*Honestly, i don't have an exact answer to this question But i think this could hepl:* 

From the standard model of particle physics, all massive particles get their masses from the Higgs field couplings and these masses are just energies bounded-intrinsically to the particles. Therefore all the masses we see around as is just the sum of intrinsically-bounded and potential energies. 
 
This bounded energy or rest mass together with other quantum numbers like spin, elctric charge is what gives a particle a structure and if string theory is true all the particles are corresponds to different modes of vibration of the respective string. In particular 1d, 2d or 3d particles or strings only means that the energy distribution is 1d, 2d or 3d respectively around that particle or string. There is no energy independent notion of object's dimensions or structure.