Short answers, pulled from the story.
Three-dimensional space is a mathematical space in which three values, called coordinates, are required to determine the position of a point. It is also known as 3D space, 3-space, or rarely tri-dimensional space, and most commonly refers to three-dimensional Euclidean space, which models physical space.
Aristotle argued that a magnitude divisible one way is a line, two ways a surface, and three ways a body. He held that beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all.
René Descartes introduced Cartesian coordinates for three-dimensional space in the 17th century in his work La Géométrie, launching analytic geometry. Pierre de Fermat independently developed similar ideas in his unpublished manuscript Ad locos planos et solidos isagoge.
There are nine regular polytopes in three dimensions: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. The five Platonic solids include the cube, octahedra, icosahedra, and dodecahedra.
The cross product is a binary operation on two vectors in three-dimensional space, denoted by the symbol times, that produces a vector perpendicular to both and normal to the plane containing them. It can compute torque on a bolt turned by a wrench or the Lorentz force on an electron moving through a magnetic field, and such a binary product exists only in three and seven dimensions.
William Rowan Hamilton described three-dimensional space using quaternions with a vanishing scalar component, and coined the terms scalar and vector within that framework. His work indirectly introduced the dot product and cross product, which Josiah Willard Gibbs later identified in their own right.
The six non-degenerate quadric surfaces are the ellipsoid, the hyperboloid of one sheet, the hyperboloid of two sheets, the elliptic cone, the elliptic paraboloid, and the hyperbolic paraboloid. The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces made up of straight lines.