Emil Molnár

Classification of Tile-Transitive 3-Simplex Tilings and Their Realizations in Homogeneous Spaces

This is a survey on the tilings (T, Γ) in the title where the vertex stabilizers in are finite spherical S2 or infinite Euclidean E2 (cocompact) plane groups. The results are collected in figures and tables and illustrated by an infinite family series Family 30 in Section 4. The obtained orbifolds, maybe after splitting procedure, are realized in seven homogeneous Riemannian 3-spaces by means of projective metrics.

Fundamental domains for planar discontinuous groups and uniform tilings

In this paper, we classify combinatorially different fundamental domains for any given planar discontinuous group and we give an algorithm for the complete enumeration of uniform tilings of any complete, simply connected, two-dimensional Riemannian manifold of constant curvature.

Combinatorial construction of tilings by barycentric simplex orbits (D symbols) and their realizations in Euclidean and other homogeneous spaces.

A new method, developed in previous works by the author (partly with co-authors), is presented which decides algorithmically, in principle by computer, whether a combinatorial space tiling (Tau, Gamma) is realizable in the d-dimensional Euclidean space E(d) (think of d = 2, 3, 4) or in other homogeneous spaces, e.g. in Thurstons 3-geometries: E(3), S(3), H(3), S(2) x R, H(2) x R, SL(2)R, Nil, Sol. Then our group Gamma will be an isometry group of a projective metric 3-sphere PiS(3) (R, < , >), acting discontinuously on its above tiling Tau. The method is illustrated by a plane example and by the well known rhombohedron tiling (Tau, Gamma), where Gamma = R3m is the Euclidean space group No. 166 in International Tables for Crystallography.

D-V cells and fundamental domains for crystallographic groups, algorithms, and graphic realizations

This work is related to graphic software in progress by our department to the computer package Carat, developed by colleagues in Aachen headed by Plesken. Carat is available via http://wwwb.math. rwth-aachende/carat/. Our software intends to help the applicants, e.g., crystallographers, and others in modelling real crystals. Furthermore, it will hopefully be developed for visualization of higher-dimensional (d = 4) and non-Euclidean (d = 2, 3) investigations. The well-known algorithms for Dirichlet-Voronoi (D-V) cell partition of n points in general position (Voronoi diagram) in E^d have the worst case complexity [1] O(dn^@?^d^2^@?^+^1)+O(d^3n^@?^d^2^@?logn). It becomes more simple for a fixed dimension d, if we assume a transitive group action on the point set. In particular, we consider a point orbit under a (crystallographic) space group @C in E^3, and determine its D-V cell D and-depending on the stabilizer of the starting point-a fundamental domain F for @C with an appropriate face pairing for a set of generators and algebraic presentation of @C This latter algorithm with its graphic implementation is our new initiative in the topic. In general, the worst case time complexity exponentially increases only by the dimension d, but it is completely satisfactory for d = 2,3,4.

Minimal presentation of crystallographic groups by fundamental polyhedra

Abstract There is a Poincares method to present a discrete isometry group G by means of a fundamental polyhedron F equipped with a face identification [1]. The identifying isometries generate the group G. The cycle relations, belonging to the edge segment equivalence classes of F, together with the occasional reflection relations, give us the presentation of G mentioned. The authors intention, to give a so-called minimal geometric presentation for each space group, has been realized in most cases, sometimes only by concave topological polyhedra [2–4]. In this paper we shall determine such a polyhedra with minimal number of (curved) faces, presenting minimally those 38 space groups which have a semi-direct decomposition G = G1 ∘ C2 is an invariant Coxeter subgroup generated by plane reflections and G1 is a so-called rod group leaving the fundamental domain of C2 and a straight line invariant [5]. This geometric presentation illustrated by figures can give us all the essential information on the structure of each space group.

On Triply Periodic Minimal Balance Surfaces

AbstractA strategy for finding all triply periodic minimal balance surfaces (TPMBS) will be sketched mainly on the basis of fundamental domains of space group pairs of index 2. Some possibilities are confirmed, but there is no TPMBS for the space group pair Ia

An Algorithm for Classification of Fundamental Polygons for a Plane Discontinuous Group

{\bar 3}

Nice tiling, nice geometry!?!

/Pa