Spheres, cubes and simple

Abstract

Abstract. In 1929 Tarski showed how to construct points in a region-based first-order logic for space representation. The resulting system, called the geometry of solids, is a cornerstone for region-based geometry and for the comparison of point-based and region-based geometries. We expand this study of the construction of points in region-based systems using different primitives, namely hyper-cubes and regular simplexes, and show that these primitives lead to equivalent systems in dimension n ≥ 2. The result is achieved by adopting a single set of definitions that works for both these classes of figures. The analysis of our logics shows that Tarski's choice to take sphere as the geometrical primitive might be intuitively justified but is not optimal from a technical viewpoint.

Spatial Symmetry Driven Pruning Strategies for Efficient Declarative Spatial Reasoning

“…Thus, a key focus within analytic spatial reasoning has been methods for managing this inherent intractibility. 8 More efficient refinements of the original CAD algorithm include partial CAD [14]. Symbolic methods for solving systems of multivariate equations include the Gröbner basis method [11] and Wu's characteristic set method [40].…”

“…(e.g. [8,36]). Building on such foundational capabilities, another outreach is in the area of computer-aided learning systems in mathematics (e.g.…”

“…Characterising aligned (a),(b) and rotated (c),(d) concentric squares using mereology (reproduced from[Borgo, 2013]) Aligned Concentric. Two squares A, B are aligned and concentric if: A is part of B and there does not exist a square P such that (a) P is covertex with B, and (b) the intersection of P and A is not a square(Figure 9).…”

“…Characterising aligned (a),(b) and rotated (c),(d) concentric squares using mereology (reproduced from[Borgo, 2013]) …”

“…Characterising aligned (a),(b) and rotated (c),(d) concentric squares using mereology (reproduced from[Borgo, 2013]) Ruler and compass method for angle bisection. Line Lc bisects the angle between lines La, L b .…”