Polyhedra enjoy a peculiar property: every geometric shape with a certain ?regularity? ? in specific terms, certain classes of (closed) topological manifolds ? can be captured by a polyhedron via triangulation, that is, by subdividing the geometric shapes into appropriate ?triangles?, called simplices (which, in the 1- and 0-dimensional case, are simply edges and vertices, respectively). Therefore, one might well wonder: what is the intermediate logic of the class of triangulable topological manifolds of a given dimension d? The main result of the present work is to give the answer to this question in the case of 1-dimensional manifolds, that is, the circle and the closed interval.