We show that a Thue-Morse aperiodic structure presents a unique kind of positional correlation between its constituents, leading to an unattenuated transmission of light as well as electrons through it. The reason for this is a resonant tunnelling, whose origin can be traced back to the presence of certain `dimers' which are not explicitly displayed in the structure. It is interesting to observe that under suitable conditions, two apparently uncorrelated constituents in a Thue-Morse sequence combine together to form a dimer and the entire system can be thought of as being composed of nested dimers only. This aspect has been analysed in terms of light propagation through a Thue-Morse multilayered system and its electronic counterpart.