The strengths and weaknesses of the popular ''projection method'' for constructing aperiodic point sets and tilings are assessed; we show that the method has become a point of departure for more general theories.

One of the terms of reference published in 1992 (Acta Cryst. A48, 922–946) in the report of the (at that time) Ad Interim Commission on Aperiodic Crystals was `to co-operate with other Commissions of the IUCr in establishing adequate guidelines and standards for articles to be published in IUCr journals reporting structural investigations of aperiodic crystals and theoretical investigations of aperiodic patterns'. It was soon recognized by the Commission that the establishment of a checklist for the publication of incommensurately modulated structures was perhaps the most urgent task. One of the first attempts to address the problem of publishing incommensurate structures was presented during a discussion organized in 1991 in Leikeitio, Spain, in a workshop dedicated to methods of structural analysis of modulated structures and quasicrystals. Since this meeting, the work of the Commission progressed interactively by numerous exchanges on the occasions of international meetings and mostly by electronic correspondence. The Checklist was accepted in 1994 by the Executive Committee of the IUCr. The opinion of the Commission on Journals was also requested and, following the comments of its members, the Checklist has been completed with an example of an incommensurate crystal structure illustrating each specific item that is required in the Checklist.

We study periodic tilings of d-dimensional space by clusters of related zonotopes. For such tilings, the fundamental region relative to a fully-dimensional translation group is the finite union of zonotopes constructed as Minkowski sums of subsets of a given set of generating vectors. For any rationally-realizable zonotope Z, the rational realization itself determines a unique periodic tiling in which one of the cells is the zonotope Z. In an effort to find analogous tilings using nonrational zonotopes, we conjecture a characterization in terms of a new family oriented matroid structures, for which we provide axioms.

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Quasicrystals, and more generally aperiodic crystals, generate fundamental mathematical problems in diffraction theory and the theory of aperiodic tilings. Their solution requires a reformulation of the basic concepts of mathematical crystallography.

New or redefined printed symbols are proposed in the light of the recently accepted redefinition of symmetry elements [de Wolff et al. (1989). Acta Cryst. A45, 494-499]. In particular, the letter e covers certain glide planes which hitherto had no unique symbol, such as those called 'either a or b'. The use of e in the Hermann- Mauguin symbol of five different space groups is recommended. For e planes projected in a direction parallel to the plane, a graphical symbol is proposed which removes the ambiguity of their present designation. The letter k is proposed for a newly defined class of glide planes which until now were without specific symbol. The symbols for symmetry operations introduced in the space-group descriptions of International Tables for Crystallography (1989), Vol. A (Dordrecht: Kluwer Academic Publishers) are recommended for general use, with modifications only for glide reflection operations.

Rereading Hermann Weyl's now-classic 1952 monograph Symmetry, one is struck both by its beauty and by its limitations. Many of the most interesting problems in contemporary symmetry theory concern local configurations, and group theory may not be the only, or the best tool for studying them.

A 'geometric element' is defined, for any given symmetry operation, as a geometric item that allows the operation (after removing any intrinsic translation) to be located and oriented.In the case of an inversion, a (screw-) rotation or a (glide-) reflection, it is respec-*

Color symmetry, introduced by Shubnikov in the 1950's, has generated research in many areas of symmetry theory, including the enumeration of the subgroups of crystallographic groups. The basic ideas of color symmetry are simple, but they are subtle; many problems remain open, particularly the problem of classifying colored patterns.

GENERALIZATIONstructure factors and the transformed one are identical, i.e.F(h)=F'(h).For the same reason, all transformations-corresponding to a given coset of G in NE(G) result in the same set of indices and related structure factors.In the present context it is $ufficient, therefore, to treat one representative symmetry operation from each coset.Two cases shall be discussed separately: (a) The coset can be represented by a pure translation (I,p), I being the identity matrix: Then all indices remain unchanged (h'=hI=h) and only the phases change (~'(h)=~(h)-2nhp).-The-numbernt of such cosets equals the Index-between the translation subgroups,of G and of NE(G).The permissible origin translations (C.Giacovazzo, Acta Cryst.(1974) A30, 390) pI ayi ng a fundamenta L pa rt in direct methods may-be derived directly as those translations of NE(G) not belonging to G itself.(b) The coset cannot pe represented by a pure translation: Then each corresponding unit cell transformation causes a mapping of the reciprocal lattice with the property, that the two structure factors with the same indices F(h) and F'(h) (referring to the original basis and the transformed basis, respectively) are not related by space-group symmetry, i.e. !F(h)! ~!F'(h)l.For this, G and NE(G) have to belong to difrerent crystal classes.If n is the index of G in NE(G) and nt is the index between the corresponding two tran~lation subgroups, then n/nt is the number of symmetrically inequivalent indexing schemes in reciprocal lattice.In the special case of a non-centrosymmetrical crystal structure without anomalous scatterers Friedel's law holds and, therefore, the number of inequivalent indexing schemes is reduced by a factor of 2 (exception: space groups from enantiomorphic pairs).In such a case different indexing schemes occur only if G and NE(G) belong to different Laue groups.

Two questions which have been independently studied (the distribution of colors in colored lattices and lattice preservation in derivative lattices) are in fact closely related. It is possible, for instance, to determine the distribution of colors in rows and nets by the lattice-preservation indices cr and cp as functions of row indices [u0, v0, W0] and net indices (h0, k0, l0), respectively. A formula is also given for the number of classes of equivalent derivative lattices of a given index n.

Color groups provide a partial classification of colored objects, but the coloring problem is much more complex. In this paper, criteria are established for determining whether a transitive pattern can be consistently colored with each pattern unit receiving a single color, and for calculating the number of such colorings. The colorings of the face-transitive poly- hedra (a class which includes the simple crystal forms) are enumerated.

Vibrational spectra of some N-methyl chloro and N-methyl bromotriazoles-1,2,4 are reported. An assignment of the normal modes is proposed. This study was done in order to find new elements to solve the problem of tautomerism in C-substituted triazoles-1,2,4.

The subgroups of finite index of any n-dimensional space group are determined by the solutions of a set of congruences analogous in form and meaning to the Frobenius congruences which characterize the space groups themselves. These congruences can be solved in any dimension in which the space groups are known.

Chemischer InformationsdienstVolume 11, Issue 31 Physical Organic Chemistry ChemInform Abstract: CRYSTAL STRUCTURE AND VIBRATIONAL SPECTRA OF 3(5)-CHLORO-1,2,3-STRIAZOLE M. SAIDI IDRISSI, M. SAIDI IDRISSISearch for more papers by this authorM. SENECHAL, M. SENECHALSearch for more papers by this authorH. SAUVAITRE, H. SAUVAITRESearch for more papers by this authorM. COTRAIT, M. COTRAITSearch for more papers by this authorC. GARRIGOU-LAGRANGE, C. GARRIGOU-LAGRANGESearch for more papers by this author M. SAIDI IDRISSI, M. SAIDI IDRISSISearch for more papers by this authorM. SENECHAL, M. SENECHALSearch for more papers by this authorH. SAUVAITRE, H. SAUVAITRESearch for more papers by this authorM. COTRAIT, M. COTRAITSearch for more papers by this authorC. GARRIGOU-LAGRANGE, C. GARRIGOU-LAGRANGESearch for more papers by this author First published: August 5, 1980 https://doi.org/10.1002/chin.198031065Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume11, Issue31August 5, 1980 RelatedInformation