Related papers: A Hierarchy of Geometric Constructions
Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step, in particular along the Process Planning pipeline, can now count on dozens…
We introduce the first geometric construction of codes in the sum-rank metric, which we called linearized Algebraic Geometry codes, using quotients of the ring of Ore polynomials with coefficients in the function field of an algebraic…
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study…
We construct a converging geometric iterated function system on the moduli space of ordered triangles, for which the involved functions have geometric meanings and contain a non-contraction map under the natural metric.
In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We…
We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…
Shape-morphing capabilities are crucial for enabling multifunctionality in both biological and artificial systems. Various strategies for shape morphing have been proposed for applications in metamaterials and robotics. However, few of…
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is…
This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the {\it generalized corner-vector method}. First we establish a uniform upper bound for the degree…
We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of…
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…
A method is developed here for building differentiable three-dimensional manifolds on multicube structures. This method constructs a sequence of reference metrics that determine differentiable structures on the cubic regions that serve as…
In this short paper we show that with a small change of the common ruler and compass construction of the regular pentadecagon, we can produce more regular polygons
The realization space of geometric constraint systems is given by the vanishing locus of polynomials corresponding to natural geometric constraints. Such geometric constraint systems arise in many real-world scenarios such as structural…
In contrast to traditional toy tracks, a patented system allows the creation of a large number of tracks with a minimal number of pieces, and whose loops always close properly. These circuits strongly resemble traditional self-avoiding…
We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches…
A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional…
While geometry with transcendental curves, like the Quadratrix of Hippias and the Spiral of Archimedes, played a significant role in our modern developments of geometry and algebra. The investigation has fallen off in the modern era despite…