  This paper explores a novel mathematical approach to extract archaeological
insights from ensembles of similar artifact shapes. We show that by considering
all the shape information in a find collection, it is possible to identify
shape patterns that would be difficult to discern by considering the artifacts
individually or by classifying shapes into predefined archaeological types and
analyzing the associated distinguishing characteristics. Recently, series of
high-resolution digital representations of artifacts have become available, and
we explore their potential on a set of 3D models of ancient Greek and Roman
sundials, with the aim of providing alternatives to the traditional
archaeological method of ``trend extraction by ordination'' (typology). In the
proposed approach, each 3D shape is represented as a point in a shape space --
a high-dimensional, curved, non-Euclidean space. By performing regression in
shape space, we find that for Roman sundials, the bend of the sundials'
shadow-receiving surface changes with the location's latitude. This suggests
that, apart from the inscribed hour lines, also a sundial's shape was adjusted
to the place of installation. As an example of more advanced inference, we use
the identified trend to infer the latitude at which a sundial, whose
installation location is unknown, was placed. We also derive a novel method for
differentiated morphological trend assertion, building upon and extending the
theory of geometric statistics and shape analysis. Specifically, we present a
regression-based method for statistical normalization of shapes that serves as
a means of disentangling parameter-dependent effects (trends) and unexplained
variability.
