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Ákos G. Horváth

Publications and source records attributed to Ákos G. Horváth.

At least 19 recordsLinked to original sources
arXiv2025

Soft cells, Kelvin's foam and the minimal surfaces of Schwarz

Recently, we introduced a new class of shapes, called soft cells which fill space as soft tilings without gaps and overlaps while minimizing the number of sharp corners. We introduced the edge bending algorithm that deforms a polyhedral tiling into a soft tiling and we proved that an infinite class of polyhedral tilings can be smoothly deformed into standard soft tilings. Here, we demonstrate that certain triply periodic minimal surfaces naturally give rise to non-standard soft tilings. By extending the edge-bending algorithm, we further establish that the soft tilings derived from the Schwarz P and Schwarz D surfaces can be continuously transformed into one another through a one-parameter family of intermediate non-standard soft tilings. Notably, by carrying its combinatorial structure, both resulting tilings belong to the first order equivalence class of the Dirichlet-Voronoi tiling on the body-centered cubic bcc lattice, highlighting a deep geometric connection underlying these minimal surface configurations. By requiring identical end-tangents for edges in a first order class, we also define second order equivalence classes among tilings and prove that there exist exactly two such classes among soft tilings which share the full symmetry group of the DV-bcc tiling. Additionally, we construct a one-parameter family of tilings bridging standard and non-standard soft tilings, explicitly including the classic Kelvin foam structure as an intermediate configuration. This construction highlights that both the soft cells themselves and the geometric methods employed in their generation provide valuable insights into the structural principles underlying natural forms. We also present the soft tiling induced by the gyroid structure.

cs.CG↗
arXiv2025

Strongly self-dual polytopes

This article aims to study the class of strongly self-dual polytopes (ssd-polytopes for short), defined in a paper by Lovász \cite{lovasz}. He described a series of such polytopes (called $L$-type polytopes), which he used to solve a combinatorial problem. From a geometrical point of view, there are interesting questions: what additional elements of this class exist, and are there any with a different structure from the $L$-type ones? We show that in dimension three, one of their faces defines $L$-type polyhedra. Illustrating the algorithm of the proof, we present an ssd-polytope of 23 vertices whose combinatorial structure differ from those of $L$-type ones. Finally, with an elementary discussion, we prove that for fewer than nine vertices, there are only fifth one ssd-polyhedra, four of them can be constructed by Lovász's method, and we can find the fifth one with "the diameter gradient flow algorithm" of Katz, Memoli and Wang \cite{katz-memoli-wang}.

math.CO↗
arXiv2024

On the coordinates of minimal vectors in a Minkowski-reduced basis

Finding the shortest vectors in a lattice is an NP-hard problem, so low-dimensional results also play an essential role in lattice reduction theory. Using Ryskov's result for the admissible centerings and Tammela's result for determining the Minkowski-reduced form, we prove that the absolute values of the coordinates of a minimal vector on a six-dimensional Minkowski-reduced basis are less than or equal to three. To sharpen P. Tammela's work, we combine some lattice geometry arguments with the aforementioned theoretical results.

math.MG↗
arXiv2024

Soft cells and the geometry of seashells

A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, non-flat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{soft tilings}. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet-Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells.

physics.app-ph↗
arXiv2023

Elementary Constructions of conic sections

In classical geometry, there is no such well-known and much-studied topic as the construction of conic sections (or briefly conics) from its five points. Its importance in many applications of mechanical engineering, civil engineering and architectural engineering, as well as other applied sciences is clear. The beauty of the topic is that it raises difficult questions that can be approached with basic tools. In this article, we provide constructions (and corresponding theories) that can be taught to high school and university students without knowledge of projective geometry. For this, we recall some important facts about conic sections that can be found in the rich literature. We use the concepts of power of a point on a circle, similarity, orthogonal affinity and inversion. We also mention famous constructions related to our questions. We begin our article at this point, where the standard teaching ends the discussion of conic sections. We therefore assume that the reader knows the basic definitions and constructions of conics, the concepts of focus, axis, tangent, leading circle and leading line.

math.HO↗
arXiv2023

The bridge between Desargues' and Pappus' theorems

In this paper, we investigate the configuration theorems of Desargues and Pappus in a synthetic geometric way. We provide a bridge between the two configurations with a third one that can be considered a specification for both. We do not use the theory of collineations or the analytic description of the plane over a ternary ring.

math.HO↗
arXiv2022

A two-vertex theorem for normal tilings

We regard a smooth, $d=2$-dimensional manifold $\mathcal{M}$ and its normal tiling $M$, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by $\bar v^{\star}$ and we prove that if $M$ is periodic then $v^{\star} \geq 2$ and we show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with $\bar v^{\star}=0$.

math.DG↗
arXiv2021

On the convex hull and homothetic convex hull functions of a convex body

The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body $K$ in Euclidean $n$-space, defined as the volume of the union of $K$ and one of its translates, and the volume of $K$ and a translate of a homothetic copy of $K$, respectively, as functions of the translation vector. In particular, we prove that the convex hull function of the body $K$ does not determine $K$. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horváth and Lángi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jerónimo-Castro about the homothetic convex hull function, and prove a homothetic variant of the translative constant volume property conjecture for $3$-dimensional convex polyhedra. We also apply our results to describe the properties of the illumination bodies of convex bodies.

math.MG↗
arXiv2020

Diameter, width and thickness in the hyperbolic plane

This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given thickness.

math.MG↗
arXiv2019

On convex bodies that are characterizable by volume function

The "old-new" concept of convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, respectively. A very interesting fact that there are many long-standing open problems connected with these functions whose serious investigation closed before the "age of computers". In this survey, we concentrate only on the three-dimensional case, we will mention the most important concepts, statements, and problems.

math.MG↗
arXiv2018

An extremal problem of regular simplices-The higher dimensional case

The new result of this paper connected with the following problem: Consider a supporting hyperplane of a regular simplex and its re ected image at this hyperplane. When will be the volume of the convex hull of these two simplices maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume achieves in that case when the hyperplane goes through on a vertex and orthogonal to the height of the simplex at this vertex. More interesting that in the higher dimensional cases this position is not optimal. We also determine the optimal position of hyperplane in the 5-dimensional case. This corrects an erroneous statement in my paper [3].

math.MG↗
arXiv2018

Gallucci's axiom revisited

In this paper we propose a well-justified synthetic approach of the projective space. We define the concepts of plane and space of incidence and also the Gallucci's axiom as an axiom to our classical projective space. To this purpose we prove from our space axioms, the theorems of Desargues, Pappus, the fundamental theorem of projectivities, and the fundamental theorem of central-axial collinearities, respectively. Our building up do not use any information on analytical projective geometry, as the concept of cross-ratio and the homogeneous coordinates of points.

math.HO↗
arXiv2017

On the constructibility of the axes of an ellipsoid

In this paper we discuss Chasles's construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar.In this paper we discuss Chasles's construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar.

math.MG↗
arXiv2017

Proper quadrics in the Euclidean $n$-space

In this paper we investigate the metric properties of quadrics and cones of the $n$-dimensional Euclidean space. As applications of our formulas we give a more detailed description of the construction of Chasles and the wire model of Staude, respectively.

math.MG↗
arXiv2016

Constructive curves in non-Euclidean planes

In this paper we overview the theory of conics and roulettes in four non-Euclidean planes. We collect the literature about these classical concepts, from the eighteenth century to the present, including papers available only on arXiv. The comparison of the four non-Euclidean planes, in terms of the known results on conics and roulettes, reflects only the very subjective view of the author.

math.MG↗
arXiv2016

Angles in normed spaces

The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including also further related aspects. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.

math.MG↗