$\sqrt{3}$-Based 1-Form Subdivision

In this paper we construct an edge based, or 1-form, subdivision scheme consistent with \(\sqrt{3}\) subdivision. It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation.