§§ 1–2: Principal Group & Transformation Hierarchy — Critical & Symbolic Notes

Exact Symbolic Verification of Invariant Shedding in geometor.klein.erlangen

Visualizing the Erlangen Subsumption Ladder

Klein’s Erlangen Hierarchy (§§ 1–3 & § 6). As the transformation group expands from the Principal Group (similarities) to Affine shears and Projective collineations, metric angles and quadrance ratios are progressively shed until only incidence and the $4$-point collinear cross-ratio survive; branching to Reciprocal Radii (Möbius circle inversion) merges lines and circles into generalized circles while preserving conformal angles and $4$-point quadrance cross-ratios.

Exact Symbolic Verification Over $\mathbb{Q}$ (geometor.klein.erlangen)

In geometor.klein.erlangen.verify_erlangen_invariants(), we instantiate rational test configurations inside geometor.model.Model (with zero floating-point approximations) and apply exact representatives of Klein’s four planar groups:

1. Principal Group (§ 1 — Rational Similarity): $(x, y) \mapsto (3x - 4y + 2,\; 4x + 3y - 1)$ (Pythagorean rotation-dilation scaling lengths by $5$ and quadrances by $25$).

2. Affine Group (§ 2 — Area-Scaling Shear): $(x, y) \mapsto (x + 2y + 1,\; 3y - 2)$ (fixes the line at infinity $w = 0$ without fixing the circular points $I, J$).

3. Projective Group (§ 3 — Full Homography): $(x, y) \mapsto \left(\frac{2x + y + 1}{x + y + 3},\; \frac{x + 3y + 2}{x + y + 3}\right)$ (moves the line at infinity to a finite horizon).

4. Reciprocal Radii (§ 6 — Möbius Circle Inversion): $(x, y) \mapsto \left(\frac{4x}{x^2 + y^2},\; \frac{4y}{x^2 + y^2}\right)$ (inverts across $x^2 + y^2 = 4$).

Using Rational Trigonometry invariants (Norman Wildberger, 2005; geometor.model.invariants), every invariant is tested by exact symbolic subtraction in SymPy:

Exact Erlangen Invariant Survival Matrix (verify_erlangen_invariants())

Algebraic Invariant	Principal (§ 1)	Affine (§ 2)	Projective (§ 3)	Inversion (§ 6)
Line Spread $s(\ell_{12}, \ell_{13})$ (Angle)	Preserved ($\frac{9}{10}$)	Broken ($\frac{81}{130}$)	Broken ($\frac{25}{74}$)	(Conformal)
Quadrance Ratio $\frac{Q(T_1, T_2)}{Q(T_1, T_3)}$	Preserved ($\frac{4}{5}$)	Broken ($\frac{1}{13}$)	Broken ($\frac{185}{117}$)	Broken ($\frac{65}{18}$)
Parallelism ($\det(\ell, \ell') = 0$)	Preserved	Preserved	Broken	Broken
Collinear Division Ratio $\frac{Q(A,B)}{Q(A,C)}$	Preserved ($\frac{1}{9}$)	Preserved ($\frac{1}{9}$)	Broken ($\frac{25}{147}$)	Broken ($\frac{5}{27}$)
Collinear Cross-Ratio $(A, B; C, D)$	Preserved ($\frac{4}{3}$)	Preserved ($\frac{4}{3}$)	Preserved ($\frac{4}{3}$)	(On circle)
4-Point Quadrance Cross-Ratio $\frac{Q_{AC} Q_{BD}}{Q_{AD} Q_{BC}}$	Preserved ($\frac{16}{9}$)	Preserved ($\frac{16}{9}$)	Preserved ($\frac{16}{9}$)	Preserved ($\frac{16}{9}$)

Philological Note on Footnote 8 (Group Inverses, 1893)

In 1872, following the finite permutation terminology of Galois, Serret, and Camille Jordan, Klein defined a group solely by closure under composition ($g_1, g_2 \in G \implies g_1 g_2 \in G$). For a finite set of bijections, closure automatically implies $g^{-1} = g^{k-1} \in G$; for infinite continuous families (such as contractions $x \mapsto \lambda x$ with $0 < \lambda < 1$), closure alone defines only a semigroup. Klein’s 1893 bracketed addition in Footnote 8 explicitly supplies the inverse axiom required by Sophus Lie's Theorie der Transformationsgruppen.