We discuss some central questions in affine algebraic geometry: the “Cancellation Problem”, the “Embedding Problem”, and the “Characterization Problem” for affine spaces. We present a general framework that produces families of “rectifiable embeddings” between affine spaces, providing evidence toward the Abhyankar–Sathaye Embedding Conjecture and yielding counterexamples to Zariski cancellation in positive characteristic. This method also gives a recipe for constructing smooth, non-affine spaces in every dimension $\ge 3$ which are $\mathbb{A}^1$-homotopically equivalent to affine spaces, yet the geometry of cylinders over them remains mysterious. As a result, they are potential counterexamples to the Zariski Cancellation Problem in characteristic zero and interesting objects for further study.