The elements of a double groupoid Q can be regarded as squares made up of two pairs of arrows, each pair being elicited from one of the base groupoids. With a view to generalizing this fertile idea, it is convenient to review the properties of hypercubes, that is, cubes in an arbitrary number of dimensions. A hypercube in dimension n contains faces of all dimensions smaller than n. The 0-faces are called vertices, and the 1-faces are called edges. The skeleton of a hypercube is the collection of its edges and vertices, the ‘flesh’ having been discarded. Skeletons can be regarded as directed graphs by adopting an ingenious binary notation for its vertices. By the same device, the vertices of a hypercube can be organized into a useful and natural hierarchical structure.