§§ 3–5: Projective Geometry, Representation, and Choice of Space-Element — Summary¶
Analytical Synopsis of Projective Extensions, Group Equivalence, and Plücker Coordinates
§ 3. Projective Geometry, Duality, and Imaginary Elements¶
Genesis of Projective Geometry: Projective geometry emerged when projection ceased to be merely a heuristic trick for transferring properties between individual figures and became the defining equivalence relation under the group of all projective collineations.
Two Internal Group Extensions: 1. Adjoining dualistic transformations (correlations exchanging points and planes in $3$-space, or points and lines in the plane), rendering dual figures identical in the group sense. 2. Adjoining complex/imaginary elements and transformations (von Staudt's Beiträge, 1856–1860), motivated not by projectivity per se (since real projectivities already form a group) but by closure under algebraic equations.
Recovery of Real Elementary Geometry: Elementary real Euclidean geometry is recovered inside complex projective geometry by simultaneously fixing the imaginary spherical circle at infinity and the real subspace $\mathbb{RP}^3$.
§ 4. Transfer of Properties by Representation (Abbildung)¶
Whenever a bijection $f : A \to A'$ conjugates a group action $(A, B)$ into $(A', B' = f B f^{-1})$, the two geometries $(A, B)$ and $(A', B')$ are isomorphic. Klein illustrates this by two canonical isomorphisms:
Binary Forms $\cong$ Conic Projectivities: Stereographic projection from a point on a conic maps $\mathbb{P}^1$ under $\operatorname{PGL}_2$ isomorphically onto the conic under planar projectivities stabilizing it.
Elementary Plane Geometry $\cong$ Punctured Quadric: Stereographic projection of a quadric surface from a point $P_0$ maps the two generators through $P_0$ to the circular points $I, J$ at infinity, identifying planar similarity geometry with projective quadric geometry fixing $P_0$.
§ 5. Arbitrariness of the Space-Element and Transitive Group-Bodies (Körper)¶
Dimension Is Relative to the Space-Element: Taking points ($\infty^3$), lines ($\infty^4$, Plücker), planes ($\infty^3$), or general quadrics ($\infty^9$) as the primary space-element changes the dimension of space at will, while preserving the mathematical substance of the geometry so long as the underlying transformation group remains unchanged.
Hesse’s Principle & Plücker’s Line Quadric: Pairing points on a conic identifies binary form theory with Cayley–Klein planar metric geometry; similarly, Plücker’s $6$ homogeneous line coordinates satisfying $p_{01} p_{23} + p_{02} p_{31} + p_{03} p_{12} = 0$ identify $3$-dimensional projective geometry ($\operatorname{PGL}_4 \rtimes \mathbb{Z}_2$) with $5$-dimensional projective metric geometry relative to the Klein quadric in $\mathbb{P}^5$.
Group-Bodies (*Körper*): Borrowing Dedekind’s algebraic term Körper, Klein defines a geometric body as a single transitive orbit $G \cdot x_0$ (requiring in 1893 that the stabilizer contain no normal subgroup of $G$). Coherent coordinate representation demands that chosen space-elements form a single Körper (homogeneous space) or decompose cleanly into Körper.