While connected rational arithmetical discrete lines and connected rational arithmetical discrete planes are entirely characterized, only partial results exist for the irrational arithmetical discrete planes. In the present paper, we focus on the connectedness of irrational arithmetical discrete planes, namely the arithmetical discrete planes with a normal vector of which the coordinates are not Q-linear dependent. Given v in R^3, we compute the lower bound of the thicknesses 2-connecting the arithmetical discrete planes with normal vector v. In particular, we show how the translation parameter operates in the connectedness of the arithmetical discrete planes.