  Finite element exterior calculus refers to the development of finite element
methods for differential forms, generalizing several earlier finite element
spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary
polynomial degree $r$, and arbitrary differential form degree $k$. The study of
finite element exterior calculus began with the $\mathcal P_r\Lambda^k$ and
$\mathcal P_r^-\Lambda^k$ families of finite element spaces on simplicial
triangulations. In their development of these spaces, Arnold, Falk, and Winther
rely on a duality relationship between $\mathcal P_r\Lambda^k$ and
$\mathring{\mathcal P}_{r+k+1}^-\Lambda^{n-k}$ and between $\mathcal
P_r^-\Lambda^k$ and $\mathring{\mathcal P}_{r+k}\Lambda^{n-k}$. In this
article, we show that this duality relationship is, in essence, Hodge duality
of differential forms on the standard $n$-sphere, disguised by a change of
coordinates. We remove the disguise, giving explicit correspondences between
the $\mathcal P_r\Lambda^k$, $\mathcal P_r^-\Lambda^k$, $\mathring{\mathcal
P}_r\Lambda^k$ and $\mathring{\mathcal P}_r^-\Lambda^k$ spaces and spaces of
differential forms on the sphere. As a direct corollary, we obtain new
pointwise duality isomorphisms between $\mathcal P_r\Lambda^k$ and
$\mathring{\mathcal P}_{r+k+1}^-\Lambda^{n-k}$ and between $\mathcal
P_r^-\Lambda^k$ and $\mathring{\mathcal P}_{r+k}\Lambda^{n-k}$, which we
illustrate with examples.
