  Isospectrality of planar domains which are obtained by successive unfolding
of a fundamental building block is studied in relation to iso-length
spectrality of the corresponding domains. Although an explicit and exact trace
formula such as Poisson's summation formula or Selberg's trace formula is not
known to exist for planar domains, equivalence between isospectrality and
iso-length spectrality in a certain setting can be proved by employing the
matrix representation of "transplantation of eigenfunctions". As an application
of the equivalence, transplantable pairs of domains, which are all isospectral
pairs of planar domains and therefore counter examples of Kac's question "can
one hear the shape of a drum?", are numerically enumerated and it is found at
least up to the domain composed of 13 building blocks transplantable pairs
coincide with those constructed by the method due to Sunada.
