The ℓ2 flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. It has been conjectured that the target dimension bounds can be refined and bounded in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03, Mat02, ABN08, CGT10]), and is still open. We prove another result in this line of work: The snowflake metric d1/2 of a doubling set S ⊂ ℓ2 can be embedded with arbitrarily low distortion into ℓD 2, for dimension D that depends solely on the doubling constant of the metric. In fact, the target dimension is polylogarithmic in the doubling constant. Our techniques are robust and extend to the more difficult spaces ℓ1 and ℓ∞, although the dimension bounds here are quantitatively inferior than those for ℓ2.