I will discuss my recent work with Semin Yoo on the functorial construction of manifolds from n-ary alternating quasigroups. Beginning with examples from the 2-dimensional case, which was introduced by Herman and Pakianathan for groups, I will show how orientable surfaces such as the sphere and torus arise as geometric instantiations of the multiplication operations of groups. I will then move on to the nonassociative case by considering surfaces built from loops. In dimensions 3 and above nonassociativity becomes critical as only nonassociative n-ary quasigroups yield manifolds via our generalized construction. I will conclude by sketching our proof that every orientable triangulable manifold may be built from some such n-ary quasigroup. Along the way we will encounter the octonions and a generalized Evans Conjecture about Latin hypercubes. Code will be provided.