  We present a systematic construction of finite element exact sequences with a
commuting diagram for the de Rham complex in one-, two- and three-space
dimensions. We apply the construction in two-space dimensions to rediscover two
families of exact sequences for triangles and three for squares, and to uncover
one new family of exact sequence for squares and two new families of exact
sequences for general polygonal elements. We apply the construction in
three-space dimensions to rediscover two families of exact sequences for
tetrahedra, three for cubes, and one for prisms; and to uncover four new
families of exact sequences for pyramids, three for prisms, and one for cubes.
