Proof of T-duality within Euclid's Geometry

Euclidean geometry contains an exact duality of scale. For the round 2-sphere S2R of radius R, both ends of the scale parameter R carry the Euclidean plane: as R → 0 the sphere collapses to a point whose tangent plane is Euclidean, and as R → ∞ the local geometry is Euclidean. The inversion R ↦→ R2 0/R is an involution that exchanges the two ends and leaves the flat structure invariant. This is the analogue, within Euclidean geometry, of the small-large duality of string theory known as T-duality. A single Riemannian space, the Apex Plane, is then exhibited, in which the two ends meet at one point: the apex, where the metric degenerates (the small end) and which is simultaneously the point at infinity of the geometry (the large end). All statements are proved from the definitions.