We show that the self-similar known as the antenna set has the property that inff dim(f(X)) = 1 (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space. Given a compact metric space X we define its conformal dimension as Cdim(X) = inf dim(f (X)), f where the infimum is over all quasisymmetric maps of X into some metric space and dim denotes Hausdorff dimension. Recall that f is quasisymmetric if there is a homeomorphism 17 of [0, oo) to itself such that ix Yt If(x) f(y)W 2, these are exactly the same as quasiconformal maps. This notion of conformal dimension was introduced by Pansu [9]. See also [4] and [10]. In this note we are interested in the question of whether the infimum above must be attained by some quasisymmetric image of X. It is known that for some sets (e.g., the usual Cantor middle thirds in IR) the conformal dimension is zero, whereas any quasisymmetric image must have strictly positive dimension. Here we answer a question of Heinonen's (Section 15, [7]) by proving Theorem 1. There is a compact, connected X C IR2 so that for every quasisymmetric map f of X into a metric space, dim(f(X)) > 1 -Cdim(X). In particular, this holds for all quasiconformal self-maps of the plane, which is already an interesting case. Our is the self-similar associated to the four conformal contractions f1 (z) =2Z, f2 (Z) = z+2 f3x(z) -aiz * 2, f4 (z) = -cviz + 2 + ati, Received by the editors November 15, 1999 and, in revised form, April 27, 2000. 2000 Mathematics Subject Classification. Primary 30C62; Secondary 28A78.