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              <p>Self-avoiding polymers in two-dimensional ͑d =2͒ melts are known to adopt compact configurations of typical size R͑N͒ϳN 1/d , with N being the chain length. Using molecular-dynamics simulations we show that the irregular shapes of these chains are characterized by a perimeter length L͑N͒ϳR͑N͒ d p of fractal dimension d p = d − ⌰ 2 =5/ 4, with ⌰ 2 =3/ 4 being a well-known contact exponent. Due to the self-similar structure of the chains, compactness and perimeter fractality repeat for subchains of all arclengths s down to a few monomers. The Kratky representation of the intramolecular form factor F͑q͒ reveals a strong nonmonotonous behavior with q 2 F͑q͒ϳ1 / ͑qN 1/d ͒ ⌰ 2 in the intermediate regime of the wave vector q. Measuring the scattering of labeled subchains the form factor may allow to test our predictions in real experiments.</p>
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