We study the structure of the Penrose tiling (PT, in short) constructed by the matching rule, and deduce directly a substitution rule from that, which gives us (i)local configuration of the tiles,(ii) elementary proofs of the aperiodicity, the locally isomorphic property, and the uncountability,(iii) alternative proof of the fact that all PT's obtained by the matching rule can be constructed via the up-down generation.
"A substitution rule for the Penrose tiling." Nihonkai Math. J. 19 (2) 111 - 135, 2008.