While connected arithmetic discrete lines are entirely char- acterized by their arithmetic thickness, only partial results exist for arith- metic discrete hyperplanes in any dimension. In the present paper, we focus on 0-connected rational arithmetic discrete planes in Z3. Thanks to an arithmetic reduction on a given integer vector n, we provide an al- gorithm which computes the thickness of the thinnest 0-connected arith- metic plane with normal vector n.