Intrinsic shape analysis in archaeology: A case study on ancient sundials

Abstract

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.

Figures (20)

• Figure 1: Front (left) and side (right) view of a conical sundial with gnomon (1) and shadow surface (2). In order to craft the shadow surface, part of a cone was cut from the stone, as indicated by the gray cone in the picture.
• Figure 2: Front (left) and side (right) view of a spherical sundial with missing gnomon. In order to craft the shadow surface, part of a round ball was cut from the stone, as indicated by the gray sphere.
• Figure 3: Manually segmented shadow surface
• Figure 5: Results of geodesic regression for shadow surfaces of spherical sundials w.r.t. latitude. The results for Roman and Greek sundials are shown in the top and bottom row, respectively. After calculating the trajectories in the space of differential coordinates, we sampled each curve at 4 points and computed the corresponding triangular meshes. The Roman geodesic models the trend for latitudes in $I_R = [40.7030, 43.3155]$ while the Greek one is defined on $I_G = [36.0917, 37.3900]$.
• Figure 6: Normalization w.r.t. some parameter $t$ for a single data group $(q_j,t_j)$, $j=1,2,3,4$, in (a part of) a shape manifold $M$: Each point represents a complete shape. The curve $\gamma$ is the result of geodesic regression w.r.t. $t$. The points $\gamma(t_j)$ are depicted in white, while the tangent vectors $v_j$ are the gray arrows. Finally, the parallel translations $w_j$ of the $v_j$ in the tangent space $T_{\gamma(t_0)}M$ at $\gamma(t_0)$ are shown in orange, yielding the normalized data $\widetilde{q}_j$.

Theorems & Definitions (1)

• Example 2.1