Abstract: Motivic homotopy theory applies ideas from topology to algebraic varieties by replacing continuous paths with polynomial paths given by the affine line. We introduce this viewpoint starting from intuitive notions of manifolds and varieties and explain what it means for a variety to be \mathbb{A}^1-contractible. In low dimensions this notion is rigid, but in dimension three exotic examples appear, such as the Koras–Russell threefolds. Motivated by the topological characterization of Euclidean space, we discuss a proposed motivic analogue for affine space using the motivic homotopy type at infinity. This illustrates how motivic homotopy theory provides a new notion of “shape” for polynomial spaces.