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.

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.

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

GENERALIZATIONS OF THE KOCH CURVE

Fractals ◽  
2008 ◽  
Vol 16 (03) ◽  
pp. 267-274 ◽  
Author(s):  
R. B. DARST ◽  
J. A. PALAGALLO ◽  
T. E. PRICE
Keyword(s):  
Iterative Method ◽  
Iterated Function System ◽  
Continuous Functions ◽  
Cantor Sets ◽  
Koch Curve ◽  
Double Points ◽  
Iterated Function ◽  
Fixed Set ◽  
Equilateral Triangles ◽  
Two Parameter

We present an iterative method to define a two-parameter family of continuous functions fa,θ: I → ℂ such that f1/3,π/3 is the Koch curve. We consider the two-cases θ = π/3 and θ = π/4 of these generalized Koch curves fa,θ(I). In each case we determine the pivotal value of a, the largest value of a for which the corresponding curve is not simple. We give characterizations of the double points of the curve (points on the curve that have two inverse images). In the case where θ = π/3 double points are vertices of equilateral triangles. When θ = π/4 the double points form Cantor sets in the plane. We conclude with a more general result that proves that if the fixed set (attractor) of an iterated function system is connected, then it is a curve.

scholarly journals Implementation of TAGE Method Using Seikkala Derivatives Applied to Two-Point Fuzzy Boundary Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
A. A. Dahalan ◽  
J. Sulaiman
Keyword(s):  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Linear Equations ◽  
Alternating Group ◽  
System Of Linear Equations ◽  
Fuzzy Boundary Value Problems ◽  
Fuzzy Boundary ◽  
Two Parameter ◽  
Number Of Iterations

Iterative methods particularly the Two-Parameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). The formulation and implementation of the TAGE method are also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the method. The results show that TAGE method is superior compared to GS method in the aspect of number of iterations, execution time, and Hausdorff distance.

scholarly journals Presentations for some direct products of groups

1983 ◽  
Vol 28 (1) ◽  
pp. 131-133 ◽  
Author(s):  
P.E. Kenne
Keyword(s):  
Direct Product ◽  
Alternating Group ◽  
Direct Products ◽  
Icosahedral Group ◽  
Products Of Groups

We give efficient presentations for the direct product of two copies of the alternating group of degree five and the direct product of the alternating group of degree five and the binary icosahedral group.

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 EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

2012 ◽  
Vol 54 (2) ◽  
pp. 371-380
Author(s):  
G. G. BASTOS ◽  
E. JESPERS ◽  
S. O. JURIAANS ◽  
A. DE A. E SILVA
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Alternating Group ◽  
Finite Abelian Groups ◽  
Quaternion Group ◽  
Icosahedral Group ◽  
Odd Order ◽  
Even Order

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

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.

Sign in / Sign up

Export Citation Format

Share Document