Frieze Patterns in the Hyperbolic Plane

1974 ◽  
Vol 17 (1) ◽  
pp. 45-50 ◽  
Author(s):  
C. W. L. Garner
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Symmetry Groups ◽  
Euclidean Case ◽  
Parallel Lines

AbstractIt is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and

scholarly journals Hyperbolization of Euclidean Ornaments

10.37236/78 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Martin von Gagern ◽  
Jürgen Richter-Gebert
Keyword(s):  
Differential Geometry ◽  
Pattern Recognition ◽  
Hyperbolic Plane ◽  
Discrete Differential Geometry ◽  
Fundamental Region ◽  
Symmetry Groups ◽  
Symmetry Detection ◽  
Conformal Deformation ◽  
Optimal Resolution ◽  
Wallpaper Groups

In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution.

Regular compound tessellations of the hyperbolic plane

1964 ◽  
Vol 278 (1373) ◽  
pp. 147-167 ◽  
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Regular Polyhedron ◽  
Alternating Group ◽  
Platonic Solids ◽  
Icosahedral Group ◽  
Analogous Compound ◽  
Equilateral Triangles ◽  
Two Parameter

Kepler, in his Harmonice mundi of 1619, extended the idea of a regular polyhedron in at least two directions. Observing that two equal regular tetrahedra can interpenetrate in such a way that their twelve edges are the diagonals of the six faces of a cube, he called this combination stella octangida . It is occasionally found in nature as twinned crystals of tetrahedrite, Cu 10 (Zn, Fe, Cu) 2 Sb 4 S 13 . In addition to this ‘compound’ of two tetrahedra inscribed in a cube, there are several other compound polyhedra, the prettiest being the compound of five tetrahedra inscribed in a dodecahedron. The icosahedral group of rotations may be described as the alternating group on these five tetrahedra. Kepler observed also that the tessellation of squares (or regular hexagons, or equilateral triangles), filling and covering the Euclidean plane, may be regarded as an infinite analogue of the spherical tessellations which are ‘blown-up’ versions of the Platonic solids. Putting these two ideas together, one naturally regards the compound polyhedra as compound tessellations of the sphere. The analogous compound tessellations of the Euclidean plane (18 two-parameter families of them) were enumerated in 1948. The present paper describes many compound tessellations of the hyperbolic plane: five one-parameter families and seventeen isolated cases. It is conjectured that this list is complete, but there remains the possibility that a few more isolated cases may still be discovered.

Quadratic Integers and Coxeter Groups

1999 ◽  
Vol 51 (6) ◽  
pp. 1307-1336 ◽  
Author(s):  
Norman W. Johnson ◽  
Asia Ivić Weiss
Keyword(s):  
Hyperbolic Plane ◽  
Modular Group ◽  
Coxeter Groups ◽  
Picard Group ◽  
Symmetry Groups ◽  
Discrete Groups ◽  
Algebraic Integers ◽  
Linear Fractional Transformations ◽  
Rings Of Algebraic Integers ◽  
Groups Of Transformations

AbstractMatrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive n-space or hyperbolic (n+1)-space Hn+1. For small n, thesemay be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of Hn+1. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group PSL2(), the Gaussian modular (Picard) group PSL2([i]), and the Eisenstein modular group PSL2([ω]).

An Analogue of Napoleon’s Theorem in the Hyperbolic Plane

2001 ◽  
Vol 44 (3) ◽  
pp. 292-297 ◽  
Author(s):  
Angela McKay
Keyword(s):  
Equilateral Triangle ◽  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Equilateral Triangles ◽  
Napoleon’S Theorem

AbstractThere is a theorem, usually attributed to Napoleon, which states that if one takes any triangle in the Euclidean Plane, constructs equilateral triangles on each of its sides, and connects the midpoints of the three equilateral triangles, one will obtain an equilateral triangle. We consider an analogue of this problem in the hyperbolic plane.

Averages for polygons formed by random lines in Euclidean and hyperbolic planes

1972 ◽  
Vol 9 (1) ◽  
pp. 140-157 ◽  
Author(s):  
L. A. Santaló ◽  
I. Yañez
Keyword(s):  
Infinite Number ◽  
Stochastic Geometry ◽  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Second Order ◽  
Countable Number ◽  
Geometrical Probability ◽  
Starting Point ◽  
The Mean ◽  
Hyperbolic Planes

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]–[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry.

scholarly journals The Three Reflections Theorem Revisited

KoG ◽  
2018 ◽  
pp. 41-48
Author(s):  
Gunter Weiss
Keyword(s):  
Projective Geometry ◽  
Euclidean Plane ◽  
Projective Planes ◽  
Common Point ◽  
Point Of View ◽  
Euclidean Case ◽  
Minkowski Metric ◽  
Point Reflection ◽  
Pass Through ◽  
The Given

It is well-known that, in a Euclidean plane, the product of three reflections is again a reflection, iff their axes pass through a common point. For this ``Three reflections Theorem'' (3RT) also non-Euclidean versions exist, see e.g. [4]. This article presents affine versions of it, considering a triplet of skew reflections with axes through a common point. It turns out that the essence of all those cases of 3RT is that the three pairs (axis, reflection direction) of the given (skew) reflections can be observed as an involutoric projectivity. For the Euclidean case and its non-Euclidean counterparts this property is automatically fulfilled. From the projective geometry point of view a (skew) reflection is nothing but a harmonic homology. In the affine situation a reflection is an indirect involutoric transformation, while ``direct'' or ``indirect'' makes no sense in projective planes. A harmonic homology allows an interpretation both, as an axial reflection and as a point reflection. Nevertheless, one might study products of three harmonic homologies, which result in a harmonic homology again. Some special mutual positions of axes and centres of the given homologies lead to elations or even to the identity, too. A consequence of the presented results are further generalisations of the 3RT, e.g. in planes with Minkowski metric, affine or projective 3-space, or in circle geometries.

The Classification of Non-Euclidean Plane Crystallographic Groups

1967 ◽  
Vol 19 ◽  
pp. 1192-1205 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Quotient Space ◽  
Fuchsian Groups ◽  
Discrete Groups ◽  
Crystallographic Groups ◽  
Algebraic Classification ◽  
Compact Quotient

This paper deals with the algebraic classification of non-euclidean plane crystallographic groups (NEC groups, for short) with compact quotient space. The groups considered are the discrete groups of motions of the Lobatschewsky or hyperbolic plane, including those which contain orientation-reversing reflections and glide-reflections. The corresponding problem for Fuchsian groups, which contain only orientable transformations, is essentially solved in the work of Fricke and Klein (6).

scholarly journals Perimeter Approximation of Convex Discs in the Hyperbolic Plane and on the Sphere

2021 ◽  
Author(s):  
Ferenc Fodor
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Plane Convex ◽  
Convex Disc ◽  
Convex Curves

AbstractEggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.

scholarly journals A Census of Vertices by Generations in Regular Tessellations of the Plane

10.37236/574 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Alice Paul ◽  
Nicholas Pippenger
Keyword(s):  
Shortest Path ◽  
Generating Functions ◽  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Infinite Graphs ◽  
Rational Generating Functions

We consider regular tessellations of the plane as infinite graphs in which $q$ edges and $q$ faces meet at each vertex, and in which $p$ edges and $p$ vertices surround each face. For ${1/p + 1/q = 1/2}$, these are tilings of the Euclidean plane; for ${1/p + 1/q < 1/2}$, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all $p\ge 3$ and $q \ge 3$ with ${1/p + 1/q\le1/2}$, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.

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