  The Weisfeiler--Lehman (WL) test is a fundamental iterative algorithm for
checking isomorphism of graphs. It has also been observed that it underlies the
design of several graph neural network architectures, whose capabilities and
performance can be understood in terms of the expressive power of this test.
Motivated by recent developments in machine learning applications to datasets
involving three-dimensional objects, we study when the WL test is {\em
complete} for clouds of euclidean points represented by complete distance
graphs, i.e., when it can distinguish, up to isometry, any arbitrary such
cloud.
  Our main result states that the $(d-1)$-dimensional WL test is complete for
point clouds in $d$-dimensional Euclidean space, for any $d\ge 2$, and that
only three iterations of the test suffice. Our result is tight for $d = 2, 3$.
We also observe that the $d$-dimensional WL test only requires one iteration to
achieve completeness.
