← klein / GEOMETOR • klein

modules¶

geometor.klein¶

GEOMETOR • klein — Dual-treatise Book Codex & exact symbolic engine for Felix Klein.

Provides exact SymPy and geometor.model implementations accompanying:

Famous Problems of Elementary Geometry (Vorträge über ausgewählte Fragen der Elementargeometrie, 1895; trans. W. W. Beman & D. E. Smith, 1897).
A Comparative Review of Recent Researches in Geometry (“The Erlangen Program”) (Vergleichende Betrachtungen über neuere geometrische Forschungen, 1872; trans. M. W. Haskell, 1893).

Submodules¶

Attributes¶

`ERLANGEN_UNITS`
`FAMOUS_PROBLEMS_UNITS`
`SISTER_CORPORA`
`__version__`

Classes¶

`CodexAuditReport`	Structural verification report for the dual 4-track Book Codex and JSON apparatus.
`ConstructibilityReport`	Immutable symbolic report on compass-and-straightedge constructibility.
`CayleyKleinReport`	Symbolic verification of Cayley–Klein projective metric additivity (Note V & § 10).
`ErlangenHierarchyReport`	Aggregate report across all four transformation groups in the Erlangen hierarchy.
`ErlangenRungResult`	Exact symbolic invariant audit for a single transformation group in Klein's hierarchy.
`Gauss17GonPeriods`	Exact algebraic and geometric realization of Gauss's and Klein's regular 17-gon.

Functions¶

`audit_codex`([repo_root]) → CodexAuditReport	Validate 4-track parity, apparatus JSON integrity, and content/code separation.
`load_citations`([repo_root]) → dict[str, dict[str, str]]	Load `resources/citations.json`.
`load_concordance`([repo_root]) → list[dict[str, Any]]	Load `resources/klein/concordance.json`.
`load_glossary`([repo_root]) → dict[str, dict[str, str]]	Load `resources/glossary.json`.
`load_persons`([repo_root]) → dict[str, dict[str, str]]	Load `resources/persons.json`.
`load_sources_manifest`([repo_root]) → dict[str, Any]	Load `resources/klein/sources_manifest.json`.
`analyze_constructibility`(poly_or_n) → ConstructibilityReport	Analyze compass-and-straightedge constructibility of a regular polygon, polynomial, or number.
`cubic_resolvent`(quartic_poly[, var]) → sympy.Poly	Return the exact depressed cubic resolvent \(R(t)\) of a degree-4 polynomial.
`cyclotomic_real_minimal_poly`(n[, var]) → sympy.Poly	Compute the exact minimal polynomial over \(\mathbb{Q}\) of \(u = 2\cos(2\pi/n)\).
`is_fermat_prime`(p) → tuple[bool, int \| None]	Test whether odd integer `p` is a Fermat prime \(2^{2^\mu} + 1\).
`is_power_of_two`(m) → tuple[bool, int \| None]	Return `(True, h)` if `m == 2**h` for `h >= 0`, else `(False, None)`.
`tabulate_polygon_constructibility`([max_n, min_n]) → list[ConstructibilityReport]	Return exact constructibility reports for all regular \(n\)-gons in `[min_n, max_n]`.
`circular_quadrance_cross_ratio`(p1, p2, p3, p4) → sympy.Expr	Compute the Möbius-invariant 4-point quadrance cross-ratio \(\frac{Q(P_1,P_3)Q(P_2,P_4)}{Q(P_1,P_4)Q(P_2,P_3)}\).
`is_concyclic`(p1, p2, p3, p4) → bool	Return `True` iff `p1, p2, p3, p4` lie on a generalized circle (circle or straight line).
`signed_area_ratio`(p1, p2, p3, q1, q2, q3) → sympy.Expr	Return the exact affine-invariant ratio of signed areas \([P_1P_2P_3] / [Q_1Q_2Q_3]\).
`verify_cayley_klein_additivity`() → CayleyKleinReport	Prove symbolic additivity of Klein's projective metric \(d(A, B) = c \log(A, B; I_1, I_2)\).
`verify_erlangen_invariants`() → ErlangenHierarchyReport	Execute an exact symbolic invariant audit across four groups of Klein's Erlangen Program.
`canonicalize_17`(expr) → sympy.Expr	Canonicalize an exact cyclotomic-17 expression onto Gauss's radical basis in \(K_{16}\).
`construct_gauss_17gon_periods`([model, ...]) → Gauss17GonPeriods	Construct Gauss's 17-gon period tower and Klein's von Staudt–Schröter single-circle 17-gon.

Package Contents¶

geometor.klein.ERLANGEN_UNITS: tuple[str, ...] = ('prefatory-intro', 'sec-01-02-principal-hierarchy', 'sec-03-05-projective-transference',...¶

geometor.klein.FAMOUS_PROBLEMS_UNITS: tuple[str, ...] = ('intro', 'part1-ch1-square-roots', 'part1-ch2-delian-trisection', 'part1-ch3-cyclotomy',...¶

geometor.klein.SISTER_CORPORA: tuple[str, ...] = ('euclid', 'kepler', 'hilbert', 'bertrand-russell', 'hoenen', 'lemoine', 'petersen', 'pappus',...¶

class geometor.klein.CodexAuditReport¶

Structural verification report for the dual 4-track Book Codex and JSON apparatus.

famous_problems_units: int¶

erlangen_units: int¶

total_track_files: int¶

missing_track_files: tuple[str, ...]¶

persons_count: int¶

citations_count: int¶

glossary_count: int¶

concordance_count: int¶

linked_sister_corpora: tuple[str, ...]¶

embedded_py_files_in_resources: tuple[str, ...]¶

property is_valid: bool¶: Return True iff all 64 track files and apparatus constraints pass.

geometor.klein.audit_codex(repo_root: pathlib.Path = REPO_ROOT) → CodexAuditReport¶: Validate 4-track parity, apparatus JSON integrity, and content/code separation.

geometor.klein.load_citations(repo_root: pathlib.Path = REPO_ROOT) → dict[str, dict[str, str]]¶: Load resources/citations.json.

geometor.klein.load_concordance(repo_root: pathlib.Path = REPO_ROOT) → list[dict[str, Any]]¶: Load resources/klein/concordance.json.

geometor.klein.load_glossary(repo_root: pathlib.Path = REPO_ROOT) → dict[str, dict[str, str]]¶: Load resources/glossary.json.

geometor.klein.load_persons(repo_root: pathlib.Path = REPO_ROOT) → dict[str, dict[str, str]]¶: Load resources/persons.json.

geometor.klein.load_sources_manifest(repo_root: pathlib.Path = REPO_ROOT) → dict[str, Any]¶: Load resources/klein/sources_manifest.json.

class geometor.klein.ConstructibilityReport¶

Immutable symbolic report on compass-and-straightedge constructibility.

target: str¶

kind: str¶

is_constructible: bool¶

degree: int | None¶

is_power_of_two: bool¶

tower_depth: int | None¶

minimal_poly: sympy.Poly | None¶

real_minimal_poly: sympy.Poly | None = None¶

euler_totient: int | None = None¶

odd_prime_factors: tuple[int, ...] = ()¶

fermat_primes: tuple[int, ...] = ()¶

non_fermat_primes: tuple[int, ...] = ()¶

repeated_odd_primes: tuple[int, ...] = ()¶

galois_obstruction: str | None = None¶

klein_locus: str = ''¶

summary: str = ''¶

geometor.klein.analyze_constructibility(poly_or_n: int | sympy.Integer | sympy.Poly | sympy.Expr) → ConstructibilityReport¶

Analyze compass-and-straightedge constructibility of a regular polygon, polynomial, or number.

Parameters:

poly_or_n –

int / sympy.Integer (\(n \ge 3\)): tests the regular \(n\)-gon via Euler’s totient \(\varphi(n) = 2^k\) and Fermat-prime factorization.
sympy.Poly or univariate sympy.Expr: factors over \(\mathbb{Q}\) and applies Klein’s degree-\(2^h\) criterion plus cubic-resolvent Galois check.
Closed-form algebraic or transcendental sympy.Expr (no free symbols): detects transcendence (\(\pi\), \(e\)) or computes the exact minimal polynomial over \(\mathbb{Q}\).

Returns:

A populated ConstructibilityReport with exact minimal polynomial, tower depth, and Galois/transcendence obstruction diagnosis.

geometor.klein.cubic_resolvent(quartic_poly: sympy.Poly, var: sympy.Symbol | None = None) → sympy.Poly¶

Return the exact depressed cubic resolvent \(R(t)\) of a degree-4 polynomial.

Depresses \(a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0\) via \(x = y - a_3/(4a_4)\) into \(y^4 + p y^2 + q y + r\) and returns \(R(t) = t^3 - 2p t^2 + (p^2 - 4r) t + q^2\) over \(\mathbb{Q}\).

geometor.klein.cyclotomic_real_minimal_poly(n: int, var: sympy.Symbol | None = None) → sympy.Poly¶

Compute the exact minimal polynomial over \(\mathbb{Q}\) of \(u = 2\cos(2\pi/n)\).

For \(n \ge 3\), \(\Phi_n(z)\) has even degree \(\varphi(n) = 2d\) and is palindromic, so \(z^{-d}\Phi_n(z)\) is an integer polynomial of degree \(d\) in \(u = z + z^{-1} = 2\cos(2\pi/n)\). For \(n = 1, 2\), returns \(u - 2\) and \(u + 2\) respectively.

geometor.klein.is_fermat_prime(p: int) → tuple[bool, int | None]¶

Test whether odd integer p is a Fermat prime \(2^{2^\mu} + 1\).

Parameters:: p – Integer candidate.
Returns:: Tuple (True, mu) when p is prime and p - 1 = 2**(2**mu), otherwise (False, None).

geometor.klein.is_power_of_two(m: int) → tuple[bool, int | None]¶: Return (True, h) if m == 2**h for h >= 0, else (False, None).

geometor.klein.tabulate_polygon_constructibility(max_n: int = 24, *, min_n: int = 3) → list[ConstructibilityReport]¶: Return exact constructibility reports for all regular \(n\)-gons in [min_n, max_n].

class geometor.klein.CayleyKleinReport¶

Symbolic verification of Cayley–Klein projective metric additivity (Note V & § 10).

cross_ratio_ab: sympy.Expr¶

cross_ratio_bc: sympy.Expr¶

cross_ratio_ac: sympy.Expr¶

multiplicative_identity_residual: sympy.Expr¶

hyperbolic_sample_distance_ab: sympy.Expr¶

hyperbolic_sample_distance_bc: sympy.Expr¶

hyperbolic_sample_distance_ac: sympy.Expr¶

additive_residual: sympy.Expr¶

class geometor.klein.ErlangenHierarchyReport¶

Aggregate report across all four transformation groups in the Erlangen hierarchy.

collinear_quadruple: tuple[sympy.geometry.Point2D, sympy.geometry.Point2D, sympy.geometry.Point2D, sympy.geometry.Point2D]¶

concyclic_quadruple: tuple[sympy.geometry.Point2D, sympy.geometry.Point2D, sympy.geometry.Point2D, sympy.geometry.Point2D]¶

triangle_vertices: tuple[sympy.geometry.Point2D, sympy.geometry.Point2D, sympy.geometry.Point2D, sympy.geometry.Point2D]¶

rungs: tuple[ErlangenRungResult, ...]¶

by_key() → dict[str, ErlangenRungResult]¶: Map group_key to its ErlangenRungResult.

class geometor.klein.ErlangenRungResult¶

Exact symbolic invariant audit for a single transformation group in Klein’s hierarchy.

group_key: str¶

group_name: str¶

erlangen_section: str¶

transformation_formula: str¶

preserved: dict[str, bool]¶

exact_values_before: dict[str, sympy.Expr]¶

exact_values_after: dict[str, sympy.Expr]¶

exact_differences: dict[str, sympy.Expr]¶

geometor.klein.circular_quadrance_cross_ratio(p1: sympy.geometry.Point2D, p2: sympy.geometry.Point2D, p3: sympy.geometry.Point2D, p4: sympy.geometry.Point2D) → sympy.Expr¶

Compute the Möbius-invariant 4-point quadrance cross-ratio \(\frac{Q(P_1,P_3)Q(P_2,P_4)}{Q(P_1,P_4)Q(P_2,P_3)}\).

Under any Euclidean similarity or any circle inversion \(P \mapsto O + R^2(P-O)/Q(O,P)\) (whose center \(O\) is distinct from \(P_1..P_4\)), the squared chord length transforms as \(Q(P_i', P_j') = R^4 Q(P_i, P_j) / [Q(O, P_i) Q(O, P_j)]\), so all four denominator factors \(Q(O, P_k)\) cancel identically! Furthermore, when \(A, B, C, D\) are collinear, this coincides with the square \((A, B; C, D)^2\) of the projective cross-ratio.

geometor.klein.is_concyclic(p1: sympy.geometry.Point2D, p2: sympy.geometry.Point2D, p3: sympy.geometry.Point2D, p4: sympy.geometry.Point2D) → bool¶: Return True iff p1, p2, p3, p4 lie on a generalized circle (circle or straight line).

geometor.klein.signed_area_ratio(p1: sympy.geometry.Point2D, p2: sympy.geometry.Point2D, p3: sympy.geometry.Point2D, q1: sympy.geometry.Point2D, q2: sympy.geometry.Point2D, q3: sympy.geometry.Point2D) → sympy.Expr¶: Return the exact affine-invariant ratio of signed areas \([P_1P_2P_3] / [Q_1Q_2Q_3]\).

geometor.klein.verify_cayley_klein_additivity() → CayleyKleinReport¶

Prove symbolic additivity of Klein’s projective metric \(d(A, B) = c \log(A, B; I_1, I_2)\).

In Note V (On the So-Called Non-Euclidean Geometry) and § 10.2 of the Erlangen Program, Klein subsumes hyperbolic, elliptic, and parabolic metrics under projective geometry by adjoining an invariant second-degree Absolute (\(\Omega = 0\)). Along any line meeting the Absolute in \(I_1, I_2\), the cross-ratio multiplicativity \((A, B; I_1, I_2) \cdot (B, C; I_1, I_2) = (A, C; I_1, I_2)\) converts into exact geodesic distance additivity \(d(A, B) + d(B, C) = d(A, C)\).

geometor.klein.verify_erlangen_invariants() → ErlangenHierarchyReport¶

Execute an exact symbolic invariant audit across four groups of Klein’s Erlangen Program.

Applies four rational planar transformations to a shared exact rational point configuration:

principal_similarity (Hauptgruppe, § 1): rational rotation \((3/5, 4/5)\), isotropic scaling \(5/2\), and translation \((3, -2)\).
affine_shear (§ 2): equiaffine shear + translation \((x, y) \mapsto (x + 2y + 1, y - 3)\).
projective_homography (§ 3): nonsingular projective collineation \((x, y) \mapsto \left(\frac{2x + y + 1}{x + y + 3}, \frac{x + 3y + 2}{x + y + 3}\right)\).
moebius_circle_inversion (Geometrie der reciproken Radien, § 6): inversion in the circle centered at \(O_{\mathrm{inv}}(-1, -2)\) of squared radius \(R^2 = 5\).

class geometor.klein.Gauss17GonPeriods¶

Exact algebraic and geometric realization of Gauss’s and Klein’s regular 17-gon.

primitive_root: int¶

exponent_cycle: tuple[int, ...]¶

x1: sympy.Expr¶

x2: sympy.Expr¶

y1: sympy.Expr¶

y2: sympy.Expr¶

y3: sympy.Expr¶

y4: sympy.Expr¶

z_periods: tuple[sympy.Expr, ...]¶

cos_2pi_17: sympy.Expr¶

sin_2pi_17: sympy.Expr¶

eq_xi: sympy.Poly¶

eq_eta: sympy.Poly¶

eq_eta_prime: sympy.Poly¶

eq_zeta: sympy.Poly¶

radical_dependence_residual: sympy.Expr¶

model: geometor.model.Model¶

fixed_circle: sympy.geometry.Circle¶

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

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

geometor.klein.canonicalize_17(expr: sympy.Expr) → sympy.Expr¶: Canonicalize an exact cyclotomic-17 expression onto Gauss’s radical basis in \(K_{16}\).

geometor.klein.construct_gauss_17gon_periods(model: geometor.model.Model | None = None, *, inscribe_polygon: bool = True, add_sides: bool = True) → Gauss17GonPeriods¶

Construct Gauss’s 17-gon period tower and Klein’s von Staudt–Schröter single-circle 17-gon.

Solves all four quadratic period equations \((\xi), (\eta), (\eta'), (\zeta)\) on the single fixed circle \(C_0: x^2 + y(y - 2) = 0\) (center C0c(0, 1), bottom O(0, 0), top A(0, 2)) using straightedge lines alone, and (when inscribe_polygon=True) inscribes all 17 exact regular heptadecagon vertices V0..``V16`` on \(C_0\) via Steiner’s straightedge parallel theorem—keeping len(model.circles) == 1 and 0 floats!

Parameters:

model – Optional existing Model (must have auto_intersect=False or will be configured with canonicalize_17). When None, creates a fresh Model("klein_von_staudt_17gon", logger=False, auto_intersect=False).
inscribe_polygon – If True (default), construct all 17 exact vertices V0..``V16``.
add_sides – If True (default) and inscribe_polygon=True, add the 17 polygon sides.

Returns:

A frozen Gauss17GonPeriods bundle containing the verified algebraic periods, minimal quadratic equations, and exact Model.

geometor.klein.__version__ = '0.1.0'¶