  In this paper we consider piecewise linear (pl) isometric embeddings of
Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a
metric space $\mathcal{P}$ which admits a triangulation $\mathcal{T}$ such that
each $n$-dimensional simplex of $\mathcal{T}$ is affinely isometric to a
simplex in $\mathbb{E}^n$. We prove that any 1-Lipschitz map from an
$n$-dimensional Euclidean polyhedron $\mathcal{P}$ into $\mathbb{E}^{3n}$ is
$\epsilon$-close to a pl isometric embedding for any $\epsilon > 0$. If we
remove the condition that the map be pl then any 1-Lipschitz map into
$\mathbb{E}^{2n + 1}$ can be approximated by a (continuous) isometric
embedding. These results are extended to isometric embedding theorems of
spherical and hyperbolic polyhedra into Euclidean space by the use of the
Nash-Kuiper $C^1$ isometric embedding theorem. Finally, we discuss how these
results extend to various other types of polyhedra.
