympy.Rational¶

harmonic_cross_ratio_2d: sympy.Expr¶

cross_ratio_invariant_verified: bool¶

delian_cube: DelianCubeCertificate¶

class geometor.durer.SpiralVoluteResult¶

Exact 12-sector Archimedean spiral (Schneckenlini, Book I, Figs. 7 & 17) over Q(sqrt(3)).

model: geometor.model.Model¶

center: sympy.geometry.Point¶

outer_radius: sympy.Integer¶

sector_count: int¶

spiral_points: tuple[sympy.geometry.Point, ...]¶

radii: tuple[sympy.Expr, ...]¶

geometor.durer.construct_durer_conic_sections(model: geometor.model.Model | None = None) → ConicSectionsResult¶

Construct Dürer’s Book I elevation + ground-plan + true-section diagram for the ellipse.

Also computes exact focal and asymptotic invariants for Dürer’s Parabola (Brennspiegellini) and Hiperbole (Gabellini).

geometor.durer.construct_durer_perspective_apparatus(model: geometor.model.Model | None = None) → PerspectiveApparatusResult¶

Project a 3D cube in Q^3 onto Dürer’s picture plane x = 0 from eye E(-12, 3, 5).

Verifies:

Exact rational screen coordinates (y', z') in Q^2 for all 8 cube vertices.
Exact convergence of all 4 depth edges at principal vanishing point V_x = (3, 5).
Exact projective cross-ratio invariance CR(P4', P8'; P6', V_x) = -1 (harmonic range!).
Exact algebraic verification of Plato’s / Sporus’s two mean proportionals (Book IV, Fig. 50) for doubling the cube (g = 1, e = 2 => c = 2^(1/3), d = 2^(2/3), c^3 = 2).

geometor.durer.construct_durer_spiral_volute(turns: int = 2, model: geometor.model.Model | None = None) → SpiralVoluteResult¶

Construct Dürer’s Book I Archimedean spiral over Q(sqrt(3)) using 12 angular sectors.

Dürer divides the outer circle into 12 radial rays (1..12, every 30°) and divides the outer radius R = 12 into 12 * turns equal radial steps r_k = k / turns, plotting P_k = (r_k * cos(k * 30°), r_k * sin(k * 30°)) with 100% exact coordinates in Q(sqrt(3)).

geometor.durer.PHI: sympy.Expr¶

class geometor.durer.HeptagonWorkshopReport¶

Exact symbolic construction & cubic-impossibility audit for Dürer’s Book II heptagon.

model: geometor.model.Model¶

center: sympy.geometry.Point¶

circumcircle: sympy.geometry.Circle¶

equilateral_chord_top: sympy.geometry.Point¶

equilateral_chord_bottom: sympy.geometry.Point¶

radius_midpoint: sympy.geometry.Point¶

workshop_vertices: tuple[sympy.geometry.Point, ...]¶

durer_chord_quadrance: sympy.Rational¶

minimal_polynomial_expr: sympy.Expr¶

minimal_polynomial_at_durer: sympy.Rational¶

eisenstein_prime: int¶

quadrature_circle_diameter_over_side: sympy.Expr¶

quadrature_implied_pi: sympy.Rational¶

workshop_catalog: tuple[WorkshopPolygonEntry, ...]¶

class geometor.durer.PentagonComparisonResult¶

Exact symbolic result comparing Dürer’s single-aperture pentagon with Euclid’s pentagon.

model: geometor.model.Model¶

base_a: sympy.geometry.Point¶

base_b: sympy.geometry.Point¶

durer_vertices: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]¶

euclid_vertices: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]¶

durer_aux_points: dict[str, sympy.geometry.Point]¶

durer_side_quadrances: tuple[sympy.Expr, sympy.Expr, sympy.Expr, sympy.Expr, sympy.Expr]¶

euclid_side_quadrances: tuple[sympy.Expr, sympy.Expr, sympy.Expr, sympy.Expr, sympy.Expr]¶

euclid_diagonal_quadrance: sympy.Expr¶

durer_base_diagonal_quadrance: sympy.Expr¶

durer_shoulder_diagonal_quadrance: sympy.Expr¶

durer_base_spread: sympy.Expr¶

euclid_base_spread: sympy.Expr¶

durer_equilateral: bool¶

durer_equiangular: bool¶

euclid_regular: bool¶

property durer_base_interior_deg: float¶: Numerical evaluation (degrees) of Dürer’s interior angle at A and B.

property base_angle_error_deg: float¶: Signed error (degrees) of Dürer’s base interior angle relative to exact 108°.

property base_angle_error_arcmin: float¶: Signed error (arcminutes) of Dürer’s base interior angle relative to exact 108°.

class geometor.durer.WorkshopPolygonEntry¶

Exact algebraic vs. trigonometric audit for one of Dürer’s Book II workshop rules.

n_sides: int¶

durer_label: str¶

classification: str¶

exact_chord_over_r: sympy.Expr¶

exact_quadrance_over_r2: sympy.Expr¶

true_chord_over_r_float: float¶

relative_chord_error: float¶

central_angle_durer_deg: float¶

central_angle_true_deg: float¶

geometor.durer.construct_durer_heptagon_workshop(model: geometor.model.Model | None = None) → HeptagonWorkshopReport¶

Construct Dürer’s workshop regular heptagon in the unit circle and audit its cubic residual.

In Book II, Figs. 11 & 15, Dürer sets the heptagon chord equal to the half-side of the inscribed equilateral triangle: s_7 = sqrt(3)/2 * R (Q_Durer = 3/4 for R = 1). The true regular heptagon chord quadrance Q_7 = 2*(1 - cos(2*pi/7)) is a root of the Eisenstein-irreducible cubic P_7(q) = q^3 - 7*q^2 + 14*q - 7 = 0, yielding the exact non-zero rational residual P_7(3/4) = -1/64.

geometor.durer.construct_durer_vs_euclid_pentagon(model: geometor.model.Model | None = None) → PentagonComparisonResult¶

Construct Dürer’s single-aperture pentagon and Euclid’s exact pentagon on base AB.

Parameters:: model (Model | None) – Optional existing geometor.model.Model (created with auto_intersect=False if omitted).
Returns:: Exact symbolic coordinates, quadrances, spreads, and regularity audit.
Return type:: PentagonComparisonResult

geometor.durer.DURER_BOOK4_SOLIDS_CATALOG: tuple[Book4SolidCatalogEntry, ...]¶

class geometor.durer.Book4SolidCatalogEntry¶

Historical & combinatorial catalog entry for Dürer’s Book IV solids (Figs. 29–43).

figure_num: int¶

slub_page: int¶

german_name_1525: str¶

english_name: str¶

family: str¶

vertex_config: str¶

vertices: int¶

edges: int¶

faces: int¶

face_summary: str¶

class geometor.durer.PolyhedralFace¶

Single regular polygon face in an unfolded 2D planar polyhedral net.

face_id: str¶

n_gon: int¶

vertices: tuple[tuple[sympy.Expr, sympy.Expr], ...]¶

class geometor.durer.PolyhedralNet¶

Exact 2D planar fold-out net for a Platonic or Archimedean polyhedron.

key: str¶

title: str¶

durer_figure: str¶

family: str¶

algebraic_field: str¶

vertices_3d: int¶

edges_3d: int¶

faces_3d: int¶

faces: tuple[PolyhedralFace, ...]¶

fold_edges: tuple[tuple[tuple[sympy.Expr, sympy.Expr], tuple[sympy.Expr, sympy.Expr]], ...]¶

cut_edges: tuple[tuple[tuple[sympy.Expr, sympy.Expr], tuple[sympy.Expr, sympy.Expr]], ...]¶

unit_edge_verified: bool¶

euler_characteristic: int¶

seam_count_verified: bool¶

non_overlapping_verified: bool¶

geometor.durer.construct_durer_polyhedral_nets() → dict[str, PolyhedralNet]¶

Construct all 6 exact symbolic planar polyhedral nets from Dürer’s Book IV.

Returns:: Mapping from 'tetrahedron', 'cube', 'octahedron', 'dodecahedron', 'icosahedron', and 'cuboctahedron' to verified exact planar nets.
Return type:: dict[str, PolyhedralNet]

geometor.durer.polyhedral_net_to_model(net: PolyhedralNet, model: geometor.model.Model | None = None) → geometor.model.Model¶: Populate a geometor.model.Model with exact vertices, face polygons, and fold/cut seams.

geometor.durer.EMBEDDED_SVG_STYLE: str = ''¶

geometor.durer.export_svg(svg_text: str, output_path: str | pathlib.Path) → pathlib.Path¶: Write svg_text to output_path creating parent directories if needed.

geometor.durer.generate_all_codex_figures(output_dir: str | pathlib.Path) → list[pathlib.Path]¶: Generate all 10 interactive .euclid-graphic SVG figures for docsrc/_static/figures/.

geometor.durer.render_model_svg(model: geometor.model.Model, *, width: int = 760, height: int = 560, padding: float = 48.0, math_bounds: tuple[float, float, float, float] | None = None, labeled_keys: dict[str, str] | None = None, extra_annotations: list[tuple[float, float, str]] | None = None, show_lines: bool = True, show_unlabeled_points: bool = False) → str¶: Serialize a Model into a crisp .euclid-graphic SVG string.

class geometor.durer.RomanLetterAResult¶

Exact symbolic construction of Dürer’s Roman Antiqua capital A in a 10 x 10 square.

model: geometor.model.Model¶

square_corners: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]¶

thick_stroke: sympy.Integer¶

thin_stroke: sympy.Rational¶

outer_serif_radius: sympy.Rational¶

inner_serif_radius: sympy.Rational¶

counter_triangle: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]¶

crossbar_quad: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]¶

serif_centers: dict[str, sympy.geometry.Point]¶

class geometor.durer.RomanLetterOQResult¶

Exact symbolic construction of Dürer’s Roman Antiqua O or Q in a 10 x 10 square.

letter: str¶

model: geometor.model.Model¶

square_corners: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]¶

center_i: sympy.geometry.Point¶

shifted_center_f: sympy.geometry.Point¶

shifted_center_g: sympy.geometry.Point¶

bowl_radius: sympy.Expr¶

circle_f: sympy.geometry.Circle¶

circle_g: sympy.geometry.Circle¶

crossover_points: tuple[sympy.geometry.Point, sympy.geometry.Point]¶

tail_points: tuple[sympy.geometry.Point, ...]¶

class geometor.durer.TexturaMinusculesResult¶

Exact modular construction of Dürer’s Gothic Textura minuscules i, n, m.

model: geometor.model.Model¶

stem_width: sympy.Integer¶

stacked_square_count: int¶

diamond_side: sympy.Expr¶

glyph_polygons: dict[str, list[tuple[tuple[sympy.Expr, sympy.Expr], ...]]]¶

geometor.durer.construct_durer_roman_letter_A(model: geometor.model.Model | None = None) → RomanLetterAResult¶

Construct Dürer’s Roman capital A inside the 10 x 10 square ABCD.

Dürer labels the framing square (rechte fierung) a top-left (0, 10), b top-right (10, 10), c bottom-left (0, 0), d bottom-right (10, 0), with midlines ef (vertical) and gh (horizontal), thick stroke 1 (1/10 of 10), thin stroke 1/3, outer foot serifs of radius 10/7, and inner foot serifs of radius 1/2.

geometor.durer.construct_durer_roman_letter_O(model: geometor.model.Model | None = None) → RomanLetterOQResult¶: Construct Dürer’s Roman capital O inside the 10 x 10 square.

geometor.durer.construct_durer_roman_letter_Q(model: geometor.model.Model | None = None) → RomanLetterOQResult¶: Construct Dürer’s Roman capital Q inside the 10 x 10 square.

geometor.durer.construct_durer_textura_minuscules(model: geometor.model.Model | None = None) → TexturaMinusculesResult¶

Construct Dürer’s modular Textura minuscules i, n, and m from unit squares.

In Book III (p. 136), Dürer constructs i first because nearly the entire Gothic minuscule alphabet is generated combinatorially from its modular stem. For n (two stems) and m (three stems), adjacent stems touch corner-to-corner at their 45° diamond caps! Wait: look at the spacing when adjacent top diamonds touch corner-to-corner: Top diamond of stem 1 has right corner at x0 + 1/3 + sqrt(2)/2 and left corner at x0 + 1/3 - sqrt(2)/2, so corner-to-corner contact spaces consecutive stems by exact pitch sqrt(2) (leaving interior counter gap sqrt(2) - 1 ≈ 0.414 < 1, matching Dürer’s exact words: “so wirt das felt schlemer zwischen den zügen dann ein zug des pustaben preyt ist”!).

geometor.durer.__version__ = '0.1.0'¶