Unit II: Non-Euclidean Geometries — Scholarly Notes¶
Exact Symbolic Geometry, Rational Trigonometry & Cross-Repository Commentary
1. Circle Inversion & Orthogonal Circles (geometor/casey & geometor/lemoine)¶
Poincaré’s Translation Dictionary (Fig. 2, Concordance PC.04) rests synthetically on two classical pillars digitized in the GEOMETOR canon:
Casey’s Orthogonal Circle Criterion (``geometor/casey``): In John Casey’s Sequel to Euclid (Casey 1888, Book III & Book VI Section V), a circle $\mathcal{C}(C, r)$ cuts the unit horizon $\Omega: x^2 + y^2 = 1$ orthogonally if and only if the power of $O$ with respect to $\mathcal{C}$ equals $R_\Omega^2 = 1$, i.e. $|OC|^2 - r^2 = 1$, which holds if and only if $\mathcal{C}$ passes through the horizon inverse $P^* = P / \|P\|^2$ of every point $P \in \mathcal{C}$.
Lemoine’s 1902 *Géométrographie* (``geometor/lemoine``): Published in Paris in 1902—the exact year of La Science et l’Hypothèse—Émile Lemoine’s Géométrographie analyzes ruler-and-compass inversion via Euclid VI.8 (Heath 1908): erecting the perpendicular chord $T_1 T_2 \perp OP$ inside $\Omega$ and taking the pole where the tangents at $T_1, T_2$ meet ray $OP$ constructs $P^*$ over the exact base field $\mathbb{Q}(P)$ without introducing a single extraneous radical! In
geometor.poincare.disk.construct_hyperbolic_geodesic(), for $P=(1/2, 1/4)$ and $Q=(-2/5, 2/5)$, this yields exact rational $P^*=(8/5, 4/5)$ and circumcenter $C=(9/16, 33/16)$ with $|OC|^2 - r^2 = \frac{1170}{256} - \frac{914}{256} = 1$.
2. Cross-Ratio Metric & Wildberger’s Rational Hyperbolic Quadrance (geometor/pappus & geometor/wildberger)¶
Poincaré defines the distance between $P, Q \in \mathbb{D}$ via Pappus’s anharmonic ratio $\chi(P, Q; U, V) = \frac{|PV|\cdot|QU|}{|PU|\cdot|QV|}$. In geometor.poincare.disk.hyperbolic_cross_ratio_and_quadrance() (Concordance PC.05), this classical logarithmic formula is united with Norman Wildberger’s transcendental-free Universal Hyperbolic Geometry (Wildberger 2005/2013, geometor/wildberger):
Whenever $P, Q \in \mathbb{Q}^2 \cap \mathbb{D}$, both the conformal quadrance $\tau(P, Q) = \tanh^2(d_H/2)$ and Wildberger’s hyperbolic quadrance $q_{\mathrm{UHG}}(P, Q) = -\sinh^2(d_H)$ remain 100% rational numbers even when the ideal endpoints $U, V$ lie in a quadratic field $\mathbb{Q}(\sqrt{d})$!
3. Postulate V, Ideal Triangles & Saccheri’s Acute Summit (geometor/euclid, geometor/elements)¶
Figures 3–5 realize in exact rational arithmetic the classic counterexamples to Euclid’s Fifth Postulate (G.1–G.28 neutral core in geometor/elements):
Horoparallels & Ultraparallels (``Fig. 3``): Over base line $L_{\mathrm{hyp}}$ with Pythagorean endpoints $U_L(4/5, -3/5), V_L(-4/5, -3/5)$, point $P(1/6, 2/5)$ admits two limiting horoparallels sharing $U_L$ or $V_L$ on $\partial\Omega$ and infinitely many ultraparallels strictly disjoint in $\overline{\mathbb{D}}$.
Rational Ideal Triangle (``Fig. 4``): Vertices $U_1(1,0), U_2(-3/5,4/5), U_3(-3/5,-4/5)$ produce orthogonal side circles with pure rational centers $(1,2), (1,-2), (-5/3,0)$, radii $(2, 2, 4/3)$, and exact tangent spreads $(0,0,0)$—realizing Gauss and Lobachevsky’s maximum finite area $\pi$ at angle sum $0$.
Saccheri Quadrilateral (``Fig. 5``): Over $A_1(-1/3,0), A_2(1/3,0)$, erecting equal hyperbolic legs ($q_{\mathrm{UHG}} = -225/64$) to $B_1(-17/39, 20/39)$ and $B_2(17/39, 20/39)$ refutes Saccheri’s Right Angle Hypothesis by an exact rational inequality: the summit spread at $B_1, B_2$ is $s = \frac{64}{145} < 1$.
4. Poincaré’s Fourth Geometry & Chromogeometry’s Red Metric¶
Poincaré’s little-known Fourth Geometry (FR pp. 60–61 / EN p. 46; Concordance PC.07) corresponds geometrically to the exterior $x^2 + y^2 > 1$ of the fundamental horizon (de Sitter / single-sheeted hyperboloid plane of signature $(+, -)$), where lines tangent to $\Omega$ are null (isotropic) lines perpendicular to themselves ($Q_{\mathrm{red}}(v, v) = 0$)—directly anticipating Minkowski spacetime (1905–1908) and Wildberger’s Red relativistic metric in Chromogeometry (geometor/wildberger).