  We study contact 3-manifolds using the techniques of sub-Riemannian geometry
and geometric measure theory, in particular establishing properties of their
Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of
Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is
locally biLipschitz equivalent to the Heisenberg group $\mathbb{H}^n$ with its
\cc metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian
structure is purely $k$-unrectifiable for $k>n$. We then extend results of
Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger and Young
(arXiv:1210.6943 [math.GT]) on the Lipschitz homotopy groups of $\mathbb{H}^1$
to an arbitrary contact 3-manifold endowed with a \cc metric, namely that for
any contact 3-manifold the first Lipschitz homotopy group is uncountably
generated and all higher Lipschitz homotopy groups are trivial. Therefore, in
the sense of Lipschitz homotopy groups, a contact 3-manifold is a
$K(\pi,1)$-space with an uncountably generated first homotopy group. Along the
way, we prove that each open distributional embedding between purely
2-unrectifiable sub-Riemannian manifolds induces an injective map on the
associated first Lipschitz homotopy groups. Therefore, each open subset of a
contact 3-manifold determines an uncountable subgroup of the first Lipschitz
homotopy group of the contact 3-manifold.
