Wing
Geometries & Spaces
Euclid’s Parallel Postulate
Euclid, ~300 BC
Through a point not on a line, Euclid permits exactly one parallel — a constraint that defines flat space.
Spherical Geometry
Spherical trigonometry (navigation), ~300 BC
On a sphere, geodesics are great circles and the angles of a triangle add to more than a flat plane allows.
Curvature Triptych
Carl Friedrich Gauss, 1827
The same triangle asks three different spaces for its angle sum — and receives three different answers.
The Poincaré Disk
Nikolai Lobachevsky & János Bolyai, 1830
In the Poincaré model, straightest paths bend inward — geodesics appear as circular arcs meeting the boundary at right angles.
Villarceau Circles
Yvon Villarceau, 1848
Two orthogonal families of planar circles slicing a torus at oblique angles.
Torus Knots
Classical knot theory, 1860
Closed curves wind around a torus with distinct topological signatures.
Clifford Torus Projection
William Kingdon Clifford, 1873
Two perpendicular circle grids on the flat torus S¹×S¹ in S³, projected into 3D.