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We study translation covers of several triply periodic polyhedral surfaces that are intrinsically Platonic. We describe their affine symmetry groups and compute the quadratic asymptotics for counting saddle connections and cylinders,…
Stochastic process models for spatiotemporal data underlying random fields find substantial utility in a range of scientific disciplines. Subsequent to predictive inference on the values of the random field (or spatial surface indexed…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
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The Earth's surface is subdivided into eight large tectonic plates and many smaller ones. We reconstruct the plate tessellation history and demonstrate that both large and small plates display two distinct hierarchical patterns, described…
The purpose of this article is to give a geometric interpretation to the so-called "twelve surfaces of Darboux", or "Darboux wreath", which appear by applying repeatedly certain simple transformations to a given infinitesimal isometric…
In this paper we investigate the evolution of the concept of area in Peano's works, taking into account the main role played by Grassmann's geometric-vector calculus and Peano's theory on derivative of measures. Geometric (1887) and…
In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…
We study word-level semantic alignment across four historical stages of Ancient Egyptian. These stages differ in script and orthography, and parallel data are scarce. We jointly train a compact encoder-decoder model with a shared byte-level…
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Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more…
The ruled surface is a typical modeling surface in computer aided geometric design. It is usually given in the standard parametric form. However, it can also be in the forms than the standard one. For these forms, it is necessary to…
Two-dimensional random surfaces are studied numerically by the dynamical triangulation method. In order to generate various kinds of random surfaces, two higher derivative terms are added to the action. The phases of surfaces in the…
We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some…
The conic sections, as well as the solids obtained by revolving these curves, and many of their surprising properties, were already studied by Greek mathematicians since at least the fourth century B.C. Some of these properties come to the…
We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an…
In this text I present some problems which led to the introduction of special kinds of graphs as tools for studying singular points of algebraic surfaces. I explain how such graphs were first described using words, and how several…
Among the first devices used to measure the time we find the sundials and the water-clocks, as told by Vitruvius in his book on the Architecture. The sundials work because of the shadows cast by a rod or pole, the gnomon, on their…
Tropical counting tools are useful for many enumerative questions. We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor…
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