Detailed Description

Affine ⊂ InnerProduct ⊂ Euclidean ⊂ Hilbert.

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geometry__Structural_Geometry

"Geometrie ist nicht die Lehre von den Dingen, sondern von der Art und Weise, wie sie koexistieren. Raum ist eine relationale Ordnung koexistierender Objekte." (Geometry is not the study of things, but of the way in which they coexist. Space is a relational order of coexisting objects.) — Bernulf Kanitscheider, 'Geometrie und Wirklichkeit' (1971).

geometry__Geometric_Taxonomy

Following the Erlangen Program and ETCS foundations, we treat geometry as the study of invariants under specific groups of morphisms:

Affine (affine): The study of parallelisms and ratios. Defined by the action of a Vector Space (Module) on a set of Points.
**Metric (inner_product, euclidean)**: The study of congruence. Introduces the Bilinear Form ⟨u, v⟩ to measure angles and the Norm ||v|| to measure the "Magnitude of Being."
Analytic (hilbert): The study of infinite convergence. Completes the metric space, allowing for Fourier analysis and Quantum Mechanical state-representations.

See also: Wikipedia: Erlangen Program, Hilbert Space; Kanitscheider, B. (1971). Geometrie und Wirklichkeit.

Note: "Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on." — Alexander Grothendieck, letter to Ronald Brown (April 12, 1983).

Roadmap (work in progress)

#286 undergraduate calculus: the Jacobian and the Fréchet derivative of a differentiable map; still open are gradient and Hessian and the integral theorems.
#272 Denotational Automatic Differentiation: the chain rule for composed differentiable maps, Jacobians from dual propagation.