  We develop a family of finite element spaces of differential forms defined on
cubical meshes in any number of dimensions. The family contains elements of all
polynomial degrees and all form degrees. In two dimensions, these include the
serendipity finite elements and the rectangular BDM elements. In three
dimensions they include a recent generalization of the serendipity spaces, and
new H(curl) and H(div) finite element spaces. Spaces in the family can be
combined to give finite element subcomplexes of the de Rham complex which
satisfy the basic hypotheses of the finite element exterior calculus, and hence
can be used for stable discretization of a variety of problems. The
construction and properties of the spaces are established in a uniform manner
using finite element exterior calculus.
