Short answers, pulled from the story.
Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of perpendicularity, parallelism, and orthogonality of lines and planes. Book XII discusses infinitesimals and the method of exhaustion for finding volume.
Three-dimensional space was described with Cartesian coordinates in the 17th century. René Descartes developed analytic geometry in his work La Géométrie during this period. Pierre de Fermat independently developed similar ideas in the manuscript Ad locos planos et solidos isagoge.
There are nine regular polytopes in three dimensions. Five convex Platonic solids exist alongside four nonconvex Kepler-Poinsot polyhedra. The symmetry group for the tetrahedron is Td with order 24 while the cube has symmetry Oh with order 48.
William Rowan Hamilton developed quaternions in the 19th century. He coined the terms scalar and vector within his geometric framework. Josiah Willard Gibbs identified dot products and cross products as distinct operations later.
At least three dimensions are required to tie a knot in string. Generic three-dimensional spaces are 3-manifolds locally resembling R3. Globally these manifolds can curve in various manners while remaining continuous.