On Maximal Homogeneous 3-Geometries and Their Visualization

Molnár, E.; Prok, I.; Szirmai, Jenő

doi:10.3390/universe3040083

Cited by 4 publications

(19 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As the titles show, this paper is in a close connection with another paper [1] of this volume. For example in [1] on the basis of projective interpretation of the eight Thurston geometries we introduce also an infinite series of hyperbolic manifolds derived not from a regular polyhedron.…”

Section: Introductionmentioning

confidence: 76%

See 1 more Smart Citation

On Maximal Homogeneous 3-Geometries—A Polyhedron Algorithm for Space Tilings

Prok

2018

Universe

Self Cite

View full text Add to dashboard Cite

Abstract:In this paper we introduce a polyhedron algorithm that has been developed for finding space groups. In order to demonstrate the problem and the main steps of the algorithm, we consider some regular plane tilings in our examples, and then we generalize the method for 3-dimensional spaces. Applying the algorithm and its computer implementation we investigate periodic, face-to-face, regular polyhedron tilings in 3-spaces of constant curvature and of the other homogeneous 3-geometries, too. We summarize and visualize the most important results, emphasizing the fixedpoint-free space groups which determine 3-dimensional manifolds.

show abstract

Section: Introductionmentioning

confidence: 76%

“…For example in [1] on the basis of projective interpretation of the eight Thurston geometries we introduce also an infinite series of hyperbolic manifolds derived not from a regular polyhedron.…”

Section: Introductionmentioning

confidence: 99%

On Maximal Homogeneous 3-Geometries—A Polyhedron Algorithm for Space Tilings

Prok

2018

Universe

Self Cite

View full text Add to dashboard Cite

show abstract

“…We first summarize the key information about projective models of Thurston geometries (see [30,31,34]). All the Thurston 3-geometries will be uniformly modelled in the projective spherical space PS 3 that can be embedded into the affine hence into Euclidean 4-space.…”

Section: On the Projective Models Of Thurston Geometriesmentioning

confidence: 99%

“…A unified approach to Thurston geometries enabling the investigations in this direction were made possible by the nice paper of E. Molnár [30] where he showed that the Thurston geometries can be uniformly modeled in the projective 3-space P 3 (or in the projective 3-sphere PS 3 ). The projective spherical model is based on linear algebra over the real vector space V 4 (for points) and its dual V 4 (for planes), up to a positive real factor, so that the proper dimension is indeed three.…”

Section: Introductionmentioning

confidence: 99%

Classical Notions and Problems in Thurston Geometries

SZİRMAİ

2023

International Electronic Journal of Geometry

View full text Add to dashboard Cite

Of the Thurston geometries, those with constant curvature geometries (Euclidean $\EUC$, hyperbolic $\HYP$, spherical $\SPH$) have been extensively studied, but the other five geometries, $\HXR$, $\SXR$, $\NIL$, $\SLR$, $\SOL$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries -- can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.

show abstract

“…See also [7,8] for the classical and modern use of projective metrical structures in geometry. In this paper, we will give an outline of this new approach, which connects naturally to the relativistic geometry of Lorentz, Einstein and Minkowski and also allows us to consider hyperbolic geometries over general fields, including finite fields.…”

Section: Two Famous Questions and A Projective/algebraic Look At Hypementioning

confidence: 99%

Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative

Wildberger

2018

Universe

View full text Add to dashboard Cite

Universal hyperbolic geometry gives a purely algebraic approach to the subject that connects naturally with Einstein's special theory of relativity. In this paper, we give an overview of some aspects of this theory relating to triangle geometry and in particular the remarkable new analogues of midpoints called sydpoints. We also discuss how the generality allows us to consider hyperbolic geometry over general fields, in particular over finite fields.Keywords: rational trigonometry; universal hyperbolic geometry; sydpoints; finite fields Two Famous Questions and a Projective/Algebraic Look at Hyperbolic GeometryWhile physicists have long pondered the question of the physical nature of the "continuum", mathematicians have struggled to similarly understand the corresponding mathematical structure. In the last decade, we have seen the emergence of rational trigonometry [1,2] as a viable alternative to traditional geometry, built not over a continuum of "real numbers", but rather algebraically over a general field, so also over the rational numbers, or over finite fields.Universal hyperbolic geometry (UHG) extends this understanding to the projective setting, yielding a new and broader approach to the Cayley-Klein framework (see [3]) for the remarkable geometry discovered now almost two centuries ago by Bolyai, Gauss and Lobachevsky as in [4][5][6]. See also [7,8] for the classical and modern use of projective metrical structures in geometry. In this paper, we will give an outline of this new approach, which connects naturally to the relativistic geometry of Lorentz, Einstein and Minkowski and also allows us to consider hyperbolic geometries over general fields, including finite fields.To avoid technicalities and make the subject accessible to a wider audience, including physicists, we aim to describe things both geometrically in a projective visual fashion, as well as algebraically in a linear algebraic setting. The Polarity of a Conic Discovered by ApolloniusWe augment the projective plane, which we may regard as a two-dimensional affine plane and a line at infinity, with a fixed conic. This conic is called the absolute in Cayley-Klein geometry. In this universal hyperbolic geometry (UHG), developed in [9-12], we take it to be a circle, typically in blue, and call it the null circle.The polarity associated with a conic was investigated by Apollonius and gives a duality between points a and lines A = a ⊥ in the space, which we also write as a = A ⊥ . Given a point a, consider any two lines through a, which meet the conic at two points each as in Figure 1. The other two diagonal points of this cyclic quadrilateral defines the dual line A. Remarkably, this construction does not depend on the choice of lines through a, as Apollonius realized.

show abstract

On Maximal Homogeneous 3-Geometries and Their Visualization

Cited by 4 publications

References 16 publications

On Maximal Homogeneous 3-Geometries—A Polyhedron Algorithm for Space Tilings

On Maximal Homogeneous 3-Geometries—A Polyhedron Algorithm for Space Tilings

Classical Notions and Problems in Thurston Geometries

Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative

Contact Info

Product

Resources

About