Short answers, pulled from the story.
Aristotle recognized the existence of three dimensions over two thousand years ago. This recognition occurred long before the mathematical framework to describe them was established. The precise language to quantify the space we inhabit remained elusive until centuries later.
René Descartes published his groundbreaking work La Géométrie to introduce a system that could describe every point in three-dimensional space by means of three coordinates. Pierre de Fermat independently developed similar ideas in the manuscript Ad locos planos et solidos isagoge. Their combined efforts laid the foundation for analytic geometry.
Edwin Bidwell Wilson compiled the textbook Vector Analysis in 1901. Josiah Willard Gibbs introduced the modern notation for the dot and cross product in his classroom teaching notes before Wilson compiled them. This work standardized the language of physics and engineering.
There are nine regular polytopes in three dimensions. These comprise the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. Euclid first constructed these shapes within a sphere in Book XIII of his Elements.
At least three dimensions are required to create a knot. A two-dimensional plane cannot support the necessary crossings without intersection. Generic three-dimensional spaces are known as 3-manifolds in differential geometry.