  A carpet is a metric space which is homeomorphic to the standard Sierpi\'nski
carpet in $\mathbb{R}^2$, or equivalently, in $S^2$. A carpet is called thin if
its Hausdorff dimension is $<2$. A metric space is called Q-Loewner if its
$Q$-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a
$(1,Q)$-Poincar\'e inequality. As we will show, $Q$-Loewner planar metric
spaces are always carpets, and admit quasisymmetric embeddings into the plane.
In this paper, for every pair $(Q,Q')$, with $1<Q<Q'< 2$ we construct
infinitely many pairwise quasi-symmetrically distinct $Q$-Loewner carpets $X$
which admit explicit snowflake embeddings, $f: X\to S^2$, for which the image,
$f(X)$, admits an explicit description and is $Q'$-Ahlfors regular. In
particular, these $f$ are quasisymmetric embeddings. By a result of Tyson, the
Hausdorff dimension of a Loewner space cannot be lowered by a quasisymmetric
homeomorphism. By definition, this means that the carpets $X$ and $f(X)$
realize their conformal dimension. Each of images $f(X)$ can be further
uniformized via post composition with a quasisymmetric homeomorphism of $S^2$,
so as to yield a circle carpet and also a square carpet. Our Loewner carpets
$X$ are constructed via what we call an admissable quotiented inverse system.
This mechanism extends the inverse limit construction for PI spaces given in
\cite{cheegerkleinerinverse}, which however, does not yield carpets. Loewner
spaces are a particular subclass of PI spaces. They have strong rigidity
properties which which do not hold for PI spaces in general.
